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A general framework is described to..."/> <meta name="twitter:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs00009-023-02300-w/MediaObjects/9_2023_2300_Fig1_HTML.png"/> <meta name="journal_id" content="9"/> <meta name="dc.title" content="Representation Varieties of Twisted Hopf Links"/> <meta name="dc.source" content="Mediterranean Journal of Mathematics 2023 20:2"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Springer"/> <meta name="dc.date" content="2023-01-29"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2023 The Author(s)"/> <meta name="dc.rights" content="2023 The Author(s)"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="In this paper, we study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the $${{\,\textrm{SL}\,}}_r({\mathbb {C}})$$ -representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representation and character varieties."/> <meta name="prism.issn" content="1660-5454"/> <meta name="prism.publicationName" content="Mediterranean Journal of Mathematics"/> <meta name="prism.publicationDate" content="2023-01-29"/> <meta name="prism.volume" content="20"/> <meta name="prism.number" content="2"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="30"/> <meta name="prism.copyright" content="2023 The Author(s)"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://link.springer.com/article/10.1007/s00009-023-02300-w"/> <meta name="prism.doi" content="doi:10.1007/s00009-023-02300-w"/> <meta name="citation_pdf_url" content="https://link.springer.com/content/pdf/10.1007/s00009-023-02300-w.pdf"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/article/10.1007/s00009-023-02300-w"/> <meta name="citation_journal_title" content="Mediterranean Journal of Mathematics"/> <meta name="citation_journal_abbrev" content="Mediterr. 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</li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1007/s00009-023-02300-w/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>In this paper, we study the representation theory of the fundamental group of the complement of a Hopf link with <i>n</i> twists. A general framework is described to analyze the <span class="mathjax-tex">${{\,\textrm{SL}\,}}_r({\mathbb {C}})$</span>-representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their <i>E</i>-polynomials are provided for ranks 2 and 3, both for the representation and character varieties.</p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1134/S0021364018110048?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1134/S0021364018110048">On the Hopf-Induced Deformation of a Topological Locus </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">28 May 2018</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2Fs10711-023-00777-z/MediaObjects/10711_2023_777_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s10711-023-00777-z?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/s10711-023-00777-z">Hyperbolic twisted torus links </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">19 February 2023</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s00208-018-1659-y?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/s00208-018-1659-y">Twisted K-theoretic Gromov–Witten invariants </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">21 February 2018</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1732358093, embedded_user: 'null' } }); </script> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=9" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>This work studies a special type of algebraic invariants of 3-dimensional links. To be precise, given a link <span class="mathjax-tex">$L \subset S^3$</span> and a complex affine algebraic group <i>G</i>, we can form the so-called <i>G</i>-representation variety of the link</p><div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R(L, G) = {{\,\textrm{Hom}\,}}(\pi _1(S^3-L), G), \end{aligned}$$</span></div></div><p>which parametrizes representations of the fundamental group of the link complement into <i>G</i>. This set can be naturally equipped with an algebraic structure in such a way that <i>R</i>(<i>L</i>, <i>G</i>) becomes a complex affine variety. In particular, its cohomology is endowed with a mixed Hodge structure from which we can compute the <i>E</i>-polynomial</p><div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(R(L,G)) = \sum _{k,p,q} (-1)^k h_c^{k,p,q}(R(L,G)) \, u^pv^q \in {\mathbb {Z}}[u,v], \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$h^{k,p,q}_{c}(R(L,G))= h^{p,q}(H_{c}^k(R(L,G)))=\dim {{\,\textrm{Gr}\,}}^{p}_{F}{{\,\textrm{Gr}\,}}^{W}_{p+q}H^{k}_{c}(R(L,G))$</span> are the compactly supported Hodge numbers of <i>R</i>(<i>L</i>, <i>G</i>). In the case that <span class="mathjax-tex">$h_c^{k,p,q}(R(L,G)) = 0$</span> for <span class="mathjax-tex">$p \ne q$</span>, it is customary to write the <i>E</i>-polynomial in the variable <span class="mathjax-tex">$q = uv$</span>.</p><p>Since the fundamental group of the link complement does not vary under diffeotopy of the link, the <i>E</i>-polynomial <i>e</i>(<i>R</i>(<i>L</i>, <i>G</i>)) is an algebraic invariant of the link <i>L</i> up to link equivalence. This <i>E</i>-polynomial provides an invariant encoding the algebraic structure of the representation variety attached to <i>L</i>, and typically differs from other classical invariants of <i>L</i> such as its Jones polynomial [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Kauffman, L.: Statistical mechanics and the Jones polynomial, Contemporary Mathematics, (1988)" href="/article/10.1007/s00009-023-02300-w#ref-CR16" id="ref-link-section-d90372366e1157">16</a>] or its <i>A</i>-polynomial [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Cooper, D., Culler, M., Gillet, H., Long, D., Shalen, P.: Plane curves associated to character varieties of 3-manifolds. Invent. Math. 118, 47–84 (1994)" href="/article/10.1007/s00009-023-02300-w#ref-CR4" id="ref-link-section-d90372366e1164">4</a>]. In fact, the geometry of the representation variety has been exploited several times in the literature to prove striking results of 3-manifolds. For instance, in the foundational work of Culler and Shalen [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Culler, M., Shalen, P.: Varieties of group representations and splitting of 3-manifolds. Ann. Math. 2(117), 109–146 (1983)" href="/article/10.1007/s00009-023-02300-w#ref-CR3" id="ref-link-section-d90372366e1167">3</a>], the authors used some simple properties of the <span class="mathjax-tex">${{\,\textrm{SL}\,}}_2({\mathbb {C}})$</span>-representation variety to provide new proofs of Thurston’s theorem stating that the space of hyperbolic structures on an acylindrical 3-manifold is compact, and of the Smith conjecture, which claims that any quotient with cyclic stabilizers of a closed oriented 3-manifold with non-trivial branch knot is not simply-connected [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Culler, M., Shalen, P.: Varieties of group representations and splitting of 3-manifolds. Ann. Math. 2(117), 109–146 (1983)" href="/article/10.1007/s00009-023-02300-w#ref-CR3" id="ref-link-section-d90372366e1214">3</a>, Section 5].</p><p>Representation varieties also play a central role in mathematical physics. In the very influential paper [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Witten, E.: Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121, 351–399 (1989)" href="/article/10.1007/s00009-023-02300-w#ref-CR28" id="ref-link-section-d90372366e1220">28</a>], Witten applied Chern–Simons theory to geometrically quantize <span class="mathjax-tex">${{\,\textrm{SU}\,}}(2)$</span>-representation varieties of knot complements, leading to a Topological Quantum Field Theory that computes the Jones polynomial of the knot. In some sense, our approach of looking at the <i>E</i>-polynomial of the representation variety can be understood as an alternative quantization of the representation varieties, more similar to Fourier–Mukai transforms in derived geometry [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Huybrechts, D.: Fourier-Mukai transforms in algebraic geometry, Clarendon Press, (2006)" href="/article/10.1007/s00009-023-02300-w#ref-CR15" id="ref-link-section-d90372366e1261">15</a>], in the sense that it consists of a pull-push construction (with identity kernel), than to path integrals as arising in Chern–Simons theory [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Witten, E.: Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121, 351–399 (1989)" href="/article/10.1007/s00009-023-02300-w#ref-CR28" id="ref-link-section-d90372366e1264">28</a>] (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="González-Prieto, A., Logares, M., Muñoz, V.: A lax monoidal Topological Quantum Field Theory for representation varieties. Bull. Sci. Math. 161, 102871 (2020)" href="/article/10.1007/s00009-023-02300-w#ref-CR11" id="ref-link-section-d90372366e1268">11</a>] for more information).</p><p>For these reasons, the computation of the <i>E</i>-polynomials <i>e</i>(<i>R</i>(<i>L</i>, <i>G</i>)) has been object of intense research in the recent years. The representation variety of torus knots for <span class="mathjax-tex">$G = {{\,\textrm{SL}\,}}_2({\mathbb {C}})$</span> was studied in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Muñoz, V.: The 
 
 
 
 $$SL(2,{\mathbb{C} })$$
 
 
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 -character varieties of torus knots. Rev. Mat. Complut. 22, 489–497 (2009)" href="/article/10.1007/s00009-023-02300-w#ref-CR25" id="ref-link-section-d90372366e1338">25</a>], and for <span class="mathjax-tex">$G={{\,\textrm{SU}\,}}(2)$</span> in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title="Marínez, J., Muñoz, V.: The SU(2)-character varieties of torus knots. Rocky Mountain J. Math. 2(45), 583–600 (2015)" href="/article/10.1007/s00009-023-02300-w#ref-CR24" id="ref-link-section-d90372366e1381">24</a>]; whereas the <span class="mathjax-tex">$G={{\,\textrm{SL}\,}}_3({\mathbb {C}})$</span> case was accomplished in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="V. Muñoz and J. Porti, Geometry of the 
 
 
 
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 -character variety of torus knots, Algebraic Geometric Topology 16 (2016) 397–426. (also 
 arXiv:1409.4784
 
 )" href="/article/10.1007/s00009-023-02300-w#ref-CR26" id="ref-link-section-d90372366e1432">26</a>], and recently the case <span class="mathjax-tex">$G={{\,\textrm{SL}\,}}_4({\mathbb {C}})$</span> in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="González-Prieto, A., Muñoz, V.: Motive of the 
 
 
 
 $$SL_4$$
 
 
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 -character variety of torus knots. J. Algebra (2022). 
 https://doi.org/10.1016/j.jalgebra.2022.06.008
 
 " href="/article/10.1007/s00009-023-02300-w#ref-CR12" id="ref-link-section-d90372366e1484">12</a>] through a computer-aided proof. More exotic knots have also been studied, as the figure eight knot in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Heusener, M., Muñoz, V., Porti, J.: The 
 
 
 
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 -character variety of the figure eight knot. Illinois J. Math. 60, 55–98 (2017)" href="/article/10.1007/s00009-023-02300-w#ref-CR14" id="ref-link-section-d90372366e1487">14</a>]. However, despite of these advances for representation varieties of knots, almost nothing is known in the case of links. The most studied case is the character variety of trivial links, i.e. representations of the free group, addressed in works such as [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Florentino, C., Lawton, S.: The topology of moduli spaces of free group representations. Math. Ann. 2(345), 453–489 (2009)" href="/article/10.1007/s00009-023-02300-w#ref-CR6" id="ref-link-section-d90372366e1490">6</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Florentino, C., Lawton, S.: Singularities of free group character varieties. Pac. J. Math. 260, 149–179 (2012)" href="/article/10.1007/s00009-023-02300-w#ref-CR7" id="ref-link-section-d90372366e1493">7</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Lawton, S.: Minimal affine coordinates for 
 
 
 
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 -character varieties of free groups. J. Algebra 320, 3773–3810 (2008)" href="/article/10.1007/s00009-023-02300-w#ref-CR18" id="ref-link-section-d90372366e1496">18</a>] (focused on the topology) and [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Cavazos, S., Lawton, S.: E-polynomial of 
 
 
 
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 -character varieties of free groups. Int. J. Math. 25, 1450058 (2014)" href="/article/10.1007/s00009-023-02300-w#ref-CR1" id="ref-link-section-d90372366e1500">1</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Florentino, C., Nozad, A., Zamora, A.: 
 
 
 
 $$E$$
 
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 -polynomials of 
 
 
 
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 and 
 
 
 
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 -character varieties of free groups, 
 arXiv:1912.05852
 
 " href="/article/10.1007/s00009-023-02300-w#ref-CR8" id="ref-link-section-d90372366e1503">8</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Lawton, S., Muñoz, V.: E-polynomial of the 
 
 
 
 $$SL(3, {\mathbb{C} })$$
 
 
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 -character variety of free groups. Pac. J. Math. 282, 173–202 (2016)" href="/article/10.1007/s00009-023-02300-w#ref-CR19" id="ref-link-section-d90372366e1506">19</a>] (computing the <i>E</i>-polynomials). Very recently, more complicated links were studied, such as the twisted Alexander polynomial for the Borromean link in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Chen, H., Yu, T.: The 
 
 
 
 $$SL(2, {\mathbb{C}})$$
 
 
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 L
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 -character variety of the Borromean link, 
 arXiv:2202.07429
 
 " href="/article/10.1007/s00009-023-02300-w#ref-CR2" id="ref-link-section-d90372366e1512">2</a>].</p><p>The aim of this work is to give the first steps towards an extension of the techniques to links. In particular, we shall focus on the “twisted” Hopf link <span class="mathjax-tex">$H_n$</span>, obtained by twisting a classical Hopf link with 2 crossings to get 2<i>n</i> crossings, as depicted in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00009-023-02300-w#Fig1">1</a>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Fig. 1"><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s00009-023-02300-w/figures/1" rel="nofollow"><picture><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs00009-023-02300-w/MediaObjects/9_2023_2300_Fig1_HTML.png" alt="figure 1" loading="lazy" width="685" height="686"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>The twisted Hopf link of <i>n</i> twists</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s00009-023-02300-w/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>The fundamental group of the link complement of <span class="mathjax-tex">$H_n$</span> can be computed through a Wirtinger presentation (Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar6">3.1</a>) giving rise to the group <span class="mathjax-tex">$\Gamma _n = \langle a, b \,|\, [a^n,b] = 1 \rangle $</span>. Therefore, the associated <i>G</i>-representation variety is</p><div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R(H_n, G) = \left\{ (A,B) \in G^2 \,|\, [A^n,B] = 1\right\} . \end{aligned}$$</span></div></div><p>In this sense, <span class="mathjax-tex">$R(H_n, G)$</span> should be understood as the variety counting “supercommuting” elements of <i>G</i>, generalizing the case <span class="mathjax-tex">$n=1$</span> of the usual Hopf link that corresponds to commuting elements, as studied in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Florentino, C., Silva, J.: Hodge–Deligne polynomials of character varieties of free abelian groups. Open Math. 19, 338–362 (2021)" href="/article/10.1007/s00009-023-02300-w#ref-CR9" id="ref-link-section-d90372366e1858">9</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Logares, M., Muñoz, V., Newstead, P.: Hodge polynomials of 
 
 
 
 $$SL(2,{\mathbb{C}})$$
 
 
 S
 L
 (
 2
 ,
 C
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 -character varieties for curves of small genus. Rev. Mat. Complut. 26, 635–703 (2013)" href="/article/10.1007/s00009-023-02300-w#ref-CR22" id="ref-link-section-d90372366e1861">22</a>].</p><p>One of the main challenges we face in the study of the geometry of <span class="mathjax-tex">$R(H_n, G)$</span> is the analysis of the map <span class="mathjax-tex">$p_n: G \rightarrow G$</span>, <span class="mathjax-tex">$A \mapsto A^n$</span>. In this paper, we propose to split this analysis into two different frameworks, that we call the <i>combinatorial</i> and the <i>geometric</i>. The combinatorial setting focuses on the study of the configuration space of possible eigenvalues, and how it can degenerate under the map <span class="mathjax-tex">$p_n$</span>. We will show in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec7">4.1</a> that understanding these degenerations can be done systematically, and eventually it is performed by means of a thorough application of the inclusion-exclusion principle.</p><p>The geometric setting is discussed in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec8">4.2</a>, where we show how the <i>E</i>-polynomial of the representation variety can be obtained from the possible Jordan forms. To this aim, both the stabilizer of <span class="mathjax-tex">$A^n$</span> in <span class="mathjax-tex">${{\,\textrm{SL}\,}}_r({\mathbb {C}})$</span> under the conjugacy action (to parametrize the possible matrices <i>B</i>) and the stabilizer of <i>A</i> (through the conjugacy orbit of the Jordan form) play a role. Moreover, in the cases in which the Jordan form is not unique, but only unique up to permutation of eigenvalues, we show how the quotient by the corresponding symmetric group can be computed via equivariant Hodge theory, as developed in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec4">2.2</a>.</p><p>To show the feasibility of this approach, we apply it to the cases of rank 2 (Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec9">5</a>) and rank 3 (Sects. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec13">7</a> and <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec15">8</a>), obtaining the main result of this paper.</p> <h3 class="c-article__sub-heading" id="FPar1">Theorem</h3> <p>The <i>E</i>-polynomials of the <span class="mathjax-tex">${{\,\textrm{SL}\,}}_r({\mathbb {C}})$</span>-representation variety of the twisted Hopf link <span class="mathjax-tex">$H_n$</span> with <i>n</i> twists for ranks <span class="mathjax-tex">$r = 2,3$</span>, are the following.</p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\big (R(H_n, {{\,\textrm{SL}\,}}_{2}({\mathbb {C}}))\big )&= \left( (n-1)q^2+ n q -n+5)\right) (q^3-q), \\ e\big (R(H_n, {{\,\textrm{SL}\,}}_{3}({\mathbb {C}}))\big )&=\, (q^3-1)(q^2-1)q^2 \Big ( \left\lfloor \frac{n}{2} \right\rfloor (q^2-q)(q^2-q-1) \\&\quad +\frac{1}{2}n^2 (q^7+2q^6+2q^5+q^4-3q^3-3q^2+2q)\\&\quad -\frac{1}{2}n (3q^7+6q^6-3q^4 -17q^3 \\&\quad -q^2+12q) + q^7+2q^6-q^5-2q^4\\&\quad -6q^3+2q^2+13q\Big ). \end{aligned}$$</span></div></div> <p>Additionally, in this paper we will go a step forward and also study the associated character varieties. The key point is that, if we want to obtain a genuine moduli space, we must identify isomorphic representations. This can be done by means of the GIT quotient of the representation variety <i>R</i>(<i>L</i>, <i>G</i>) under the adjoint action of <i>G</i>, giving rise to the so-called character variety</p><div id="Equ112" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/lw176/springer-static/image/art%3A10.1007%2Fs00009-023-02300-w/MediaObjects/9_2023_2300_Equ112_HTML.png" class="u-display-block" alt=""></div></div><p>It is well known [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Lubotzky, A., Magid, A.: Varieties of representations of finitely generated groups, Mem. Am. Math. Soc. 58 (1985)" href="/article/10.1007/s00009-023-02300-w#ref-CR23" id="ref-link-section-d90372366e2903">23</a>] that every representation is equivalent, under the GIT quotient, to a semi-simple representation. The semi-simple representations are those that are direct sums of irreducible ones. Hence <span class="mathjax-tex">${\mathfrak {M}}(L,G)$</span> is stratified according to partitions of <i>r</i>, where <span class="mathjax-tex">$G={{\,\textrm{SL}\,}}_r({\mathbb {C}})$</span>, corresponding to representations that are sums of irreducible representations of the ranks given in the partition. The <i>E</i>-polynomial of the reducible locus <span class="mathjax-tex">${\mathfrak {M}}^{\textrm{red}}(L,G)$</span> is computed inductively from the irreducible representations of lower ranks.</p><p>In the case of the twisted Hopf link <span class="mathjax-tex">$H_n$</span>, to compute the <i>E</i>-polynomial <span class="mathjax-tex">$e({\mathfrak {M}}^{\textrm{irr}}(H_n,G))$</span> we use the characterization that a representation is irreducible when <i>A</i>, <i>B</i> do not both leave invariant a proper subspace. This strategy is accomplished for rank 2 (Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec12">6</a>) and rank 3 (Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec18">9</a>), leading to the following result.</p> <h3 class="c-article__sub-heading" id="FPar2">Theorem</h3> <p>The <i>E</i>-polynomials of the <span class="mathjax-tex">${{\,\textrm{SL}\,}}_r({\mathbb {C}})$</span>-character variety of the twisted Hopf link <span class="mathjax-tex">$H_n$</span> with <i>n</i> twists for ranks <span class="mathjax-tex">$r = 2,3$</span>, are the following.</p><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( {\mathfrak {M}}(H_n, {{\,\textrm{SL}\,}}_{2}({\mathbb {C}}))\right)&= q^2+1+(n-1)(q^2-q+1), \\ e({\mathfrak {M}}(H_n, {{\,\textrm{SL}\,}}_3({\mathbb {C}})))&= q^{4} + \frac{1}{2} {\left( q^{6} + 2 q^{5} - 4 q^{4} + q^{3} + 3 q^{2} - 3 q + 2\right) } {\left( n^{2} - 3 n + 2\right) } \\&\quad - {\left( q^{3} - 2 q - 1\right) } \left\lfloor \frac{n-1}{2} \right\rfloor + 3 {\left( q^{4} - q^{3} + q^{2} - q + 1\right) } {\left( n - 1\right) } \\&\quad - (q^{2} + 1){\left( n - 2\right) } + q^{3}(n -1). \end{aligned}$$</span></div></div> <p>It is worth mentioning that the strategies of computation described in this paper are not restricted to low rank, and work verbatim for arbitrary rank. However, the combinatorial analysis becomes exponentially more involved with increasing rank, so the higher rank cases are untreatable with a direct counting. An interesting future work would be to algorithmize the procedure of solving the combinatorial problem, so that the higher rank cases could be addressed via a computer aided-proof, as done in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="González-Prieto, A., Muñoz, V.: Motive of the 
 
 
 
 $$SL_4$$
 
 
 S
 
 L
 4
 
 
 
 -character variety of torus knots. J. Algebra (2022). 
 https://doi.org/10.1016/j.jalgebra.2022.06.008
 
 " href="/article/10.1007/s00009-023-02300-w#ref-CR12" id="ref-link-section-d90372366e3750">12</a>] for torus knots.</p><p>Finally, we would like to point out that this work is cornerstone to the understanding of representation varieties of general 3-manifolds. Recall that the Lickorish–Wallace theorem [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Lickorish, W.: A representation of orientable combinatorial 3-manifolds. Ann. Math. 76, 531–540 (1962)" href="/article/10.1007/s00009-023-02300-w#ref-CR21" id="ref-link-section-d90372366e3756">21</a>] states that any closed orientable connected 3-manifold can be obtained by applying Dehn surgery around a link <span class="mathjax-tex">$L \subset S^3$</span>. This highlights the importance of (i) studying representation varieties for general links, not only knots; and (ii) the key role that the maps <span class="mathjax-tex">$p_n(A)=A^n$</span> play in this project, since they appear as part of the automorphism of the fundamental group of the torus around which surgery takes place.</p></div></div></section><section data-title="Representation Varieties and Character Varieties"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Representation Varieties and Character Varieties</h2><div class="c-article-section__content" id="Sec2-content"><p>Let <span class="mathjax-tex">$\Gamma $</span> be a finitely generated group, and let <i>G</i> be a complex reductive Lie group. A <i>representation</i> of <span class="mathjax-tex">$\Gamma $</span> in <i>G</i> is a homomorphism <span class="mathjax-tex">$\rho : \Gamma \rightarrow G$</span>. Consider a presentation <span class="mathjax-tex">$\Gamma =\langle \gamma _1,\ldots , \gamma _k | \{r_{\lambda }\}_{\lambda \in \Lambda } \rangle $</span>, where <span class="mathjax-tex">$\Lambda $</span> is the (possibly infinite) indexing set of relations of <span class="mathjax-tex">$\Gamma $</span>. Then <span class="mathjax-tex">$\rho $</span> is completely determined by the <i>k</i>-tuple <span class="mathjax-tex">$(A_1,\ldots , A_k)=(\rho (\gamma _1),\ldots , \rho (\gamma _k))$</span> subject to the relations <span class="mathjax-tex">$r_\lambda (A_1,\ldots , A_k)=\textrm{Id}$</span>, for all <span class="mathjax-tex">$\lambda \in \Lambda $</span>. The <i>representation variety</i> is</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R(\Gamma ,G)= & {} \, {{\,\textrm{Hom}\,}}(\Gamma , G) \nonumber \\= & {} \, \{(A_1,\ldots , A_k) \in G^k \, | \, r_\lambda (A_1,\ldots , A_k)=\textrm{Id},\forall \lambda \, \}\subset G^{k}\, . \end{aligned}$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>Therefore, <span class="mathjax-tex">$R(\Gamma ,G)$</span> is an affine algebraic set. Even though <span class="mathjax-tex">$\Lambda $</span> may be an infinite set, <span class="mathjax-tex">$ R(\Gamma ,G)$</span> is defined by finitely many equations, as a consequence of the noetherianity of the coordinate ring of <span class="mathjax-tex">$G^k$</span>.</p><p>We say that two representations <span class="mathjax-tex">$\rho $</span> and <span class="mathjax-tex">$\rho '$</span> are equivalent if there exists <span class="mathjax-tex">$g \in G$</span> such that <span class="mathjax-tex">$\rho '(\gamma )=g^{-1} \rho (\gamma ) g$</span>, for every <span class="mathjax-tex">$\gamma \in \Gamma $</span>. The moduli space of representations, also known as the <i>character variety</i>, is the GIT quotient</p><div id="Equ113" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/lw177/springer-static/image/art%3A10.1007%2Fs00009-023-02300-w/MediaObjects/9_2023_2300_Equ113_HTML.png" class="u-display-block" alt=""></div></div><p>Recall that by definition of the GIT quotient for an affine variety, if we write <span class="mathjax-tex">$ R(\Gamma ,G)={{\,\textrm{Spec}\,}}A$</span>, then <span class="mathjax-tex">$M (\Gamma ,G)={{\,\textrm{Spec}\,}}A^{G}$</span>, where <span class="mathjax-tex">$A^G$</span> is the finitely generated <i>k</i>-algebra of invariant elements of <i>A</i> under the induced action of <i>G</i>.</p><p>A representation <span class="mathjax-tex">$\rho $</span> is <i>reducible</i> if there exists some proper linear subspace <span class="mathjax-tex">$W\subset V$</span> such that for all <span class="mathjax-tex">$\gamma \in \Gamma $</span> we have <span class="mathjax-tex">$\rho (\gamma )(W)\subset W$</span>; otherwise <span class="mathjax-tex">$\rho $</span> is <i>irreducible</i>. If <span class="mathjax-tex">$\rho $</span> is reducible, then there is a flag of subspaces <span class="mathjax-tex">$0=W_0\subsetneq W_1\subsetneq \ldots \subsetneq W_r=V$</span> such that <span class="mathjax-tex">$\rho $</span> leaves <span class="mathjax-tex">$W_i$</span> invariant, and it induces an irreducible representation <span class="mathjax-tex">$\rho _i$</span> in the quotient <span class="mathjax-tex">$V_i=W_i/W_{i-1}$</span>, <span class="mathjax-tex">$i=1,\ldots ,r$</span>. Then <span class="mathjax-tex">$\rho $</span> and <span class="mathjax-tex">${\hat{\rho }}=\bigoplus \rho _i$</span> define the same point in the quotient <span class="mathjax-tex">${\mathfrak {M}}(\Gamma ,G)$</span>. We say that <span class="mathjax-tex">${\hat{\rho }}$</span> is a semi-simple representation, and that <span class="mathjax-tex">$\rho $</span> and <span class="mathjax-tex">${\hat{\rho }}$</span> are S-equivalent. The space <span class="mathjax-tex">${\mathfrak {M}}(\Gamma ,G)$</span> parametrizes semi-simple representations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Lubotzky, A., Magid, A.: Varieties of representations of finitely generated groups, Mem. Am. Math. Soc. 58 (1985)" href="/article/10.1007/s00009-023-02300-w#ref-CR23" id="ref-link-section-d90372366e5423">23</a>, Thm. 1.28] up to conjugation.</p><p>The name ‘character variety’ for <span class="mathjax-tex">$ {\mathfrak {M}}(\Gamma ,G)$</span> is justified by the following fact. Suppose now that <span class="mathjax-tex">$G={{\,\textrm{SL}\,}}_r({\mathbb {C}})$</span>. Given a representation <span class="mathjax-tex">$\rho : \Gamma \rightarrow G$</span>, we define its <i>character</i> as the map <span class="mathjax-tex">$\chi _\rho : \Gamma \rightarrow {\mathbb {C}}$</span>, <span class="mathjax-tex">$\chi _\rho (g)={{\,\textrm{tr}\,}}\rho (g)$</span>. Note that two equivalent representations <span class="mathjax-tex">$\rho $</span> and <span class="mathjax-tex">$\rho '$</span> have the same character. There is a character map <span class="mathjax-tex">$\chi : R(\Gamma ,G)\rightarrow {\mathbb {C}}^\Gamma $</span>, <span class="mathjax-tex">$\rho \mapsto \chi _\rho $</span>, whose image</p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathfrak {X}}(\Gamma ,G)=\chi (R(\Gamma ,G)) \end{aligned}$$</span></div></div><p>leads to a natural algebraic map</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathfrak {M}}(\Gamma ,G)\rightarrow {\mathfrak {X}}(\Gamma ,G). \end{aligned}$$</span></div><div class="c-article-equation__number"> (2) </div></div><p>It turns out that this map is an isomorphism for <span class="mathjax-tex">$G = {{\,\textrm{SL}\,}}_{r}({\mathbb {C}})$</span> cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Lubotzky, A., Magid, A.: Varieties of representations of finitely generated groups, Mem. Am. Math. Soc. 58 (1985)" href="/article/10.1007/s00009-023-02300-w#ref-CR23" id="ref-link-section-d90372366e5960">23</a>, Chapter 1]. This is the same as to say that <span class="mathjax-tex">$A^G$</span> is generated by the traces <span class="mathjax-tex">$\chi _\rho $</span>, <span class="mathjax-tex">$\rho \in R(\Gamma ,G)$</span>. In other words, in this case <span class="mathjax-tex">$ {\mathfrak {M}}(\Gamma ,G)$</span> is made of characters, justifying its name. However, for other reductive groups the map (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ2">2</a>) may not be an isomorphism, as for <span class="mathjax-tex">$G = \textrm{SO}_2({\mathbb {C}})$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Florentino, C., Lawton, S.: Singularities of free group character varieties. Pac. J. Math. 260, 149–179 (2012)" href="/article/10.1007/s00009-023-02300-w#ref-CR7" id="ref-link-section-d90372366e6125">7</a>, Appendix A]. For a general discussion on this issue, see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Lawton, S., Sikora, A.: Varieties of characters, Algebr. Represent. Theory 20, 1133–1141 (2017)" href="/article/10.1007/s00009-023-02300-w#ref-CR20" id="ref-link-section-d90372366e6128">20</a>].</p><h3 class="c-article__sub-heading" id="Sec3"><span class="c-article-section__title-number">2.1 </span>Hodge Structures and <i>E</i>-Polynomials</h3><p>A pure Hodge structure of weight <i>k</i> consists of a finite dimensional complex vector space <i>H</i> with a real structure, and a decomposition <span class="mathjax-tex">$H=\bigoplus _{k=p+q} H^{p,q}$</span> such that <span class="mathjax-tex">$H^{q,p}=\overline{H^{p,q}}$</span>, the bar meaning complex conjugation on <i>H</i>. A Hodge structure of weight <i>k</i> gives rise to the so-called Hodge filtration, which is a descending filtration <span class="mathjax-tex">$F^{p}=\bigoplus _{s\ge p}H^{s,k-s}$</span>. We define <span class="mathjax-tex">${{\,\textrm{Gr}\,}}^{p}_{F}(H):=F^{p}/ F^{p+1}=H^{p,k-p}$</span>.</p><p>A mixed Hodge structure consists of a finite dimensional complex vector space <i>H</i> with a real structure, an ascending (weight) filtration <span class="mathjax-tex">$\cdots \subset W_{k-1}\subset W_k \subset \cdots \subset H$</span> (defined over <span class="mathjax-tex">${\mathbb {R}}$</span>) and a descending (Hodge) filtration <i>F</i> such that <i>F</i> induces a pure Hodge structure of weight <i>k</i> on each <span class="mathjax-tex">${{\,\textrm{Gr}\,}}^{W}_{k}(H)=W_{k}/W_{k-1}$</span>. We define <span class="mathjax-tex">$H^{p,q}:= {{\,\textrm{Gr}\,}}^{p}_{F}{{\,\textrm{Gr}\,}}^{W}_{p+q}(H)$</span> and write <span class="mathjax-tex">$h^{p,q}$</span> for the <i>Hodge number</i> <span class="mathjax-tex">$h^{p,q} :=\dim H^{p,q}$</span>.</p><p>Let <i>Z</i> be any quasi-projective algebraic variety (possibly non-smooth or non-compact). The cohomology groups <span class="mathjax-tex">$H^k(Z)$</span> and the cohomology groups with compact support <span class="mathjax-tex">$H^k_c(Z)$</span> are endowed with mixed Hodge structures [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Deligne, P.: Théorie de Hodge II, Publ. Math. I.H.E.S. 40 5–57 (1971)" href="/article/10.1007/s00009-023-02300-w#ref-CR5" id="ref-link-section-d90372366e6839">5</a>]. We define the <i>Hodge numbers</i> of <i>Z</i> by <span class="mathjax-tex">$h^{k,p,q}_{c}(Z)= h^{p,q}(H_{c}^k(Z))=\dim {{\,\textrm{Gr}\,}}^{p}_{F}{{\,\textrm{Gr}\,}}^{W}_{p+q}H^{k}_{c}(Z)$</span> . The <i>E</i>-polynomial is defined as</p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(Z):=\sum _{p,q,k} (-1)^{k}h^{k,p,q}_{c}(Z) u^{p}v^{q}. \end{aligned}$$</span></div></div><p>The key property of Hodge–Deligne polynomials that permits their calculation is that they are additive for stratifications of <i>Z</i>. If <i>Z</i> is a complex algebraic variety and <span class="mathjax-tex">$Z=\bigsqcup _{i=1}^{n}Z_{i}$</span>, where all <span class="mathjax-tex">$Z_i$</span> are locally closed in <i>Z</i>, then <span class="mathjax-tex">$e(Z)=\sum _{i=1}^{n}e(Z_{i})$</span>. Also <span class="mathjax-tex">$e(X\times Y)=e(X)e(Y)$</span> or, more generally, <span class="mathjax-tex">$e(X) = e(F)e(B)$</span> for any fiber bundle <span class="mathjax-tex">$F \rightarrow X \rightarrow B$</span> in the Zariski topology [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="González-Prieto, A.: Pseudo-quotients of algebraic actions and their applications to character varieties, 
 arXiv:1807.08540
 
 " href="/article/10.1007/s00009-023-02300-w#ref-CR10" id="ref-link-section-d90372366e7434">10</a>, Proposition 4.6]. Moreover, by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Logares, M., Muñoz, V., Newstead, P.: Hodge polynomials of 
 
 
 
 $$SL(2,{\mathbb{C}})$$
 
 
 S
 L
 (
 2
 ,
 C
 )
 
 
 -character varieties for curves of small genus. Rev. Mat. Complut. 26, 635–703 (2013)" href="/article/10.1007/s00009-023-02300-w#ref-CR22" id="ref-link-section-d90372366e7437">22</a>, Remark 2.5] if <span class="mathjax-tex">$G\rightarrow X\rightarrow B$</span> is a principal fiber bundle with <i>G</i> a connected algebraic group, then <span class="mathjax-tex">$e(X)=e(G)e(B)$</span>.</p><p>When <span class="mathjax-tex">$h_c^{k,p,q}=0$</span> for <span class="mathjax-tex">$p\ne q$</span>, the polynomial <i>e</i>(<i>Z</i>) depends only on the product <i>uv</i>. This will happen in all the cases that we shall investigate here. In this situation, it is conventional to use the variable <span class="mathjax-tex">$q=uv$</span>. If this happens, we say that the variety is <i>of balanced type</i>. Some cases that we shall need are:</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">$e({\mathbb {C}}^r)=q^r$</span>.</p> </li> <li> <p><span class="mathjax-tex">$e({\mathbb {C}}^*)=q-1$</span>.</p> </li> <li> <p><span class="mathjax-tex">$e({{\,\textrm{GL}\,}}_r({\mathbb {C}}))= (q^r-1)(q^r-q)\cdots (q^r-q^{r-1})$</span>.</p> </li> <li> <p><span class="mathjax-tex">$e({{\,\textrm{SL}\,}}_r({\mathbb {C}}))=e({{\,\textrm{PGL}\,}}_r({\mathbb {C}})) = (q^r-1)(q^r-q) \cdots (q^r-q^{r-2})q^{r-1}$</span>.</p> </li> <li> <p><span class="mathjax-tex">$e({\mathbb {P}}^r)=q^r+\ldots + q^2 + q+1$</span>.</p> </li> <li> <p><span class="mathjax-tex">$\textrm{Sym}^r({\mathbb {P}}^1)={\mathbb {P}}^r$</span> hence <span class="mathjax-tex">$e(\textrm{Sym}^r({\mathbb {P}}^1))=q^r+\ldots +q+1$</span>.</p> </li> <li> <p>By [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Gusein-Zade, S., Luengo, I., Melle-Hernández, A.: On the power structure over the Grothendieck ring of varieties and its applications. Proc. Steklov Inst. Math. 258, 53–64 (2007)" href="/article/10.1007/s00009-023-02300-w#ref-CR13" id="ref-link-section-d90372366e8290">13</a>], we have that <span class="mathjax-tex">$\zeta _{{\mathbb {P}}^n}(t)=\sum _{r\ge 0} e(\textrm{Sym}^r({\mathbb {P}}^n)) t^r$</span> satisfies the formula </p><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \zeta _{{\mathbb {P}}^n}(t)=\prod _{i=0}^n \frac{1}{1-q^it}\, . \end{aligned}$$</span></div></div><p> From this, we extract <span class="mathjax-tex">$e(\textrm{Sym}^2({\mathbb {P}}^2))=q^4+q^3+2q^2+q+1$</span>, and <span class="mathjax-tex">$e(\textrm{Sym}^3({\mathbb {P}}^2))=q^6+q^5+2q^4+2q^3+2q^2+q+1$</span>.</p> </li> </ul><h3 class="c-article__sub-heading" id="Sec4"><span class="c-article-section__title-number">2.2 </span>Equivariant <i>E</i>-Polynomial</h3><p>We enhance the definition of <i>E</i>-polynomial to the case where there is an action of a finite group (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Lawton, S., Muñoz, V.: E-polynomial of the 
 
 
 
 $$SL(3, {\mathbb{C} })$$
 
 
 S
 L
 (
 3
 ,
 C
 )
 
 
 -character variety of free groups. Pac. J. Math. 282, 173–202 (2016)" href="/article/10.1007/s00009-023-02300-w#ref-CR19" id="ref-link-section-d90372366e8731">19</a>, Section 2]).</p> <h3 class="c-article__sub-heading" id="FPar3">Definition 2.1</h3> <p>Let <i>X</i> be a complex quasi-projective variety on which a finite group <i>F</i> acts. Then <i>F</i> also acts on the cohomology <span class="mathjax-tex">$H^*_c(X)$</span> respecting the mixed Hodge structure. So <span class="mathjax-tex">$[H^*_c(X)]\in R(F)$</span>, the representation ring of <i>F</i>. The <i>equivariant E-polynomial</i> is defined as</p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_F(X)=\sum _{p,q,k} (-1)^k [H^{k,p,q}_c(X)] \, u^pv^q \in R(F)[u,v]. \end{aligned}$$</span></div></div> <p>Note that the map <span class="mathjax-tex">$\dim : R(F)\rightarrow {\mathbb {Z}}$</span> recovers the usual <i>E</i>-polynomial as <span class="mathjax-tex">$\dim (e_F(X))=e(X)$</span>. Moreover, let <i>T</i> be the trivial representation. Let <span class="mathjax-tex">$\ell $</span> be the number of irreducible representations of <i>F</i>, which coincides with the number of conjugacy classes of <i>F</i>. Let <span class="mathjax-tex">$T=T_1,T_2,\ldots , T_\ell $</span> be the irreducible representations in <i>R</i>(<i>F</i>). Write <span class="mathjax-tex">$e_F(X)= \sum _{j=1}^\ell a_j T_j$</span>. Then <span class="mathjax-tex">$e(X/F)= a_1$</span>, the coefficient of <i>T</i> in <span class="mathjax-tex">$e_F(X)$</span>.</p><p>We need specifically the case of the symmetric group <span class="mathjax-tex">$S_r$</span>. For instance, for an action of <span class="mathjax-tex">$S_2$</span>, there are two irreducible representations <i>T</i>, <i>N</i>, where <i>T</i> is the trivial representation, and <i>N</i> is the non-trivial representation. Then <span class="mathjax-tex">$e_{S_2}(X)=aT+bN$</span>. Clearly <span class="mathjax-tex">$e(X) = a+b$</span>, <span class="mathjax-tex">$e(X/S_2) = a$</span>. Therefore,</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned}&e_{S_2}(X)=aT+bN, \\&a=e(X/S_2), \\&b=e(X)-e(X/S_2). \end{aligned} \end{aligned}$$</span></div><div class="c-article-equation__number"> (3) </div></div><p>Note that if <span class="mathjax-tex">$X,X'$</span> are spaces with <span class="mathjax-tex">$S_2$</span>-actions, then writing <span class="mathjax-tex">$e_{S_2}(X)=aT+bN$</span>, <span class="mathjax-tex">$e_{S_2}(X')=a'T+b'N$</span>, we have <span class="mathjax-tex">$e_{S_2}(X\times X')= (aa'+bb') T+ (ab'+ba')N$</span> and so <span class="mathjax-tex">$e((X\times X')/S_2)=aa'+bb'$</span>.</p><p>We shall use later also the case of the symmetric group <span class="mathjax-tex">$F=S_3$</span>. Denote by <span class="mathjax-tex">$\alpha =(1,2,3)$</span> the 3-cycle and <span class="mathjax-tex">$\tau =(1,2)$</span> a transposition. There are three irreducible representations <i>T</i>, <i>S</i>, <i>D</i>, where <i>T</i> is the trivial one, <i>S</i> is the sign representation, and <i>D</i> is the standard representation. The sign representation is one-dimensional <span class="mathjax-tex">$S={\mathbb {R}}$</span>, where <span class="mathjax-tex">$\alpha \cdot x=x$</span> and <span class="mathjax-tex">$\tau \cdot x=-x$</span>. The standard representation is two-dimensional <span class="mathjax-tex">$D={\mathbb {R}}^2={\mathbb {C}}$</span>, where <span class="mathjax-tex">$\tau \cdot z={\overline{z}}$</span>, <span class="mathjax-tex">$\alpha \cdot z= e^{2\pi i/3} z$</span>. The multiplicative table of <span class="mathjax-tex">$R(S_3)$</span> is easily checked to be given by</p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} T\otimes T&=T,{} & {} T\otimes S =S, \\ T\otimes D&=D,{} & {} S\otimes S =T, \\ S\otimes D&=D,{} & {} D\otimes D =T +S + D. \end{aligned}$$</span></div></div><p>Let <i>X</i> be a variety with an <span class="mathjax-tex">$S_3$</span>-action. Then <span class="mathjax-tex">$e_{S_3}(X) =a T+bS + cD$</span>. Then <span class="mathjax-tex">$e(X)=a+b+2c$</span> and <span class="mathjax-tex">$e(X/S_3)=a$</span>. For the transposition <span class="mathjax-tex">$\tau =(1,2)\in S_3$</span>, we have <span class="mathjax-tex">$T^\tau ={\mathbb {R}}$</span>, <span class="mathjax-tex">$S^\tau =0$</span> and <span class="mathjax-tex">$D^\tau ={\mathbb {R}}$</span>. Thus, <span class="mathjax-tex">$e(X/\langle \tau \rangle )=a+c$</span>. This implies that</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned}&e_{S_3}(X) =a T+bS + cD , \\&a = e(X/S_3), \\&b= e(X)-2 e(X/\langle \tau \rangle )+e(X/S_3), \\&c = e(X/\langle \tau \rangle )- e(X/S_3). \end{aligned} \end{aligned}$$</span></div><div class="c-article-equation__number"> (4) </div></div><p>An interesting case that we will also apply in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec15">8</a> is the following.</p> <h3 class="c-article__sub-heading" id="FPar4">Proposition 2.2</h3> <p>Let <i>G</i> be a complex algebraic group equipped with an action of a finite group <span class="mathjax-tex">$\rho : F \rightarrow Inn (G)$</span> acting by inner automorphisms. If <i>G</i> is connected, then</p><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_{F}(G) = e(G)T. \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar5">Proof</h3> <p>Since <span class="mathjax-tex">$Inn (G) = G/Z(G)$</span>, if <i>G</i> is connected then <span class="mathjax-tex">$Inn (G)$</span> also is so. Hence, any inner automorphism is connected to the identity through a path, meaning that any inner automorphism is homotopic to the identity. Then, for all <span class="mathjax-tex">$\tau \in F$</span>, the map <span class="mathjax-tex">$\tau \cdot : G \rightarrow G$</span> is null-homotopic, so it induces a trivial action in cohomology. <span class="mathjax-tex">$\square $</span></p> </div></div></section><section data-title="Twisted Hopf Links"><div class="c-article-section" id="Sec5-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec5"><span class="c-article-section__title-number">3 </span>Twisted Hopf Links</h2><div class="c-article-section__content" id="Sec5-content"><p>In this paper, we shall focus on the <i>twisted Hopf link</i>, which is the link formed by two circles knotted as the Hopf link but with <i>n</i> twists, as depicted in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00009-023-02300-w#Fig2">2</a>. We will denote this knot by <span class="mathjax-tex">$H_n$</span>. Notice that the link <span class="mathjax-tex">$H_1$</span> is the usual Hopf link.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Fig. 2"><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s00009-023-02300-w/figures/2" rel="nofollow"><picture><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs00009-023-02300-w/MediaObjects/9_2023_2300_Fig2_HTML.png" alt="figure 2" loading="lazy" width="685" height="686"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>The twisted Hopf link of <i>n</i> twists with oriented strands</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s00009-023-02300-w/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar6">Proposition 3.1</h3> <p>The fundamental group of the (complement of the) twisted Hopf link with <span class="mathjax-tex">$n \ge 1$</span> twists is</p><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Gamma _n:=\pi _1(S^3-H_n) = \langle a, b \,|\, [a^n,b] = 1 \rangle , \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$[a^n,b] = a^nba^{-n}b^{-1}$</span> is the group commutator.</p> <h3 class="c-article__sub-heading" id="FPar7">Proof</h3> <p>Let us compute the Wirtinger presentation of the fundamental group of <span class="mathjax-tex">$H_n$</span>, which is a presentation of the fundamental group of the complement of a knot that can be obtained algorithmically from the crossings of a planar representation of the knot [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Rolfsen, D. Knots and links, Publish or Perish, (1990)" href="/article/10.1007/s00009-023-02300-w#ref-CR27" id="ref-link-section-d90372366e11968">27</a>, Chapter III.D]. Let us orient the two strands of <span class="mathjax-tex">$H_n$</span> as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00009-023-02300-w#Fig2">2</a>. From these orientation, we observe that <span class="mathjax-tex">$\pi _1(S^3-H_n)$</span> is generated by 2<i>n</i> elements, namely <span class="mathjax-tex">$x_1, y_1, \ldots , x_n, y_n$</span>, corresponding to the 2<i>n</i> arcs of overpassing strands. For the relations, the knot has <i>n</i> double crossings of the form of Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00009-023-02300-w#Fig3">3</a>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Fig. 3"><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s00009-023-02300-w/figures/3" rel="nofollow"><picture><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs00009-023-02300-w/MediaObjects/9_2023_2300_Fig3_HTML.png" alt="figure 3" loading="lazy" width="590" height="791"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p>A crossing of the <span class="mathjax-tex">$H_n$</span></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s00009-023-02300-w/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>From each of these crossings, we obtain two relations <span class="mathjax-tex">$y_kx_k = x_{k+1}y_k$</span> and <span class="mathjax-tex">$x_{k+1}y_k = y_{k+1}x_{k+1}$</span> for <span class="mathjax-tex">$k = 1, 2, \ldots , n$</span>, writing <span class="mathjax-tex">$x_{n+1} = x_1$</span> and <span class="mathjax-tex">$y_{n+1} = y_1$</span>. Therefore, we get that</p><div id="Equ35" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} \pi _1(S^3-H_n)=&{} \langle x_1, y_1, \ldots , x_n, y_n \,|\, y_kx_k = x_{k+1}y_k, x_{k+1}y_k,\\&\,\, {} y_{k+1}x_{k+1} \text{ for } 1\le k \le n\rangle . \end{aligned} \end{aligned}$$</span></div></div><p>From this relations, we can solve for <span class="mathjax-tex">$y_k$</span> and <span class="mathjax-tex">$x_k$</span>, for <span class="mathjax-tex">$k\ge 2$</span>, from <span class="mathjax-tex">$x_1$</span> and <span class="mathjax-tex">$y_1$</span>. Thus, the group can be also written as</p><div id="Equ36" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \pi _1(S^3-H_n) \cong \langle x_1, y_1 \,|\, (x_1y_1)^n = (y_1x_1)^n\rangle . \end{aligned}$$</span></div></div><p>Making the change <span class="mathjax-tex">$a = x_1y_1$</span> and <span class="mathjax-tex">$b = y_1$</span> we get the desired presentation. <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar8">Remark 3.2</h3> <p>For <span class="mathjax-tex">$n = 1$</span>, the group <span class="mathjax-tex">$\pi _1(S^3-H_1) = {\mathbb {Z}}\times {\mathbb {Z}}$</span> coincides with the fundamental group of the 2-dimensional torus, which is generated by two commuting elements. In some sense, <span class="mathjax-tex">$\pi _1(S^3-H_n)$</span> generalizes this result by considering ‘supercommutation’ relations instead, of the form <span class="mathjax-tex">$a^nb=ba^n$</span>.</p> <p>Using the description (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ1">1</a>) we directly get from Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar6">3.1</a> that the <i>G</i>-representation variety of the twisted Hopf link with <i>n</i> twists is</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R(\Gamma _n, G) = \left\{ (A, B) \in G^2 \,|\, [A^n,B] = 1\right\} . \end{aligned}$$</span></div><div class="c-article-equation__number"> (5) </div></div><p>To emphasize the role of the twisted Hopf link in the representation variety, throughout this paper we shall denote <span class="mathjax-tex">$R(H_n, G) = R(\Gamma _n, G)$</span>.</p></div></div></section><section data-title="The ${{\,\textrm{SL}\,}}_{r}({\mathbb {C}})$-Representation Variety of the Twisted Hopf Link"><div class="c-article-section" id="Sec6-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec6"><span class="c-article-section__title-number">4 </span>The <span class="mathjax-tex">${{\,\textrm{SL}\,}}_{r}({\mathbb {C}})$</span>-Representation Variety of the Twisted Hopf Link</h2><div class="c-article-section__content" id="Sec6-content"><h3 class="c-article__sub-heading" id="Sec7"><span class="c-article-section__title-number">4.1 </span>The Combinatorial Setting</h3><p>Given <span class="mathjax-tex">$r \ge 1$</span>, let us consider the space of possible eigenvalues of a matrix of <span class="mathjax-tex">${{\,\textrm{SL}\,}}_{r}({\mathbb {C}})$</span>,</p><div id="Equ37" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Delta ^r = \{(\alpha _1, \ldots , \alpha _r) \in ({\mathbb {C}}^*)^r \,|\, \alpha _1\cdots \alpha _r = 1\}. \end{aligned}$$</span></div></div><p>We can see <span class="mathjax-tex">$\Delta ^r$</span> as a (coarse) configuration space, which is naturally stratified by equalities <span class="mathjax-tex">$\alpha _i=\alpha _j$</span>. Here, we only need to consider a simple case of the Fulton–MacPherson stratification. Given an equivalence relation <span class="mathjax-tex">$\sigma $</span> on <span class="mathjax-tex">$\{1, \ldots , r\}$</span> (equivalently, a partition of the set <span class="mathjax-tex">$\{1, \ldots , r\}$</span>), let us denote by <span class="mathjax-tex">$\Delta ^r_\sigma \subset \Delta ^r$</span> the collection of <span class="mathjax-tex">$(\alpha _1, \ldots , \alpha _r)$</span> such that <span class="mathjax-tex">$\alpha _i = \alpha _j$</span> if and only if <span class="mathjax-tex">$i \sim _\sigma j$</span>. Observe that if <span class="mathjax-tex">$\sigma = \{\upsilon _1, \ldots , \upsilon _s\}$</span>, then there is a natural action of the group <span class="mathjax-tex">$S_\sigma := S_{t_1} \times \ldots \times S_{t_r}$</span> on <span class="mathjax-tex">$\Delta _\sigma ^r$</span> by permutation of blocks, where <span class="mathjax-tex">$t_i$</span> is the number of subsets <span class="mathjax-tex">$\upsilon _j$</span> of size <span class="mathjax-tex">$|\upsilon _j| = i$</span>.</p><p>Two partitions <span class="mathjax-tex">$\sigma = \{\upsilon _1, \ldots , \upsilon _s\}$</span> and <span class="mathjax-tex">$\sigma ' = \{\upsilon _1', \ldots , \upsilon _s'\}$</span> (with the same number of equivalence classes) are said to be equivalent if there exists a permutation of <span class="mathjax-tex">$\{1, \ldots , r\}$</span> sending <span class="mathjax-tex">$\upsilon _i$</span> to <span class="mathjax-tex">$\upsilon _i'$</span> for all <i>i</i>. In this manner, if we relabel the indices in such a way that <span class="mathjax-tex">$\upsilon _1 = \{1, \ldots , r_1\}, \upsilon _2 = \{r_1+1, \dots , r_1 + r_2\}$</span> and so on, we have a simple description</p><div id="Equ38" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Delta _\sigma ^r = \left\{ (\lambda _1, \lambda _2, \ldots , \lambda _s) \in ({\mathbb {C}}^*)^s \,|\, \lambda _1^{r_1} \lambda _2^{r_2} \ldots \lambda _s^{r_s} = 1, \lambda _i \ne \lambda _j \text { for } i \ne j\right\} . \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar9">Example 4.1</h3> <p>If <span class="mathjax-tex">$\sigma = \{\{1,2,3\}, \{4\}, \{5,6,7\}\}$</span>, then</p><div id="Equ39" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Delta ^5_\sigma = \{(\lambda _1, \lambda _2, \lambda _3) \,|\, \lambda _1^3\lambda _2\lambda _3^3 = 1, \lambda _1 \ne \lambda _2, \lambda _1 \ne \lambda _3, \lambda _2 \ne \lambda _3\}, \end{aligned}$$</span></div></div><p>which is equipped with the action of <span class="mathjax-tex">$S_\sigma = S_2$</span> given by the map <span class="mathjax-tex">$(\lambda _1, \lambda _2, \lambda _3) \mapsto (\lambda _3, \lambda _2, \lambda _1)$</span>. The partition <span class="mathjax-tex">$\sigma $</span> is equivalent, for instance, to the partition <span class="mathjax-tex">$\sigma ' = \{\{1,2,3\}, \{4,5,6\}, \{7\}\}$</span>.</p> <p>For <span class="mathjax-tex">$n\ge 1$</span>, there is a natural map <span class="mathjax-tex">$p_n: \Delta ^r \rightarrow \Delta ^r$</span> given by <span class="mathjax-tex">$p_n(\alpha _1, \ldots , \alpha _r) = (\alpha _1^n, \ldots , \alpha _r^n)$</span>. We say that <span class="mathjax-tex">$\sigma '$</span> <i>refines</i> <span class="mathjax-tex">$\sigma $</span> if the partition of <span class="mathjax-tex">$\sigma '$</span> is obtained from that of <span class="mathjax-tex">$\sigma $</span> by extra subdivisions. We indicate this as <span class="mathjax-tex">$\sigma ' \rightarrow \sigma $</span>. If <span class="mathjax-tex">$\sigma '$</span> is a refinement of <span class="mathjax-tex">$\sigma $</span>, let us denote</p><div id="Equ40" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Delta ^r_{\sigma ' \rightarrow \sigma } = \Delta ^r_{\sigma '} \cap p_n^{-1}(\Delta ^r_{\sigma }). \end{aligned}$$</span></div></div><p>A key observation is that we can compute <span class="mathjax-tex">$e(\Delta ^r_{\sigma ' \rightarrow \sigma })$</span> recursively. For instance, for notational simplicity, let us suppose that <span class="mathjax-tex">$\sigma = \{\upsilon _1, \ldots , \upsilon _s\}$</span>, <span class="mathjax-tex">$\sigma ' = \{\upsilon _1', \ldots , \upsilon _{s+t}'\}$</span>, with <span class="mathjax-tex">$\upsilon _{s} = \upsilon _{s}' \cup \upsilon _{s+1}' \cup \ldots \cup \upsilon _{s+t}'$</span> and <span class="mathjax-tex">$\upsilon _i = \upsilon _i'$</span> for <span class="mathjax-tex">$i < s$</span>. In that case, we get that if <span class="mathjax-tex">$(\lambda _1, \ldots , \lambda _{s+t}) \in \Delta ^r_{\sigma ' \rightarrow \sigma }$</span>, since <span class="mathjax-tex">$\lambda _{s}^n = \lambda _{s+1}^n = \ldots = \lambda _{s+t}^n$</span>, we must have <span class="mathjax-tex">$\lambda _{s+k} = \lambda _s\varepsilon _k$</span> for all <span class="mathjax-tex">$k = 1, \ldots , t$</span> and some pairwise different roots of unit <span class="mathjax-tex">$\varepsilon _k \in \mu _{n}^*$</span>. Here, we denote <span class="mathjax-tex">$\mu _n=\{e^{2\pi i k /n} |\, k=0,1,\ldots , n-1\}$</span> and <span class="mathjax-tex">$\mu _n^* = \mu _n - \{1\}$</span>. Hence,</p><div id="Equ41" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Delta ^r_{\sigma ' \rightarrow \sigma } = \left\{ (\lambda _1, \ldots , \lambda _s, \varepsilon _1, \ldots , \varepsilon _t) \in ({\mathbb {C}}^*)^s \times (\mu _n^*)^t \,\left| \, \begin{matrix}\lambda _1^{r_1} \cdots \lambda _s^{r_s} \varepsilon _1^{r'_{s+1}} \ldots \varepsilon _t^{r'_{s+t}} = 1, \\ \lambda _i \ne \lambda _j \varepsilon \text { for } i \ne j, \varepsilon \in \mu _n , \\ \varepsilon _{k} \ne \varepsilon _{l} \text { for } k \ne l \end{matrix}\right. \right\} . \end{aligned}$$</span></div></div><p>Now, observe that if we remove any of the two later inequality conditions, we get a space of the form <span class="mathjax-tex">$\Delta ^r_{{\tilde{\sigma }}' \rightarrow {\tilde{\sigma }}}$</span> for some coarser <span class="mathjax-tex">${\tilde{\sigma }}'$</span> and <span class="mathjax-tex">${\tilde{\sigma }}$</span>. Therefore, we can simply compute the <i>E</i>-polynomial of the space</p><div id="Equ42" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left\{ (\lambda _1, \ldots , \lambda _s, \varepsilon _1, \ldots , \varepsilon _t) \in ({\mathbb {C}}^*)^s \times (\mu _n^*)^t \,\left| \, \lambda _1^{r_1} \cdots \lambda _s^{r_s} \varepsilon _1^{r'_{s+1}} \ldots \varepsilon _t^{r'_{s+t}} = 1\right. \right\} , \end{aligned}$$</span></div></div><p>and then remove the strata corresponding to the different equalities using the inclusion–exclusion principle.</p><p>Finally, observe that the action of <span class="mathjax-tex">$S_{\sigma '}$</span> does not restrict to an action on <span class="mathjax-tex">$\Delta ^r_{\sigma ' \rightarrow \sigma }$</span>. Instead, we find that there is a maximal subgroup <span class="mathjax-tex">$S_{\sigma ' \rightarrow \sigma } < S_{\sigma '}$</span> acting on <span class="mathjax-tex">$\Delta ^r_{\sigma ' \rightarrow \sigma }$</span> namely, those permutations that preserve the refinement.</p><h3 class="c-article__sub-heading" id="Sec8"><span class="c-article-section__title-number">4.2 </span>The Geometric Setting</h3><p>In this section, we show how the combinatorial set-up previously developed can be used to control important strata that appear in the representation varieties of Hopf links. To control the Jordan structure of the matrices, we need the following definition.</p> <h3 class="c-article__sub-heading" id="FPar10">Definition 4.2</h3> <p>A <i>Jordan type</i> of rank <span class="mathjax-tex">$r \ge 1$</span> is a tuple <span class="mathjax-tex">$\xi = (\sigma , \kappa )$</span> where <span class="mathjax-tex">$\sigma = \{\upsilon _1, \ldots , \upsilon _s\}$</span> is a partition of <span class="mathjax-tex">$\{1, \ldots , r\}$</span> and <span class="mathjax-tex">$\kappa = \{\tau _1, \ldots , \tau _s\}$</span> is a collection where <span class="mathjax-tex">$\tau _i$</span> is a partition of <span class="mathjax-tex">$\upsilon _i$</span> into <i>linearly ordered</i> sets.</p> <p>A type <span class="mathjax-tex">$\xi ' = (\sigma ', \kappa ')$</span> is said to <i>refine</i> <span class="mathjax-tex">$\xi = (\sigma , \kappa )$</span>, and we shall denote it by <span class="mathjax-tex">$\xi ' \rightarrow \xi $</span>, if <span class="mathjax-tex">$\sigma '$</span> is a refinement of <span class="mathjax-tex">$\sigma $</span> and for any <span class="mathjax-tex">$\upsilon _i \in \sigma $</span> that decomposes as <span class="mathjax-tex">$\upsilon _i = \upsilon _{i_1}' \cup \ldots \cup \upsilon _{i_t}'$</span> in <span class="mathjax-tex">$\sigma '$</span> we have that <span class="mathjax-tex">$\tau _i = \tau '_{i_1} \cup \ldots \cup \tau '_{i_t}$</span>.</p> <p>The rationale behind a Jordan type is that it codifies the block structure of a Jordan matrix. On the one hand, the partition <span class="mathjax-tex">$\sigma $</span> identifies the multiplicities of the eigenvalues: each of the sets <span class="mathjax-tex">$\upsilon _i$</span> of the partition corresponds to a collection of equal eigenvalues (each number identifies the column of the eigenvalue). On the other hand, <span class="mathjax-tex">$\kappa $</span> determines the inner structure of the Jordan blocks for each eigenvalue: each set of the partition <span class="mathjax-tex">$\tau _i$</span> corresponds to a block of the Jordan matrix associated to an eigenvalue. The total order within this set is needed to identify the eigenvector column (the last element) and the off-diagonal elements of the Jordan form: if <i>a</i> is a successor of <i>b</i>, then there is a 1 at the (<i>a</i>, <i>b</i>)-entry of the matrix.</p> <h3 class="c-article__sub-heading" id="FPar11">Example 4.3</h3> <p>To clarify this association, let us provide several examples.</p><ul class="u-list-style-bullet"> <li> <p>The type <span class="mathjax-tex">$\xi _1=\big (\sigma _1=\{ \{1,2\}, \{3,4\}, \{5,6\} \} ,\tau _1=\{ \{(1,2)\}$</span>, <span class="mathjax-tex">$\{(3), (4) \}, \{(5,6)\} \}\big )$</span> corresponds to Jordan matrices of the form </p><div id="Equ114" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/lw430/springer-static/image/art%3A10.1007%2Fs00009-023-02300-w/MediaObjects/9_2023_2300_Equ114_HTML.png" class="u-display-block" alt=""></div></div> </li> <li> <p>The type <span class="mathjax-tex">$\xi _2=\big (\sigma _2=\{ \{1,2, 3,4\}, \{5,6\} \} ,\tau _2=\{ \{(1,2), (3), (4) \}, \{(5,6)\} \}\big )$</span> corresponds to Jordan matrices of the form </p><div id="Equ115" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/lw330/springer-static/image/art%3A10.1007%2Fs00009-023-02300-w/MediaObjects/9_2023_2300_Equ115_HTML.png" class="u-display-block" alt=""></div></div> </li> <li> <p>The type <span class="mathjax-tex">$\xi _3=\big (\sigma _3=\{ \{1,5\}, \{3,4\}, \{2,6\} \} ,\tau _3=\{ \{(1,5)\}$</span>, <span class="mathjax-tex">$\{(3), (4) \}, \{(2,6)\} \}\big )$</span> corresponds to Jordan matrices of the form </p><div id="Equ116" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/lw430/springer-static/image/art%3A10.1007%2Fs00009-023-02300-w/MediaObjects/9_2023_2300_Equ116_HTML.png" class="u-display-block" alt=""></div></div> </li> </ul> <p>There is a natural action of the symmetric group <span class="mathjax-tex">$S_r$</span> on the set of types by ‘permutation of columns’: relabel each element of <span class="mathjax-tex">$\{1, \ldots , r\}$</span> according to the permutation both in <span class="mathjax-tex">$\sigma $</span> and <span class="mathjax-tex">$\tau _i$</span>. We will say that two types are <i>equivalent</i> if they lie in the same <span class="mathjax-tex">$S_r$</span>-orbit. Notice that, by definition, in principle, for a type <span class="mathjax-tex">$\xi = (\sigma , \kappa )$</span>, the partition <span class="mathjax-tex">$\sigma $</span> of <span class="mathjax-tex">$\{1,\ldots , r\}$</span> may not be into segments (i.e. equal eigenvalues may be sparse in the matrix). However, by ‘putting together’ the Jordan blocks, there always exists an equivalent type for which <span class="mathjax-tex">$\sigma $</span> and each <span class="mathjax-tex">$\tau _i$</span> are made of segments, and the total order in <span class="mathjax-tex">$\tau _i$</span> agrees with the natural order in <span class="mathjax-tex">$\{1, \ldots , r\}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar12">Example 4.4</h3> <p>The types <span class="mathjax-tex">$\xi _1$</span> and <span class="mathjax-tex">$\xi _3$</span> of Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar11">4.3</a> are equivalent under the permutation <span class="mathjax-tex">$\varphi = (2\,5)$</span>. However, <span class="mathjax-tex">$\xi _1$</span> and <span class="mathjax-tex">$\xi _2$</span> are not equivalent.</p> <p>Given a type <span class="mathjax-tex">$\xi $</span> of rank <i>r</i>, let us denote by <span class="mathjax-tex">${\mathcal {A}}_\xi $</span> the collection of Jordan matrices (with 1’s as off-diagonal elements according to the total orders given by <span class="mathjax-tex">$\kappa $</span>) of <span class="mathjax-tex">${{\,\textrm{SL}\,}}_{r}({\mathbb {C}})$</span> whose block structure is <span class="mathjax-tex">$\xi $</span>. If we allow any non-zero off-diagonal element in the entries determined by <span class="mathjax-tex">$\kappa $</span>, we will refer to these matrices as generalized Jordan matrices. The space of generalized Jordan matrices will be denoted <span class="mathjax-tex">${\mathcal {A}}^g_\xi $</span>. We also consider <span class="mathjax-tex">${\tilde{{\mathcal {A}}}}_{\xi } = {{\,\textrm{SL}\,}}_{r}({\mathbb {C}}) \cdot {\mathcal {A}}_\xi $</span>, where the action is by conjugation. If we pick a collection <span class="mathjax-tex">$\xi _1, \ldots , \xi _N$</span> of non-equivalent representatives of all the types of rank <i>r</i>, we get a decomposition</p><div id="Equ43" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\,\textrm{SL}\,}}_{r}({\mathbb {C}}) = {\tilde{{\mathcal {A}}}}_{\xi _1} \sqcup \ldots \sqcup {\tilde{{\mathcal {A}}}}_{\xi _N}. \end{aligned}$$</span></div></div><p>Given a type <span class="mathjax-tex">$\xi =(\sigma ,\kappa )$</span>, we have a group <span class="mathjax-tex">$S_\xi < S_\sigma $</span>. This is the group of permutations of <span class="mathjax-tex">$\sigma =(\upsilon _1,\ldots , \upsilon _s)$</span> that permute blocks <span class="mathjax-tex">$\upsilon _i$</span> whose decompositions <span class="mathjax-tex">$\tau _i$</span> are equivalent (of the same number of sets and of the same sizes). The group <span class="mathjax-tex">$S_\xi $</span> acts on <span class="mathjax-tex">${\mathcal {A}}_\xi $</span> as follows: let <span class="mathjax-tex">$\varphi \in S_\xi $</span> and <span class="mathjax-tex">$A\in {\mathcal {A}}_\xi $</span>. So <span class="mathjax-tex">$\varphi $</span> is a permutation of the blocks <span class="mathjax-tex">$\upsilon _i$</span> of <span class="mathjax-tex">$\sigma $</span>, that is, of the eigenvalues <span class="mathjax-tex">$\lambda _i$</span>. For such eigenvalues, the Jordan forms match exactly since the corresponding decompositions <span class="mathjax-tex">$\tau _i$</span> are equivalent; therefore, the matrix <span class="mathjax-tex">$A'$</span> with the eigenvalues <span class="mathjax-tex">$\lambda _{\varphi (i)}$</span> lies in <span class="mathjax-tex">${\mathcal {A}}_\xi $</span> as well. Moreover, as this is given by a change of the basis by permutation of the vectors, there is a matrix <span class="mathjax-tex">$P_\varphi $</span> such that <span class="mathjax-tex">$A'=P_\varphi ^{-1}AP_\varphi $</span>. This <span class="mathjax-tex">$P_\varphi $</span> is well defined in <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_r({\mathbb {C}})/{{\,\textrm{Stab}\,}}(A)$</span> (quotient by action on the left). The action on <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_r({\mathbb {C}})/{{\,\textrm{Stab}\,}}(A)$</span> is by product of <span class="mathjax-tex">$P_\varphi $</span> on the right.</p><p>Now, we consider the map</p><div id="Equ44" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} p_n: {{\,\textrm{SL}\,}}_{r}({\mathbb {C}}) \rightarrow {{\,\textrm{SL}\,}}_{r}({\mathbb {C}}) , \quad p_n(A) = A^n, \end{aligned}$$</span></div></div><p>and we set</p><div id="Equ45" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned}&{{\mathcal {A}}}_{\xi ' \rightarrow \xi } = {\mathcal {A}}^g_{\xi '} \cap p_n^{-1}({\mathcal {A}}_{\xi }) , \\&{\tilde{{\mathcal {A}}}}_{\xi ' \rightarrow \xi } = {{\,\textrm{SL}\,}}_{r}({\mathbb {C}}) \cdot {{\mathcal {A}}}_{\xi ' \rightarrow \xi }. \end{aligned} \end{aligned}$$</span></div></div><p>Moreover, if <span class="mathjax-tex">$\xi '$</span> refines <span class="mathjax-tex">$\xi $</span>, then there is a group <span class="mathjax-tex">$S_{\xi ' \rightarrow \xi }$</span> of permutations of <span class="mathjax-tex">$\xi '$</span> (that is, permutations <span class="mathjax-tex">$\varphi $</span> of <span class="mathjax-tex">$\sigma '$</span> that respect the blocks of <span class="mathjax-tex">$\xi '$</span>) that induce permutations of <span class="mathjax-tex">$\xi $</span> (it is enough that they induce a permutation on <span class="mathjax-tex">$\sigma $</span> under the refinement <span class="mathjax-tex">$\sigma '\rightarrow \sigma $</span>).</p> <h3 class="c-article__sub-heading" id="FPar13">Example 4.5</h3> <p>Suppose we have types <span class="mathjax-tex">$\xi '=\big (\sigma '=\{ \{1,2\}, \{3,4\}, \{(5),(6)\} \} ,\tau '=\{ \{(1,2)\}$</span>, <span class="mathjax-tex">$\{3,4 \}, \{(5,6)\} \}\big )$</span>, <span class="mathjax-tex">$\xi =\big (\sigma =\{ \{1,2, 3,4\}, \{5,6\} \} ,\tau =\{ \{(1,2), (3), (4) \}, \{(5,6)\} \}\big )$</span> (cf. Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar11">4.3</a>). So <span class="mathjax-tex">$\xi '\rightarrow \xi $</span>. This corresponds to matrices</p><div id="Equ117" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/lw499/springer-static/image/art%3A10.1007%2Fs00009-023-02300-w/MediaObjects/9_2023_2300_Equ117_HTML.png" class="u-display-block" alt=""></div></div><p>The group <span class="mathjax-tex">$S_{\xi '} =S_2$</span> permuting <span class="mathjax-tex">$\{1,2\}$</span> and <span class="mathjax-tex">$\{5,6\}$</span>, whereas <span class="mathjax-tex">$S_{\xi '\rightarrow \xi }=\{1\}$</span>. Note also that <span class="mathjax-tex">$S_{\sigma '}=S_3$</span>.</p> <h3 class="c-article__sub-heading" id="FPar14">Lemma 4.6</h3> <p>For any types <span class="mathjax-tex">$\xi = (\sigma , \kappa )$</span> and <span class="mathjax-tex">$\xi ' = (\sigma ', \kappa ')$</span> we have that <span class="mathjax-tex">${\mathcal {A}}_{\xi ' \rightarrow \xi } = \emptyset $</span> if <span class="mathjax-tex">$\xi '$</span> does not refine <span class="mathjax-tex">$\xi $</span>, and <span class="mathjax-tex">${\mathcal {A}}_{\xi ' \rightarrow \xi } \cong \Delta ^r_{\sigma ' \rightarrow \sigma }$</span> if it does.</p> <h3 class="c-article__sub-heading" id="FPar15">Proof</h3> <p>A direct computation shows that, since taking powers preserves the Jordan block structure, if <span class="mathjax-tex">$A \in {\mathcal {A}}^g_{\xi '}$</span>, then <span class="mathjax-tex">$A^n$</span> can only lie in strata of the form <span class="mathjax-tex">${\mathcal {A}}^g_{\xi }$</span> when <span class="mathjax-tex">$\xi '$</span> refines <span class="mathjax-tex">$\xi $</span>. Hence, <span class="mathjax-tex">$p_n^{-1}({\mathcal {A}}_{\xi }) \cap {\mathcal {A}}^g_{\xi '} = \emptyset $</span> if <span class="mathjax-tex">$\xi '$</span> does not refine <span class="mathjax-tex">$\xi $</span>.</p> <p>On the other hand, given <span class="mathjax-tex">$A \in {\mathcal {A}}_{\xi ' \rightarrow \xi }$</span> with <span class="mathjax-tex">$\xi '$</span> refining <span class="mathjax-tex">$\xi $</span>, we observe that the off-diagonal elements of <i>A</i> are fixed to coincide with the Jordan structure. In this manner, the only freedom we have is to choose the eigenvalues of <i>A</i>, and this is precisely <span class="mathjax-tex">$ \Delta ^r_{\sigma ' \rightarrow \sigma }$</span>. <span class="mathjax-tex">$\square $</span></p> <p>Let us denote by</p><div id="Equ46" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R(H_n, {{\,\textrm{SL}\,}}_{r}({\mathbb {C}}))_{\xi \rightarrow \xi '} := \left\{ (A, B) \in {\tilde{{\mathcal {A}}}}_{\xi ' \rightarrow \xi } \times {{\,\textrm{SL}\,}}_{r}({\mathbb {C}}) \,|\, [A^n,B] = \textrm{Id}\right\} . \end{aligned}$$</span></div></div><p>Observe that we have a natural stratification</p><div id="Equ47" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R(H_n, {{\,\textrm{SL}\,}}_{r}({\mathbb {C}})) = \bigsqcup _{\xi ', \xi } R(H_n,{{\,\textrm{SL}\,}}_{r}({\mathbb {C}}))_{\xi ' \rightarrow \xi } , \end{aligned}$$</span></div></div><p>where each of the indices <span class="mathjax-tex">$\xi '$</span> and <span class="mathjax-tex">$\xi $</span> runs over a collection of non-equivalent representatives of all the types of rank <i>r</i>. We denote by <span class="mathjax-tex">${{\,\textrm{Stab}\,}}(\xi ')$</span> the stabilizer in <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_{r}({\mathbb {C}})$</span> of any matrix of <span class="mathjax-tex">${\mathcal {A}}_{\xi '}$</span>, and <span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi )$</span> the stabilizer in <span class="mathjax-tex">${{\,\textrm{SL}\,}}_{r}({\mathbb {C}})$</span> of any matrix of <span class="mathjax-tex">${\mathcal {A}}_{\xi }$</span>. Note that the action of an element <span class="mathjax-tex">$\varphi \in S_{\xi '\rightarrow \xi }$</span> is by permutation of columns in either <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_r({\mathbb {C}})$</span>, <span class="mathjax-tex">${{\,\textrm{SL}\,}}_r({\mathbb {C}})$</span>, <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_{r}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi ')$</span> and <span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi )$</span>.</p> <h3 class="c-article__sub-heading" id="FPar16">Proposition 4.7</h3> <p>For any types <span class="mathjax-tex">$\xi = (\sigma , \kappa )$</span> and <span class="mathjax-tex">$\xi ' = (\sigma ', \kappa ')$</span>, we have that</p><div id="Equ48" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R(H_n, {{\,\textrm{SL}\,}}_{r}({\mathbb {C}}))_{\xi ' \rightarrow \xi } \cong \left( {\mathcal {A}}_{\xi ' \rightarrow \xi } \times \big ({{\,\textrm{PGL}\,}}_{r}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi ')\big ) \times \widetilde{{{\,\textrm{Stab}\,}}}(\xi ) \right) / S_{\xi ' \rightarrow \xi } . \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar17">Proof</h3> <p>Take <span class="mathjax-tex">$(A, B) \in {\tilde{{\mathcal {A}}}}_{\xi ' \rightarrow \xi } \times {{\,\textrm{SL}\,}}_{r}({\mathbb {C}})$</span> such that <span class="mathjax-tex">$[A^n,B] = \textrm{Id}$</span>. Then <span class="mathjax-tex">$A \in {{\,\textrm{SL}\,}}_r({\mathbb {C}}) \cdot {\mathcal {A}}_{\xi ' \rightarrow \xi }$</span>, that is <span class="mathjax-tex">$A=P^{-1}A_0P$</span>, with <span class="mathjax-tex">$A_0\in {\mathcal {A}}_{\xi ' \rightarrow \xi }={\mathcal {A}}^g_{\xi '} \cap p_n^{-1}({\mathcal {A}}_{\xi })$</span>, i.e. <span class="mathjax-tex">$A_0\in {\mathcal {A}}^g_{\xi '}$</span> and <span class="mathjax-tex">$A_0^n\in {\mathcal {A}}_{\xi }$</span>. The matrix <i>P</i> is determined up to <span class="mathjax-tex">${{\,\textrm{Stab}\,}}(A_0)$</span>. On the other hand, <i>B</i> commutes with <span class="mathjax-tex">$A^n=P^{-1}A_0^n P$</span>, hence <span class="mathjax-tex">$B_0=P B P^{-1}$</span> lies in <span class="mathjax-tex">${{\,\textrm{Stab}\,}}(A_0^n)={\widetilde{{{\,\textrm{Stab}\,}}}}(\xi )$</span>.</p> <p>The pair (<i>A</i>, <i>B</i>) is determined by <span class="mathjax-tex">$(A_0,P,B_0)$</span>. This is not unique, the matrix <span class="mathjax-tex">$A_0$</span> can be changed by an equivalent matrix <span class="mathjax-tex">$A_0'$</span> via an element <span class="mathjax-tex">$\varphi \in S_{\xi '}$</span>. The associated triple is <span class="mathjax-tex">$(A_0'=P_\varphi ^{-1}A_0P_\varphi , P'=P_\varphi ^{-1}P, B_0'=P_\varphi ^{-1}B_0 P_\varphi )$</span>. In order for <span class="mathjax-tex">$B_0'$</span> to lie in the stabilizer of a matrix of <span class="mathjax-tex">${\mathcal {A}}_\xi $</span>, it is needed that whenever two eigenvalues <span class="mathjax-tex">$\lambda _i,\lambda _j$</span> satisfy <span class="mathjax-tex">$\lambda _i^n=\lambda _j^n$</span>, the permutation <span class="mathjax-tex">$\varphi $</span> moves then to eigenvalues <span class="mathjax-tex">$\lambda _i',\lambda _j'$</span> such that <span class="mathjax-tex">$(\lambda _i')^n=(\lambda _j')^n$</span>. This means that <span class="mathjax-tex">$\varphi $</span> respects the partition <span class="mathjax-tex">$\sigma $</span>, that is, it lies in <span class="mathjax-tex">$S_{\xi '\rightarrow \xi }$</span>. <span class="mathjax-tex">$\square $</span></p> </div></div></section><section data-title="Representation Variety of the Twisted Hopf Link of Rank 2"><div class="c-article-section" id="Sec9-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec9"><span class="c-article-section__title-number">5 </span>Representation Variety of the Twisted Hopf Link of Rank 2</h2><div class="c-article-section__content" id="Sec9-content"><p>In this section, we shall compute the <i>E</i>-polynomial of the <span class="mathjax-tex">${{\,\textrm{SL}\,}}_{2}({\mathbb {C}})$</span>-representation variety of the twisted Hopf link <span class="mathjax-tex">$H_n$</span>, that is, the space <span class="mathjax-tex">$R(H_n,{{\,\textrm{SL}\,}}_2({\mathbb {C}}))$</span>. For that purpose, we analyze the combinatorial and geometric settings as outlined in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec6">4</a>.</p><h3 class="c-article__sub-heading" id="Sec10"><span class="c-article-section__title-number">5.1 </span>Combinatorial Setting</h3><p>In rank 2 we have only 2 possible partitions, up to equivalence. These are</p><div id="Equ49" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sigma _1 = \{\{1,2\}\}, \qquad \sigma _{2} = \{\{1\},\{2\}\}. \end{aligned}$$</span></div></div><p>The first one corresponds to <span class="mathjax-tex">${{\,\textrm{SL}\,}}_2({\mathbb {C}})$</span> matrices with equal eigenvalues, and the second one with distinct eigenvalues. Hence, we have</p><div id="Equ50" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(\Delta ^2_{\sigma _1})&= e(\{\lambda \in {\mathbb {C}}^*|\, \lambda ^2 = 1\}) = e(\mu _2)=2, \\ e(\Delta ^2_{\sigma _2})&= e(\{(\lambda _1, \lambda _2) \in ({\mathbb {C}}^*)^2\,|\, \lambda _1\lambda _2 = 1, \lambda _1 \ne \lambda _2\}) = e({\mathbb {C}}^* - \Delta ^2_{\sigma _1})\\&= e({\mathbb {C}}^* - \mu _2)=q-3. \end{aligned}$$</span></div></div><p>Only the stratum <span class="mathjax-tex">$\Delta ^2_{\sigma _2}$</span> has a non-trivial action of <span class="mathjax-tex">$S_2$</span>, given by <span class="mathjax-tex">$(\lambda _1, \lambda _2) \mapsto (\lambda _2, \lambda _1)$</span>.</p><p>For the degenerations, we have <span class="mathjax-tex">$e(\Delta ^2_{\sigma _1 \rightarrow \sigma _1}) = e(\Delta ^2_{\sigma _1}) = e(\mu _2) = 2$</span>. In addition,</p><div id="Equ51" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( \Delta ^2_{\sigma _2 \rightarrow \sigma _1}\right)&= e\left( \{(\lambda _1, \lambda _2) \in ({\mathbb {C}}^*)^2\,| \, \lambda _1\lambda _2 = 1, \lambda _1 \ne \lambda _2, \lambda _1^n = \lambda _2^n\}\right) \\&= e\left( \{(\lambda _1, \varepsilon ) \in {\mathbb {C}}^* \times \mu _n\,|\, \lambda _1^2\varepsilon = 1, \varepsilon \ne 1\}\right) \\&= e\left( \{(\lambda _1, \varepsilon ) \in {\mathbb {C}}^* \times \mu _n\,|\, \lambda _1^2\varepsilon = 1\}\right) - e\left( \{\lambda _1 \in {\mathbb {C}}^* \,|\, \lambda _1^2 = 1\}\right) \\&= e\left( \mu _{2n}\right) - e\left( \mu _2\right) = 2n-2. \end{aligned}$$</span></div></div><p>And</p><div id="Equ52" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( \Delta ^2_{\sigma _2 \rightarrow \sigma _2}\right)&= e\left( \{(\lambda _1, \lambda _2) \in ({\mathbb {C}}^*)^2\,| \, \lambda _1\lambda _2 = 1, \lambda _1 \ne \lambda _2, \lambda _1^n \ne \lambda _2^n\}\right) \\&= e\left( \Delta ^2_{\sigma _2}\right) - e\left( \{(\lambda _1, \lambda _2) \in ({\mathbb {C}}^*)^2\,| \, \lambda _1\lambda _2 = 1, \lambda _1 \ne \lambda _2, \lambda _1^n = \lambda _2^n\}\right) \\&= q-3 - e\left( \Delta ^2_{\sigma _2 \rightarrow \sigma _1}\right) = q-3 - (2n-2) = q-2n-1. \end{aligned}$$</span></div></div><p>Now, let us analyze the quotients by <span class="mathjax-tex">$S_2$</span>. Recall that the action of <span class="mathjax-tex">$S_2$</span> on <span class="mathjax-tex">${\mathbb {C}}^*$</span> given by <span class="mathjax-tex">$\lambda \mapsto \lambda ^{-1}$</span> has quotient <span class="mathjax-tex">${\mathbb {C}}^*/S_2 ={\mathbb {C}}$</span>, given by the invariant function <span class="mathjax-tex">$s=\lambda +\lambda ^{-1}$</span>. We have that</p><div id="Equ53" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(\Delta ^2_{\sigma _2}/S_2)&= e(({\mathbb {C}}^* - \mu _2)/S_2) = e({\mathbb {C}}-\{\pm 2\})=q-2. \\ e\left( \Delta ^2_{\sigma _2 \rightarrow \sigma _1}/S_2\right)&= e\left( \mu _{2n}/S_2\right) - e\left( \Delta ^2_{\sigma _1}/S_2\right) = n+1-2=n-1.\\ e\left( \Delta ^2_{\sigma _2 \rightarrow \sigma _2}/S_2\right)&= e\left( \Delta ^2_{\sigma _2}/S_2\right) - e\left( \Delta ^2_{\sigma _2 \rightarrow \sigma _1}/S_2\right) = (q-2) - (n-1) = q-n-1. \end{aligned}$$</span></div></div><p>Therefore, using (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ3">3</a>) we get the equivariant <i>E</i>-polynomials:</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} e_{S_2}(\Delta ^2_{\sigma _2})&= (q-2)T - N. \\ e_{S_2}\left( \Delta ^2_{\sigma _2 \rightarrow \sigma _1}\right)&= (n-1)T + (n-1)N.\\ e_{S_2}\left( \Delta ^2_{\sigma _2 \rightarrow \sigma _2}\right)&= (q-n-1) T -n N. \end{aligned} \end{aligned}$$</span></div><div class="c-article-equation__number"> (6) </div></div><h3 class="c-article__sub-heading" id="Sec11"><span class="c-article-section__title-number">5.2 </span>Geometric Setting</h3><p>From the two partitions <span class="mathjax-tex">$\sigma _1$</span> and <span class="mathjax-tex">$\sigma _2$</span>, we can create three types (up to equivalence), namely</p><div id="Equ54" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \xi _1 = (\sigma _1, \{1,2\}), \qquad \xi _2 = (\sigma _1, \{(1,2)\}), \qquad \xi _3 = (\sigma _2, \{\{1\}, \{2\}\}). \end{aligned}$$</span></div></div><p>Recall that <span class="mathjax-tex">$\xi _1$</span> is the type of the diagonalizable matrices with equal eigenvalues (namely, <span class="mathjax-tex">$\pm \textrm{Id}$</span>), <span class="mathjax-tex">$\xi _2$</span> is the type of the two Jordan type matrices <span class="mathjax-tex">$J_{\pm }={\tiny \left( \begin{array}{cc} \pm 1 &{} 0 \\ 1 &{}\pm 1 \end{array}\right) }$</span>, and <span class="mathjax-tex">$\xi _3$</span> is the type of diagonalizable matrices with different eigenvalues.</p><p>Denote <span class="mathjax-tex">$R_{\xi _i \rightarrow \xi _j}= R(H_n,{{\,\textrm{SL}\,}}_2({\mathbb {C}}))_{\xi _i \rightarrow \xi _j}$</span>. Taking into account that the only refinement relations are <span class="mathjax-tex">$\xi _i \rightarrow \xi _i$</span>, <span class="mathjax-tex">$i=1,2,3$</span>, and <span class="mathjax-tex">$\xi _3 \rightarrow \xi _1$</span>, we have</p><div id="Equ55" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R(H_n,{{\,\textrm{SL}\,}}_2({\mathbb {C}})) = R_{\xi _1 \rightarrow \xi _1} \sqcup R_{\xi _2 \rightarrow \xi _2} \sqcup R_{\xi _3 \rightarrow \xi _3} \sqcup R_{\xi _3 \rightarrow \xi _1}. \end{aligned}$$</span></div></div><p>Counting for each stratum we have the following:</p><ul class="u-list-style-bullet"> <li> <p>By Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar14">4.6</a> and Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar16">4.7</a>, </p><div id="Equ56" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R_{\xi _1 \rightarrow \xi _1} = \Delta ^2_{\sigma _1 \rightarrow \sigma _1} \times \big ( {{\,\textrm{PGL}\,}}_{2}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _1) \big ) \times {\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _1). \end{aligned}$$</span></div></div><p> Now <span class="mathjax-tex">${{\,\textrm{Stab}\,}}(\xi _1) ={{\,\textrm{PGL}\,}}_2({\mathbb {C}})$</span>, <span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _1) = {{\,\textrm{SL}\,}}_{2}({\mathbb {C}})$</span> and <span class="mathjax-tex">$e(\Delta ^2_{\sigma _1 \rightarrow \sigma _1})=2$</span>, hence </p><div id="Equ57" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( R_{\xi _1 \rightarrow \xi _1}\right) = e(\Delta ^2_{\sigma _1 \rightarrow \sigma _1}) e({{\,\textrm{SL}\,}}_{2}({\mathbb {C}})) = 2(q^3-q). \end{aligned}$$</span></div></div> </li> <li> <p>We have also </p><div id="Equ58" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R_{\xi _2 \rightarrow \xi _2} = \Delta ^2_{\sigma _1 \rightarrow \sigma _1}\times \big ( {{\,\textrm{PGL}\,}}_{2}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _2)\big )\times {\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _2). \end{aligned}$$</span></div></div><p> Now <span class="mathjax-tex">${{\,\textrm{Stab}\,}}(\xi _2) ={\tiny \left\{ \left( \begin{array}{cc} 1 &{} 0 \\ a &{} 1\end{array}\right) \right\} } \cong {\mathbb {C}}$</span> and <span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _2) ={\tiny \left\{ \left( \begin{array}{cc} \pm 1 &{} 0 \\ a &{} \pm 1\end{array}\right) \right\} } \cong {\mathbb {C}}\times \mu _2$</span>. Hence, </p><div id="Equ59" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( R_{\xi _2 \rightarrow \xi _2}\right) = e(\Delta ^2_{\sigma _1 \rightarrow \sigma _1}) \frac{q^3-q}{q} \, 2q = 4(q^3-q). \end{aligned}$$</span></div></div> </li> <li> <p>We continue with <span class="mathjax-tex">$ R_{\xi _3 \rightarrow \xi _3}={\tilde{R}}_{\xi _3 \rightarrow \xi _3} /S_2$</span>, with </p><div id="Equ60" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\tilde{R}}_{\xi _3 \rightarrow \xi _3} = \Delta ^2_{\sigma _2 \rightarrow \sigma _2}\times \big ( {{\,\textrm{PGL}\,}}_{2}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _3)\big ) \times {\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _3). \end{aligned}$$</span></div></div><p> To compute this, note that <span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _3) = {\tiny \left\{ \left( \begin{array}{cc} \alpha &{} 0 \\ 0 &{} \alpha ^{-1} \end{array}\right) \right\} } \cong {\mathbb {C}}^*$</span>. The action of <span class="mathjax-tex">$S_2$</span> is given by <span class="mathjax-tex">$\alpha \mapsto \alpha ^{-1}$</span>. the quotient <span class="mathjax-tex">${\mathbb {C}}^*/S_2\cong {\mathbb {C}}$</span> via the invariant function <span class="mathjax-tex">$s=\alpha +\alpha ^{-1}$</span>. So <span class="mathjax-tex">$e({\mathbb {C}}^*/S_2)=q$</span>, and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ3">3</a>) yields the equivariant <i>E</i>-polynomial </p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_{S_2}({\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _3))= qT-N . \end{aligned}$$</span></div><div class="c-article-equation__number"> (7) </div></div><p> Also <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_2({\mathbb {C}})/{{\,\textrm{Stab}\,}}(\xi _3)={{\,\textrm{PGL}\,}}_2({\mathbb {C}})/{\mathcal {D}}$</span>, where <span class="mathjax-tex">${\mathcal {D}}$</span> are the diagonal matrices. So <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_2({\mathbb {C}})/{\mathcal {D}}\cong ({\mathbb {P}}^1)^2-\Delta $</span>, where <span class="mathjax-tex">$\Delta ={\mathbb {P}}^1$</span> is the diagonal. Hence, <span class="mathjax-tex">$e({{\,\textrm{PGL}\,}}_2({\mathbb {C}})/{\mathcal {D}})=(q+1)^2-(q+1)=q^2+q$</span>. For the <span class="mathjax-tex">$S_2$</span>-quotient, we have <span class="mathjax-tex">$({{\,\textrm{PGL}\,}}_2({\mathbb {C}})/{\mathcal {D}})/S_2 \cong \textrm{Sym}^2({\mathbb {P}}^1)-\Delta $</span>, where <span class="mathjax-tex">$\Delta ={\mathbb {P}}^1$</span>. Hence, <span class="mathjax-tex">$e(({{\,\textrm{PGL}\,}}_2({\mathbb {C}})/{\mathcal {D}})/S_2)= (q^2+q+1)-(q+1)=q^2$</span>. Using (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ3">3</a>), this produces the equivariant <i>E</i>-polynomial </p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_{S_2}({{\,\textrm{PGL}\,}}_2({\mathbb {C}})/{\mathcal {D}})= q^2T + qN . \end{aligned}$$</span></div><div class="c-article-equation__number"> (8) </div></div><p> Using the equivariant <i>E</i>-polynomial of <span class="mathjax-tex">$\Delta ^2_{\sigma _2 \rightarrow \sigma _2}$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ6">6</a>), and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ7">7</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ8">8</a>), we have </p><div id="Equ61" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_{S_2}\big ({\tilde{R}}_{\xi _3 \rightarrow \xi _3} \big )&= ((q-n-1) T -n N) \otimes (q^2 T + qN) \otimes (qT-N) \\&= \left( (q-n-1)(q^3-q)\right) T - n(q^3-q)N. \end{aligned}$$</span></div></div><p> Therefore, <span class="mathjax-tex">$e(R_{\xi _3 \rightarrow \xi _3})= (q-n-1)(q^3-q)$</span>, the coefficient of <i>T</i> in the expression above.</p> </li> <li> <p>We end up with <span class="mathjax-tex">$R_{\xi _3 \rightarrow \xi _1}={\tilde{R}}_{\xi _3 \rightarrow \xi _1}/S_2$</span>, where </p><div id="Equ62" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\tilde{R}}_{\xi _3 \rightarrow \xi _1} = \Delta ^2_{\sigma _2 \rightarrow \sigma _1} \times \big ({{\,\textrm{PGL}\,}}_{2}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _3) \big )\times {\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _1). \end{aligned}$$</span></div></div><p> By Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar4">2.2</a>, <span class="mathjax-tex">$e_{S_2}({{\,\textrm{SL}\,}}_2({\mathbb {C}}))=(q^3-q)T$</span>. Using (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ6">6</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ8">8</a>), we get </p><div id="Equ63" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_{S_2}( {\tilde{R}}_{\xi _3 \rightarrow \xi _1})&= ((n-1) T + (n-1) N) \otimes (q^2 T + qN) \otimes (q^3-q)T\\&= (n-1)(q^3-q)(q^2+q) T + (n-1)(q^3-q)(q^2+q)N, \end{aligned}$$</span></div></div><p> which produces <span class="mathjax-tex">$e( R_{\xi _3 \rightarrow \xi _1}) =(n-1)(q^3-q)(q^2+q)$</span>.</p> </li> </ul><p>Adding up all the contributions we finally get</p><div id="Equ64" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e \big (R(H_n,{{\,\textrm{SL}\,}}_2({\mathbb {C}}))\big ) = \left( (n-1)q^2+ n q -n+5)\right) (q^3-q). \end{aligned}$$</span></div></div></div></div></section><section data-title="Rank 2 Character Variety of the Twisted Hopf Link"><div class="c-article-section" id="Sec12-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec12"><span class="c-article-section__title-number">6 </span>Rank 2 Character Variety of the Twisted Hopf Link</h2><div class="c-article-section__content" id="Sec12-content"><p>In this section, we compute the <i>E</i>-polynomial <span class="mathjax-tex">$e({\mathfrak {M}}(H_n,G))$</span>, for <span class="mathjax-tex">$G={{\,\textrm{SL}\,}}_2({\mathbb {C}})$</span>. First, we deal with reducible representations (<i>A</i>, <i>B</i>). These are S-equivalent to representations of the form</p><div id="Equ65" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left( \left( \begin{array}{cc} \lambda &{} 0 \\ 0 &{} \lambda ^{-1} \end{array} \right) , \left( \begin{array}{cc} \mu &{} 0 \\ 0 &{} \mu ^{-1} \end{array} \right) \right) , \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$(\lambda ,\mu ) \in ({\mathbb {C}}^*)^2$</span>. These are defined modulo the action <span class="mathjax-tex">$(\lambda ,\mu ) \sim (\lambda ^{-1},\mu ^{-1})$</span>. The equivariant <i>E</i>-polynomial of <span class="mathjax-tex">${\mathbb {C}}^*$</span> is <span class="mathjax-tex">$e_{S_2}({\mathbb {C}}^*)=qT-N$</span>. Hence <span class="mathjax-tex">$e_{S_2}(({\mathbb {C}}^*)^2)=(qT-N)^2=(q^2+1)T -2q N$</span>, and</p><div id="Equ66" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e({\mathfrak {M}}^{\textrm{red}}(H_n,G)) = q^2+1. \end{aligned}$$</span></div></div><p>Now we move to the irreducible representations, which form a space <span class="mathjax-tex">$R^{\textrm{irr}}(H_n,G)\subset R(H_n,G)$</span>. As the action is free, <span class="mathjax-tex">${\mathfrak {M}}^{\textrm{irr}}(H_n,G)=R^{\textrm{irr}}(H_n,G)/{{\,\textrm{PGL}\,}}_2({\mathbb {C}})$</span>.</p> <h3 class="c-article__sub-heading" id="FPar18">Lemma 6.1</h3> <p>Let <span class="mathjax-tex">$r\ge 2$</span>. </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>For a type <span class="mathjax-tex">$\xi $</span> and <span class="mathjax-tex">$G={{\,\textrm{SL}\,}}_r({\mathbb {C}})$</span>, the space <span class="mathjax-tex">$R^{\textrm{irr}}(H_n,G)_{\xi \rightarrow \xi }=\emptyset $</span>.</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>For types <span class="mathjax-tex">$\xi '\rightarrow \xi $</span>, if they are not diagonalizable, then <span class="mathjax-tex">$R^{\textrm{irr}}(H_n,G)_{\xi ' \rightarrow \xi }=\emptyset $</span>.</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>For a type <span class="mathjax-tex">$\xi $</span> not corresponding to a multiple of the identity, <span class="mathjax-tex">$R^{\textrm{irr}}(H_n,G)_{\xi ' \rightarrow \xi }=\emptyset $</span>.</p> </li> </ol> <h3 class="c-article__sub-heading" id="FPar19">Proof</h3> <ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>Let <span class="mathjax-tex">$(A,B) \in R(H_n,G)_{\xi \rightarrow \xi }$</span>, then <i>A</i> and <span class="mathjax-tex">$A^n$</span> are of the same type. Therefore, the eigenvectors of <i>A</i> and <span class="mathjax-tex">$A^n$</span> are the same. If <i>A</i> and <span class="mathjax-tex">$A^n$</span> are a multiple of the identity, then any vector subspace <span class="mathjax-tex">$W\subsetneq {\mathbb {C}}^r$</span> fixed by <i>B</i> is also fixed by <i>A</i>. Otherwise, take an eigenvalue <span class="mathjax-tex">$\lambda $</span> of <i>A</i> such that the eigenspace <span class="mathjax-tex">$W=E_\lambda (A)=E_{\lambda ^n}(A^n) \subsetneq {\mathbb {C}}^r$</span> is a proper subspace. Now <span class="mathjax-tex">$[A^n,B]=\textrm{Id}$</span> implies that <span class="mathjax-tex">$B(W) =W$</span>, and hence (<i>A</i>, <i>B</i>) is a reducible representation.</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>Let <span class="mathjax-tex">$(A,B) \in R(H_n,G)_{\xi ' \rightarrow \xi }$</span>, and suppose that <span class="mathjax-tex">$\lambda $</span> is an eigenvalue of <i>A</i> such that the Jordan <span class="mathjax-tex">$\lambda $</span>-block is not diagonal. Then <span class="mathjax-tex">$E_\lambda (A)$</span> is a proper subspace of the <span class="mathjax-tex">$\lambda $</span>-block, and hence </p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} W=E_{\lambda ^n}(A^n) =\bigoplus _{\varepsilon \in \mu _n} E_{\lambda \varepsilon }(A) \subsetneq {\mathbb {C}}^r\, . \end{aligned}$$</span></div><div class="c-article-equation__number"> (9) </div></div><p> Again <span class="mathjax-tex">$[A^n,B]=\textrm{Id}$</span> implies that <span class="mathjax-tex">$B(W) =W$</span>, and hence (<i>A</i>, <i>B</i>) is a reducible representation.</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>The last item is similar, taking <span class="mathjax-tex">$\lambda $</span> an eigenvalue of <i>A</i>, then <i>W</i> defined in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ9">9</a>) is also a proper subspace of <span class="mathjax-tex">${\mathbb {C}}^r$</span>. <span class="mathjax-tex">$\square $</span></p> </li> </ol> <p>This result implies that for <span class="mathjax-tex">$G={{\,\textrm{SL}\,}}_2({\mathbb {C}})$</span> we only have to look at <span class="mathjax-tex">$R^{\textrm{irr}}_{\xi _3 \rightarrow \xi _1}$</span>. Take an irreducible <span class="mathjax-tex">$(A,B) \in R^{\textrm{irr}}_{\xi _3 \rightarrow \xi _1}$</span>. The matrix <i>A</i> can be put in diagonal form with eigenvalues <span class="mathjax-tex">$(\lambda ,\lambda \varepsilon )$</span>, <span class="mathjax-tex">$\lambda ^n = \pm 1$</span>, <span class="mathjax-tex">$\lambda ^2\varepsilon =1$</span>, <span class="mathjax-tex">$\lambda \ne \pm 1$</span>. This is the same as to say <span class="mathjax-tex">$\lambda \in \mu _{2n}-\mu _2$</span>. The action of interchanging eigenvalues is an <span class="mathjax-tex">$S_2$</span>-action free on <span class="mathjax-tex">$\mu _{2n}-\mu _2$</span>. Therefore there are <span class="mathjax-tex">$n-1$</span> possibilities. Taking a suitable basis, then</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} (A,B)= \left( \left( \begin{array}{cc} \lambda &{} 0 \\ 0 &{} \lambda \varepsilon \end{array} \right) , \left( \begin{array}{cc} a &{} b \\ c &{} d \end{array} \right) \right) , \end{aligned}$$</span></div><div class="c-article-equation__number"> (10) </div></div><p>As the pair is irreducible, then (1, 0) and (0, 1) are not eigenvectors of <i>B</i>, or equivalently, <span class="mathjax-tex">$b,c\ne 0$</span>. This is the same as <span class="mathjax-tex">$bc\ne 0$</span>. The action of <span class="mathjax-tex">${\mathcal {D}}\subset {{\,\textrm{PGL}\,}}_2({\mathbb {C}})$</span> moves <span class="mathjax-tex">$(b,c)\mapsto (\varpi ^{2} b, \varpi ^{-2}c)$</span>, so we can set <span class="mathjax-tex">$b=1$</span>, and hence <span class="mathjax-tex">$c=1-ad$</span>. Summing up,</p><div id="Equ67" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathfrak {M}}^{\textrm{irr}}(H_n,G)) \cong \big (\mu _{2n}-\mu _2\big )/S_2 \times \{(a,d) \in {\mathbb {C}}^2 \, | \, ad \ne 1\}. \end{aligned}$$</span></div></div><p>The set <span class="mathjax-tex">$ad=1$</span> is a hyperbola, isomorphic to <span class="mathjax-tex">${\mathbb {C}}^*$</span>. Therefore</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e({\mathfrak {M}}^{\textrm{irr}}(H_n,G))=(n-1)(q^2-q+1). \end{aligned}$$</span></div><div class="c-article-equation__number"> (11) </div></div><p>Finally</p><div id="Equ68" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e({\mathfrak {M}}(H_n,{{\,\textrm{SL}\,}}_2({\mathbb {C}})))=q^2+1+(n-1)(q^2-q+1). \end{aligned}$$</span></div></div></div></div></section><section data-title="Rank 3 Representation Variety of the Twisted Hopf Link: Combinatorial Setting"><div class="c-article-section" id="Sec13-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec13"><span class="c-article-section__title-number">7 </span>Rank 3 Representation Variety of the Twisted Hopf Link: Combinatorial Setting</h2><div class="c-article-section__content" id="Sec13-content"><p>In this section and the next one, we shall compute the <i>E</i>-polynomial of the <span class="mathjax-tex">${{\,\textrm{SL}\,}}_{3}({\mathbb {C}})$</span>-representation variety of the twisted Hopf link <span class="mathjax-tex">$H_n$</span>. For that purpose, we will analyze the combinatorial and geometric settings as described in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec6">4</a>. We start in this section by analyzing the combinatorial setting.</p><p>Let <span class="mathjax-tex">$G={{\,\textrm{SL}\,}}_3({\mathbb {C}})$</span>. In this rank, up to equivalence, we have 3 possible partitions, namely</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sigma _1 = \{\{1,2,3\}\}, \qquad \sigma _{2} = \{\{1,2\},\{3\}\}, \qquad \sigma _3 = \{\{1\}, \{2\}, \{3\}\}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (12) </div></div><p>They correspond to the configurations</p><div id="Equ69" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Delta ^3_{\sigma _1}&= \{\lambda _1 \in ({\mathbb {C}}^*)^3\,|\, \lambda _1^3 = 1\} = \mu _3, \\ \Delta ^3_{\sigma _2}&= \{(\lambda _1, \lambda _2) \in ({\mathbb {C}}^*)^2\,|\, \lambda _1^2\lambda _2 = 1, \lambda _1\ne \lambda _2\} = {\mathbb {C}}^* - \Delta ^3_{\sigma _1} = {\mathbb {C}}^* - \mu _3, \\ \Delta ^3_{\sigma _3}&= \{(\lambda _1, \lambda _2, \lambda _3) \in {\mathbb {C}}^*|\, \lambda _1\lambda _2\lambda _3 = 1, \lambda _1\ne \lambda _2\ne \lambda _3\ne \lambda _1\}. \end{aligned}$$</span></div></div><p>Only the stratum <span class="mathjax-tex">$\Delta ^3_{\sigma _3}$</span> is equipped with a non-trivial action. In this case, <span class="mathjax-tex">$S_3$</span> acts on <span class="mathjax-tex">$\Delta ^3_{\sigma _3}$</span> by permutation of <span class="mathjax-tex">$\lambda _1, \lambda _2$</span> and <span class="mathjax-tex">$\lambda _3$</span>. First, <span class="mathjax-tex">$\Delta ^3_{\sigma _3}\cong ({\mathbb {C}}^*)^2- \big ( \{(\lambda _1,\lambda _1)\}\cup \{(\lambda _1,\lambda _1^{-2})\}\cup \{(\lambda _1^{-2},\lambda _1)\}\big )$</span>. The curves intersect in 3 points <span class="mathjax-tex">$\{(\varpi ,\varpi )| \varpi \in \mu _3\}$</span>, therefore</p><div id="Equ70" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(\Delta ^3_{\sigma _3})=(q-1)^2-3(q-1)+3\cdot 3-3= q^2-5q+10. \end{aligned}$$</span></div></div><p>With respect to the refinements, for those over <span class="mathjax-tex">$\sigma _1$</span> we have</p><div id="Equ71" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( \Delta ^3_{\sigma _1 \rightarrow \sigma _1}\right)&= e\left( \Delta ^3_{\sigma _1}\right) = 3. \\ e\left( \Delta ^3_{\sigma _2 \rightarrow \sigma _1}\right)&= e\left( \{(\lambda _1, \varepsilon ) \in {\mathbb {C}}^* \times \mu _n \,|\, \lambda _1^3\varepsilon = 1, \varepsilon \ne 1\}\right) \\&= e\left( \{(\lambda _1, \varepsilon ) \in {\mathbb {C}}^* \times \mu _n \,|\, \lambda _1^3\varepsilon = 1\}\right) - e\left( \{\lambda _1 \in {\mathbb {C}}^* \,|\, \lambda _1^3 = 1\}\right) \\&= e(\mu _{3n}) - e(\mu _3) = 3n-3. \\ e\left( \Delta ^3_{\sigma _3 \rightarrow \sigma _1}\right)&= e\left( \{(\lambda _1, \varepsilon _1, \varepsilon _2) \in {\mathbb {C}}^* \times (\mu _n)^2 \,|\, \lambda _1^3\varepsilon _1\varepsilon _2 = 1, \varepsilon _1 \ne 1, \varepsilon _2 \ne 1, \varepsilon _1 \ne \varepsilon _2\}\right) \\&= e\left( \{(\lambda _1^3\varepsilon _1\varepsilon _2 = 1\}\right) - e\left( \{\lambda _1^3\varepsilon _2 = 1, \varepsilon _1=1\}\right) - e\left( \{\lambda _1^3\varepsilon _1 = 1, \varepsilon _2=1\}\right) \\& - e\left( \{\lambda _1^3\varepsilon _1^2 = 1, \varepsilon _1=\varepsilon _2\}\right) + 3\,e\left( \{\lambda _1^3 = 1, \varepsilon _1=\varepsilon _2=1\}\right) \\& - e\left( \{\lambda _1^3 = 1, \varepsilon _1=\varepsilon _2=1\}\right) \\&= e(\mu _{3n} \times \mu _n) - e(\mu _{3n}) - e(\mu _{3n}) - e(\mu _{3n})+ 3\,e(\mu _3) - e(\mu _3)\\&= 3n^2 - 9n +6. \end{aligned}$$</span></div></div><p>Moreover, for those over <span class="mathjax-tex">$\sigma _2$</span> we have</p><div id="Equ72" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( \Delta ^3_{\sigma _2 \rightarrow \sigma _2}\right)&= e\left( \{(\lambda _1, \lambda _2) \in ({\mathbb {C}}^*)^2 \,|\, \lambda _1^2\lambda _2 = 1, \lambda _1^n \ne \lambda _2^n\}\right) \\&= e\left( \Delta ^3_{\sigma _2}\right) - e\left( \Delta ^3_{\sigma _2 \rightarrow \sigma _1}\right) = (q-4) - (3n-3) = q-3n-1. \\ e\left( \Delta ^3_{\sigma _3 \rightarrow \sigma _2}\right)&= e\big ( \{(\lambda _1, \lambda _2, \varepsilon ) \in ({\mathbb {C}}^*)^2 \times \mu _n^* \,|\, \lambda _1^2\lambda _2\varepsilon = 1, \lambda _2 \ne \lambda _1\epsilon , \forall \epsilon \in \mu _n\} \big ) \\&= e(\{\lambda _1^2\lambda _2\varepsilon =1\}) - e(\{\epsilon \lambda _1^3\varepsilon =1,\epsilon \in \mu _n, \varepsilon \in \mu _n^*\}) \\&= e({\mathbb {C}}^* \times \mu _n^*) - e(\mu _{3n}\times \mu _n^*) = (n-1)(q-3n-1). \end{aligned}$$</span></div></div><p>Finally, to compute <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _3}$</span>, let us consider the three possible partitions equivalent to <span class="mathjax-tex">$\sigma _2$</span>, namely</p><div id="Equ73" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sigma _2 = \{\{1,2\},\{3\}\}, \qquad \sigma _2' = \{\{1\},\{2,3\}\}, \qquad \sigma _2'' = \{\{1,3\},\{2\}\}. \end{aligned}$$</span></div></div><p>Notice that we have a decomposition</p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} \Delta ^3_{\sigma _3} = \Delta ^3_{\sigma _3\rightarrow \sigma _3} \sqcup \Delta ^3_{\sigma _3\rightarrow \sigma _2} \sqcup \Delta ^3_{\sigma _3\rightarrow \sigma _2'} \sqcup \Delta ^3_{\sigma _3\rightarrow \sigma _2''} \sqcup \Delta ^3_{\sigma _3\rightarrow \sigma _1}. \end{aligned}\nonumber \\ \end{aligned}$$</span></div><div class="c-article-equation__number"> (13) </div></div><p>Since <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _2} \cong \Delta ^3_{\sigma _3\rightarrow \sigma _2'} \cong \Delta ^3_{\sigma _3\rightarrow \sigma _2''}$</span>, we get</p><div id="Equ74" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( \Delta ^3_{\sigma _3 \rightarrow \sigma _3}\right)&= e\left( \Delta ^3_{\sigma _3}\right) - 3e\left( \Delta ^3_{\sigma _3 \rightarrow \sigma _2}\right) - e\left( \Delta ^3_{\sigma _3\rightarrow \sigma _1}\right) \\&= q^2 - (3n + 2)q +6n^2 + 3n +1. \end{aligned}$$</span></div></div><h3 class="c-article__sub-heading" id="Sec14"><span class="c-article-section__title-number">7.1 </span>Equivariant <i>E</i>-polynomials</h3><p>First, let us analyze the action of <span class="mathjax-tex">$S_3$</span> on <span class="mathjax-tex">$\Delta ^3_{\sigma _3}$</span>. For the quotient <span class="mathjax-tex">$\Delta ^3_{\sigma _3}/S_3$</span>, we note that <span class="mathjax-tex">$ ({\mathbb {C}}^*)^2/S_3$</span> is parametrized by <span class="mathjax-tex">$(s=\lambda _1+\lambda _2+\lambda _3, p=\lambda _1\lambda _2+\lambda _1\lambda _3+\lambda _2\lambda _3)\in {\mathbb {C}}^2$</span>. The image of the three lines that we have to remove in <span class="mathjax-tex">$\Delta ^2_{\sigma _3}$</span> is just one line <span class="mathjax-tex">${\mathbb {C}}^*$</span> in <span class="mathjax-tex">$\Delta ^3_{\sigma _3}/S_3$</span>, so <span class="mathjax-tex">$e(\Delta ^3_{\sigma _3}/S_3)=q^2-q+1$</span>. Finally, <span class="mathjax-tex">$\Delta ^3_{\sigma _3}/\langle \tau \rangle $</span> is parametrized by <span class="mathjax-tex">$(u=\lambda _1+\lambda _2, v=\lambda _1\lambda _2) \in {\mathbb {C}}\times {\mathbb {C}}^*$</span>, removing the lines <span class="mathjax-tex">$\{(\lambda _1, \lambda _1)\} \cup \{(\lambda _1,\lambda _1^{-2})\}$</span>, which intersect in 3 points. Thus <span class="mathjax-tex">$e(\Delta ^3_{\sigma _3}/\langle \tau \rangle )=q^2-q-2(q-1)+3=q^2-3q+5$</span>. This produces:</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_{S^3}(\Delta ^3_{\sigma _3}) = (q^2-q+1) T+ S + (-2q+4)D. \end{aligned}$$</span></div><div class="c-article-equation__number"> (14) </div></div><p>The configuration spaces with actions are <span class="mathjax-tex">$\Delta ^3_{\sigma _3 \rightarrow \sigma _1}$</span>, <span class="mathjax-tex">$\Delta ^3_{\sigma _3 \rightarrow \sigma _2}$</span> and <span class="mathjax-tex">$\Delta ^3_{\sigma _3 \rightarrow \sigma _3}$</span>, which can be analyzed as follows.</p><ul class="u-list-style-bullet"> <li> <p>For the first one, we observe that the action of <span class="mathjax-tex">$S_3$</span> and <span class="mathjax-tex">$\tau =(1,2)$</span> on <span class="mathjax-tex">$\Delta ^3_{\sigma _3 \rightarrow \sigma _1}$</span> are free (they interchange different eigenvalues). Since <span class="mathjax-tex">$\Delta ^3_{\sigma _3 \rightarrow \sigma _1}$</span> is just a finite collection of points, we directly get that <span class="mathjax-tex">$e\left( \Delta ^3_{\sigma _3 \rightarrow \sigma _1}/S_3\right) = (3n^2 - 9n +6)/6 = \frac{1}{2} (n^2 - 3n + 2)$</span> and <span class="mathjax-tex">$e\left( \Delta ^3_{\sigma _3 \rightarrow \sigma _1}/\langle \tau \rangle \right) = \frac{1}{2} (3n^2 - 9n +6)$</span>. Therefore, using (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ4">4</a>) we have </p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} e_{S_3}\left( \Delta ^3_{\sigma _3 \rightarrow \sigma _1}\right)&= \frac{1}{2} \left( n^2 - 3n + 2\right) T + \frac{1}{2} \left( n^2 - 3n + 2\right) S \nonumber \\ {}&\quad + (n^2 - 3n + 2)D. \end{aligned}\nonumber \\ \end{aligned}$$</span></div><div class="c-article-equation__number"> (15) </div></div> </li> <li> <p>For the second space, observe that the action of <span class="mathjax-tex">$S_3$</span> on <span class="mathjax-tex">$\Delta ^3_{\sigma _3}$</span> does not restrict to <span class="mathjax-tex">$\Delta ^3_{\sigma _3 \rightarrow \sigma _2}$</span>. Instead, the acting subgroup is <span class="mathjax-tex">$S_{\sigma _3 \rightarrow \sigma _2} = \langle \tau \rangle < S_3$</span> with action given by <span class="mathjax-tex">$\tau \cdot (\lambda _1, \lambda _1 \varepsilon , \lambda _2) = (\lambda _1 \varepsilon , \lambda _1, \lambda _2)$</span>, in terms of the eigenvalues, or equivalently, <span class="mathjax-tex">$\tau \cdot (\lambda _1, \lambda _2, \varepsilon ) = (\lambda _1\varepsilon , \lambda _2, \varepsilon ^{-1})$</span> on <span class="mathjax-tex">$({\mathbb {C}}^*)^2\times \mu _n^*$</span>. Let us focus first on the natural extension of this action to <span class="mathjax-tex">$\{\lambda _1^2\lambda _2\varepsilon =1\} \cong {\mathbb {C}}^* \times \mu _{n}^*$</span>, given as <span class="mathjax-tex">$\tau \cdot (\lambda _1, \varepsilon ) = (\lambda _1\varepsilon , \varepsilon ^{-1})$</span>. These are <span class="mathjax-tex">$n-1$</span> different punctured lines and the action depends on the value of <span class="mathjax-tex">$\varepsilon $</span>. If <span class="mathjax-tex">$\varepsilon = -1$</span> (which can only happen if <i>n</i> is even) the action is <span class="mathjax-tex">$\lambda _1 \mapsto -\lambda _1$</span> whose quotient is <span class="mathjax-tex">${\mathbb {C}}^*$</span>; for <span class="mathjax-tex">$\varepsilon \ne \pm 1$</span>, the action interchanges the pair of lines <span class="mathjax-tex">${\mathbb {C}}^* \times \{\varepsilon \}$</span> to <span class="mathjax-tex">${\mathbb {C}}^* \times \{\varepsilon ^{-1}\}$</span> so the quotient is just one of them. In this way, </p><div id="Equ75" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(({\mathbb {C}}^* \times \mu _n^*)/\langle \tau \rangle ) = \left\lfloor \frac{n}{2} \right\rfloor (q-1), \end{aligned}$$</span></div></div><p> where <span class="mathjax-tex">$\left\lfloor x \right\rfloor $</span> is the floor function (the greatest integer less than or equal to <i>x</i>). Now we have to remove, <span class="mathjax-tex">$\{\epsilon \lambda _1^3\varepsilon =1, \epsilon \in \mu _n,\varepsilon \in \mu _n^*\} \cong \mu _{3n}\times \mu _n^*$</span>, and the action is <span class="mathjax-tex">$\tau \cdot (\lambda _1, \varepsilon )=(\lambda _1\varepsilon , \varepsilon ^{-1})$</span>. If <span class="mathjax-tex">$\varepsilon \ne -1$</span>, the action is clearly free, and if <span class="mathjax-tex">$\varepsilon =-1$</span> then the action is free as well. Hence this accounts form <span class="mathjax-tex">$3n(n-1)/2$</span> points. Therefore, putting all together we get </p><div id="Equ76" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( \Delta ^3_{\sigma _3 \rightarrow \sigma _2} / \langle \tau \rangle \right)&= \left\lfloor \frac{n}{2} \right\rfloor (q-1) - \frac{3}{2}n(n-1). \end{aligned}$$</span></div></div><p> Thus, we finally find that </p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_{S_2}\left( \Delta ^3_{\sigma _3 \rightarrow \sigma _2}\right)= & {} \left( \left\lfloor \frac{n}{2} \right\rfloor (q-1) - \frac{3n(n-1)}{2} \right) T \nonumber \\{} & {} + \left( \left\lfloor \frac{n-1}{2} \right\rfloor (q-1) - \frac{3n(n-1)}{2} \right) N. \end{aligned}$$</span></div><div class="c-article-equation__number"> (16) </div></div> </li> <li> <p>To study the remaining configuration, <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _3}$</span>, observe that regarding decomposition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ13">13</a>) the action of <span class="mathjax-tex">$S_3$</span> on <span class="mathjax-tex">$\Delta ^3_{\sigma _3}$</span> leaves invariant <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _3}$</span>, <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _1}$</span>, and <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _2} \sqcup \Delta ^3_{\sigma _3\rightarrow \sigma _2'} \sqcup \Delta ^3_{\sigma _3\rightarrow \sigma _2''} $</span>. For this later action, we have </p><div id="Equ77" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left( \Delta ^3_{\sigma _3\rightarrow \sigma _2} \sqcup \Delta ^3_{\sigma _3\rightarrow \sigma _2'} \sqcup \Delta ^3_{\sigma _3\rightarrow \sigma _2''} \right) /S_3 = \Delta ^3_{\sigma _3\rightarrow \sigma _2} / \langle \tau \rangle . \end{aligned}$$</span></div></div><p> Hence, using the previous computations we get </p><div id="Equ78" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( \Delta ^3_{\sigma _3\rightarrow \sigma _3}/S_3\right)&= e\left( \Delta ^3_{\sigma _3}/S_3\right) - e\left( \Delta ^3_{\sigma _3\rightarrow \sigma _2} / \langle \tau \rangle \right) - e\left( \Delta ^3_{\sigma _3\rightarrow \sigma _1}/S_3\right) \\&= q^{2} - q - \left\lfloor \frac{n}{2} \right\rfloor (q-1) + n^2. \end{aligned}$$</span></div></div><p> Similarly, for the action of <span class="mathjax-tex">$\tau $</span> we have that <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _3}$</span>, <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _2} $</span> and <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _1}$</span> are invariant, whereas it permutes <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _2'}$</span> and <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _2''}$</span>. Hence, </p><div id="Equ79" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left( \Delta ^3_{\sigma _3\rightarrow \sigma _2'} \sqcup \Delta ^3_{\sigma _3\rightarrow \sigma _2''} \right) /\langle \tau \rangle = \Delta ^3_{\sigma _3\rightarrow \sigma _2'} \cong \Delta ^3_{\sigma _3\rightarrow \sigma _2}, \end{aligned}$$</span></div></div><p> and therefore </p><div id="Equ80" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e\left( \Delta ^3_{\sigma _3\rightarrow \sigma _3}/\langle \tau \rangle \right)&= e\left( \Delta ^3_{\sigma _3}/\langle \tau \rangle \right) - e\left( \Delta ^3_{\sigma _3\rightarrow \sigma _2} / \langle \tau \rangle \right) - e\left( \Delta ^3_{\sigma _3\rightarrow \sigma _2}\right) \\&\quad - e\left( \Delta ^3_{\sigma _3\rightarrow \sigma _1}/\langle \tau \rangle \right) \\&= q^{2} -\left\lfloor \frac{3n+4}{2} \right\rfloor (q-1) + 3n^2-1. \end{aligned}$$</span></div></div><p> In this way, using (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ4">4</a>), we get </p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} e_{S_3}\left( \Delta ^3_{\sigma _3\rightarrow \sigma _3}\right) =&\left( q^{2} - q - \left\lfloor \frac{n}{2} \right\rfloor (q-1) + n^2 \right) T - \left( \left\lfloor \frac{n-1}{2} \right\rfloor (q-1)- n^2 \right) S \nonumber \\ {}&- \left( (n+1) (q-1) - 2n^2 \right) D. \end{aligned}\nonumber \\ \end{aligned}$$</span></div><div class="c-article-equation__number"> (17) </div></div> </li> </ul></div></div></section><section data-title="Rank 3 Representation Variety of the Twisted Hopf Link: Geometric setting"><div class="c-article-section" id="Sec15-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec15"><span class="c-article-section__title-number">8 </span>Rank 3 Representation Variety of the Twisted Hopf Link: Geometric setting</h2><div class="c-article-section__content" id="Sec15-content"><p>From the three partitions <span class="mathjax-tex">$\sigma _1,\sigma _2,\sigma _3$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ12">12</a>), we can create six types (up to equivalence), namely</p><div id="Equ81" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\xi _1 = (\sigma _1, \{1,2,3\}), \quad \xi _2 = (\sigma _1, \{(1,2),3\}), \quad \xi _3 = (\sigma _1, \{(1,2,3)\}), \\&\xi _4=(\sigma _2, \{\{1,2\},3\}), \quad \xi _5=(\sigma _2,\{\{(1,2)\},3\}), \quad \xi _6 = (\sigma _3,\{\{1\},\{2\},\{3\} \}). \end{aligned}$$</span></div></div><p>The type <span class="mathjax-tex">$\xi _j$</span> is given by the matrix <span class="mathjax-tex">$A_j$</span>, <span class="mathjax-tex">$j=1,2,3,4,5,6$</span>, where</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">$A_1=\left( \begin{array}{ccc} \lambda _1 &{} 0 &{} 0 \\ 0&{} \lambda _1 &{} 0 \\ 0 &{} 0&{} \lambda _1 \end{array} \right) $</span>, with <span class="mathjax-tex">$\lambda _1\in \mu _3$</span>.</p> </li> <li> <p><span class="mathjax-tex">$A_2=\left( \begin{array}{ccc} \lambda _1 &{} 0 &{} 0 \\ 1&{} \lambda _1 &{} 0 \\ 0 &{} 0&{} \lambda _1 \end{array} \right) $</span>, with <span class="mathjax-tex">$\lambda _1\in \mu _3$</span>.</p> </li> <li> <p><span class="mathjax-tex">$A_3=\left( \begin{array}{ccc} \lambda _1 &{} 0 &{} 0 \\ 1&{} \lambda _1 &{} 0 \\ 0 &{} 1&{} \lambda _1 \end{array} \right) $</span>, with <span class="mathjax-tex">$\lambda _1\in \mu _3$</span>.</p> </li> <li> <p><span class="mathjax-tex">$A_4=\left( \begin{array}{ccc} \lambda _1 &{} 0 &{} 0 \\ 0&{} \lambda _1 &{} 0 \\ 0 &{} 0&{} \lambda _2 \end{array} \right) $</span>, with <span class="mathjax-tex">$\lambda _1\ne \lambda _2$</span>. Here <span class="mathjax-tex">$\lambda _2=\lambda _1^{-2}$</span>, so <span class="mathjax-tex">$\lambda _1\in {\mathbb {C}}^*-\mu _3$</span>.</p> </li> <li> <p><span class="mathjax-tex">$A_5=\left( \begin{array}{ccc} \lambda _1 &{} 0 &{} 0 \\ 1&{} \lambda _1 &{} 0 \\ 0 &{} 0&{} \lambda _2 \end{array} \right) $</span>, with <span class="mathjax-tex">$\lambda _2=\lambda _1^{-2}$</span>, and <span class="mathjax-tex">$\lambda _1\in {\mathbb {C}}^*-\mu _3$</span>.</p> </li> <li> <p><span class="mathjax-tex">$A_6=\left( \begin{array}{ccc} \lambda _1 &{} 0 &{} 0 \\ 0&{} \lambda _2 &{} 0 \\ 0 &{} 0&{} \lambda _3 \end{array} \right) $</span>, with <span class="mathjax-tex">$\lambda _i\ne \lambda _j$</span> for <span class="mathjax-tex">$i\ne j$</span>, <span class="mathjax-tex">$\lambda _1\lambda _2\lambda _3=1$</span>.</p> </li> </ul><p>Only in the case <span class="mathjax-tex">$\xi _6$</span> we have an action of the group <span class="mathjax-tex">$S_3$</span> given by the permutation of the eigenvalues <span class="mathjax-tex">$\lambda _i$</span>.</p><p>Denote <span class="mathjax-tex">$R_{\xi _i\rightarrow \xi _j}=R(H_n,{{\,\textrm{SL}\,}}_3({\mathbb {C}}))_{\xi _i\rightarrow \xi _j}$</span>. Taking into account the possible refinement relations, we have</p><div id="Equ82" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R(H_n,{{\,\textrm{SL}\,}}_3({\mathbb {C}})) = R_{\xi _4 \rightarrow \xi _1} \sqcup R_{\xi _6 \rightarrow \xi _4} \sqcup R_{\xi _6 \rightarrow \xi _1}\sqcup R_{\xi _5 \rightarrow \xi _2} \sqcup \bigsqcup _{1\le i\le 6} R_{\xi _i \rightarrow \xi _i}. \end{aligned}$$</span></div></div><p>Using Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar16">4.7</a>, we have</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R_{\xi _i \rightarrow \xi _j} \cong \left( {\mathcal {A}}_{\xi _i \rightarrow \xi _j} \times \big ({{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _i)\big ) \times {\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _j) \right) / S_{\xi _i \rightarrow \xi _j} . \end{aligned}$$</span></div><div class="c-article-equation__number"> (18) </div></div><h3 class="c-article__sub-heading" id="Sec16"><span class="c-article-section__title-number">8.1 </span>The Stabilizers</h3><p>We start by studying <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/{{\,\textrm{Stab}\,}}(\xi _i)$</span> and <span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _j)$</span>. Recall that <span class="mathjax-tex">$e({{\,\textrm{PGL}\,}}_3({\mathbb {C}}))=e({{\,\textrm{SL}\,}}_3({\mathbb {C}}))= (q^3-1)(q^3-q)q^2$</span>.</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _1)={{\,\textrm{SL}\,}}_3({\mathbb {C}})$</span>.</p> </li> <li> <p><span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _2)=\left\{ { \left( \begin{array}{ccc} \beta _1 &{} 0 &{} 0 \\ a&{} \beta _1 &{} b \\ c &{} 0&{} \beta _1^{-2} \end{array} \right) }\Big |\beta _1\in {\mathbb {C}}^*\right\} \cong {\mathbb {C}}^*\times {\mathbb {C}}^3$</span>. Then <span class="mathjax-tex">$e({\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _2))=(q-1)q^3$</span>, and <span class="mathjax-tex">$e({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{{\,\textrm{Stab}\,}}(\xi _2))= (q^3-1)(q+1)$</span>.</p> </li> <li> <p><span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _3)=\left\{ { \left( \begin{array}{ccc} \beta _1 &{} 0 &{} 0 \\ a&{} \beta _1 &{} 0 \\ b &{} a&{} \beta _1 \end{array} \right) }\Big |\beta _1\in \mu _3\right\} \cong \mu _3 \times {\mathbb {C}}^2$</span>. Thus <span class="mathjax-tex">$e({\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _3))=3q^2$</span>, <span class="mathjax-tex">${{\,\textrm{Stab}\,}}(\xi _3)=q^2$</span>, and <span class="mathjax-tex">$e({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{{\,\textrm{Stab}\,}}(\xi _3))=(q^3-1)(q^2-1)q$</span>.</p> </li> <li> <p><span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _4)=\left\{ { \left( \begin{array}{c|c} B' &{} 0 \\ 0&{} (\det B')^{-1} \end{array} \right) }\Big | B'\in {{\,\textrm{GL}\,}}_2({\mathbb {C}})\right\} $</span>. Thus <span class="mathjax-tex">$e({\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _4))=(q^2-1)(q^2-q)$</span>, and <span class="mathjax-tex">$e({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{{\,\textrm{Stab}\,}}(\xi _4))= (q^2+q+1)q^2$</span>.</p> </li> <li> <p><span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _5)=\left\{ { \left( \begin{array}{ccc} \beta _1 &{} 0 &{} 0 \\ a&{} \beta _1 &{} 0 \\ 0 &{} 0&{} \beta _1^{-2} \end{array} \right) }\Big |\beta _1\in {\mathbb {C}}^*\right\} \cong {\mathbb {C}}^*\times {\mathbb {C}}$</span>. Thus <span class="mathjax-tex">$e({\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _5))=q(q-1)$</span>, and <span class="mathjax-tex">$e({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{{\,\textrm{Stab}\,}}(\xi _5))=(q^3-1)(q^3+q^2)$</span>.</p> </li> <li> <p><span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _6)=\left\{ {\left( \begin{array}{ccc} \beta _1 &{} 0 &{} 0 \\ 0&{} \beta _2 &{} 0 \\ 0 &{} 0&{} \beta _3 \end{array} \right) } \Big | \beta _1\beta _2\beta _3=1\right\} ={\mathcal {D}}\cong \Delta ^3_{\sigma _3}= ({\mathbb {C}}^*)^2$</span>. Then <span class="mathjax-tex">$e({\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _6))=(q-1)^2$</span>, and <span class="mathjax-tex">$e({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{{\,\textrm{Stab}\,}}(\xi _6))=(q^2+q+1)(q+1)q^3$</span>.</p> </li> </ul><p>Only in the case of <span class="mathjax-tex">$\xi _6$</span> there is an action of <span class="mathjax-tex">$S_3$</span>. Let us give the equivariant <i>E</i>-polynomials. The quotient <span class="mathjax-tex">${\mathcal {D}}/S_3$</span> is parametrized by <span class="mathjax-tex">$(s=\beta _1+\beta _2+\beta _3, p=\beta _1\beta _2+\beta _1\beta _3+\beta _2\beta _3)\in {\mathbb {C}}^2$</span>, hence <span class="mathjax-tex">$e({\mathcal {D}}/S_3)=q^2$</span>. Also <span class="mathjax-tex">$\Delta ^3_{\sigma _3}/\langle \tau \rangle = ({\mathbb {C}}^*/{\mathbb {Z}}_2)\times {\mathbb {C}}^* \cong {\mathbb {C}}\times {\mathbb {C}}^*$</span>, parametrized by <span class="mathjax-tex">$u=\beta _1+\beta _2$</span>, <span class="mathjax-tex">$v=\beta _1\beta _2$</span>, <span class="mathjax-tex">$\beta _3=v^{-1}$</span>, and hence <span class="mathjax-tex">$e({\mathcal {D}}/\langle \tau \rangle )=q^2-q$</span>. All together (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ4">4</a>) yields</p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_{S_3}({\mathcal {D}})= q^2 T+ S -q D. \end{aligned}$$</span></div><div class="c-article-equation__number"> (19) </div></div><p>The quotient <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}}$</span> is parametrized by the column vectors of the matrix in <span class="mathjax-tex">${{\,\textrm{GL}\,}}_3({\mathbb {C}})$</span> up to scalar. This means that</p><div id="Equ83" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}}=\{([v_1],[v_2],[v_3]) \in ({\mathbb {P}}^2)^3 \text { linearly independent}\}= ({\mathbb {P}}^2)^3- {\mathcal {L}}, \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">${\mathcal {L}}$</span> is the subspace of three linearly dependent points of <span class="mathjax-tex">${\mathbb {P}}^2$</span>. This is <span class="mathjax-tex">${\mathcal {L}}={\mathcal {L}}^0 \sqcup {\mathcal {L}}^1$</span>, according to whether they are coincident, or they span a line. In the first case <span class="mathjax-tex">${\mathcal {L}}^0\cong {\mathbb {P}}^2$</span>, and in the second, <span class="mathjax-tex">${\mathcal {L}}^1 \cong {{\,\textrm{Gr}\,}}({\mathbb {P}}^1,{\mathbb {P}}^2) \times (({\mathbb {P}}^1)^3-{\mathcal {L}}^0)$</span>, given by choosing a line and three not-all-equal points of it. The Grassmannian of lines is the dual projective plane, so</p><div id="Equ84" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e( {{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}})&= (q^2+q+1)^3\\&\quad - \Big ((q^2+q+1) + (q^2+q+1)((q+1)^3-(q+1)) \Big ) \\&= q^6+2q^5+2q^4+q^3. \end{aligned}$$</span></div></div><p>This agrees with <span class="mathjax-tex">$e({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}})=(q^3-1)(q^3-q)q^2/(q-1)^2$</span>.</p><p>For the quotient, <span class="mathjax-tex">$({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}})/S_3 = \textrm{Sym}^3({\mathbb {P}}^2)- {\mathcal {L}}$</span>, where now <span class="mathjax-tex">${\mathcal {L}}={\mathbb {P}}^2 \sqcup ({{\,\textrm{Gr}\,}}({\mathbb {P}}^1, {\mathbb {P}}^2) \times (\textrm{Sym}^3({\mathbb {P}}^1)-{\mathbb {P}}^1))$</span>. Then</p><div id="Equ85" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}})/S_3) =&\, (q^6+q^5+2q^4+2q^3+2q^2+q+1) \\ {}&- \Big ((q^2+q+1) + (q^2+q+1)((q^3+q^2+q+1)-(q+1)) \Big ) \\ =&\, q^6. \end{aligned}$$</span></div></div><p>Finally, <span class="mathjax-tex">$({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}})/\langle \tau \rangle = \textrm{Sym}^2({\mathbb {P}}^2) \times {\mathbb {P}}^2 - {\mathcal {L}}$</span>, where <span class="mathjax-tex">${\mathcal {L}}={\mathbb {P}}^2 \sqcup ( {{\,\textrm{Gr}\,}}({\mathbb {P}}^1, {\mathbb {P}}^2) \times (\textrm{Sym}^2({\mathbb {P}}^1)\times {\mathbb {P}}^1-{\mathbb {P}}^1))$</span>. Therefore,</p><div id="Equ86" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}})/\langle \tau \rangle ) =&\, (q^4+q^3+2q^2+q+1)(q^2+q+1) \\&- \Big ((q^2+q+1) + (q^2+q+1)((q^2+q+1)(q+1)-(q+1)) \Big ) \\ =&\, q^6+q^5+q^4. \end{aligned}$$</span></div></div><p>All together, understanding <span class="mathjax-tex">$S_2 = \langle \tau \rangle $</span>, this gives</p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} e_{S_3}({{\,\text {PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}}) = q^6 T+q^3 S + (q^5+q^4)D, \end{aligned} \end{aligned}$$</span></div><div class="c-article-equation__number"> (20) </div></div><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} e_{S_2}({{\,\text {PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}}) = (q^6+q^5+q^4) T+ (q^5 + q^4 + q^3)N. \end{aligned} \end{aligned}$$</span></div><div class="c-article-equation__number"> (21) </div></div><p>Finally, for the action of <span class="mathjax-tex">$S_3$</span> on <span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _1)= {{\,\textrm{SL}\,}}_3({\mathbb {C}})$</span> and the action of <span class="mathjax-tex">$S_2$</span> on <span class="mathjax-tex">${\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _4) \cong {{\,\textrm{GL}\,}}_2({\mathbb {C}})$</span>, by Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar4">2.2</a> we have that</p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_{S_3}({{\,\textrm{SL}\,}}_3({\mathbb {C}})) = (q^8 - q^6 - q^5 + q^3)T, \quad e_{S_2}({{\,\textrm{GL}\,}}_2({\mathbb {C}})) = (q^4-q^3-q^2+q) T .\nonumber \\ \end{aligned}$$</span></div><div class="c-article-equation__number"> (22) </div></div> <h3 class="c-article__sub-heading" id="FPar20">Remark 8.1</h3> <p>There is a locally trivial fibration <span class="mathjax-tex">${\mathcal {D}}\rightarrow {{\,\textrm{SL}\,}}_3({\mathbb {C}}) \rightarrow {{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}}$</span>. This is an equivariant fibration, which can be seen as follows: the base <span class="mathjax-tex">$B={{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}}$</span> parametrizes subsets of 3 non-collinear points in <span class="mathjax-tex">${\mathbb {P}}^2$</span>, which is an open subset of <span class="mathjax-tex">$\textrm{Sym}^3({\mathbb {P}}^2)$</span>. Given one triplet, there is a line <span class="mathjax-tex">$\ell \subset {\mathbb {P}}^2$</span> missing it, hence it lies in <span class="mathjax-tex">$\textrm{Sym}^3 ({\mathbb {P}}^2- \ell )$</span>. The fibration over <span class="mathjax-tex">$B\cap \textrm{Sym}^3 ({\mathbb {P}}^2- \ell )$</span> is trivial and <span class="mathjax-tex">$S_3$</span>-invariant, which shows the claim. In particular, it holds <span class="mathjax-tex">$e_{S_3}({{\,\textrm{SL}\,}}_3({\mathbb {C}}))= e_{S_3}({\mathcal {D}}) \otimes e_{S_3}({{\,\textrm{PGL}\,}}_3({\mathbb {C}})/{\mathcal {D}})$</span>. This can also be checked from (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ19">19</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ20">20</a>). Similar observations apply for <span class="mathjax-tex">${{\,\textrm{GL}\,}}_2({\mathbb {C}})$</span>.</p> <h3 class="c-article__sub-heading" id="Sec17"><span class="c-article-section__title-number">8.2 </span>Adding Up All Contributions</h3><p>Now we move to the computation of the <i>E</i>-polynomials of the strata (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ18">18</a>). For <span class="mathjax-tex">$i=1,\ldots , 5$</span>, we have</p><div id="Equ87" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} R_{\xi _i \rightarrow \xi _i} \cong \Delta ^3_{\sigma _{(i)} \rightarrow \sigma _{(i)}} \times \big ({{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _i)\big ) \times {\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _i) , \end{aligned}$$</span></div></div><p>using Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar14">4.6</a>, where <span class="mathjax-tex">$\sigma _{(i)}$</span> denotes the partition associated to <span class="mathjax-tex">$\xi _i$</span>. Therefore</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">$e(R_{\xi _1 \rightarrow \xi _1} )=e( \Delta ^3_{\sigma _1 \rightarrow \sigma _1}) e({{\,\textrm{PGL}\,}}_3({\mathbb {C}}))= 3(q^3-1)(q^3-q)q^2$</span>.</p> </li> <li> <p><span class="mathjax-tex">$e(R_{\xi _2\rightarrow \xi _2} )=e( \Delta ^3_{\sigma _1 \rightarrow \sigma _1}) e({{\,\textrm{PGL}\,}}_3({\mathbb {C}})) =3(q^3-1)(q^3-q)q^2$</span>.</p> </li> <li> <p><span class="mathjax-tex">$e(R_{\xi _3\rightarrow \xi _3} )=e( \Delta ^3_{\sigma _1 \rightarrow \sigma _1}) 3\,e({{\,\textrm{PGL}\,}}_3({\mathbb {C}}))= 9(q^3-1)(q^3-q)q^2$</span>.</p> </li> <li> <p><span class="mathjax-tex">$e(R_{\xi _4 \rightarrow \xi _4} )=e( \Delta ^3_{\sigma _2 \rightarrow \sigma _2}) e({{\,\textrm{PGL}\,}}_3({\mathbb {C}})) =(q-3n-1)(q^3-1)(q^3-q)q^2$</span>.</p> </li> <li> <p><span class="mathjax-tex">$e(R_{\xi _5 \rightarrow \xi _5} )=e( \Delta ^3_{\sigma _2 \rightarrow \sigma _2}) e({{\,\textrm{PGL}\,}}_3({\mathbb {C}})) =(q-3n-1)(q^3-1)(q^3-q)q^2$</span>.</p> </li> </ul><p>The remaining five strata are analyzed one by one:</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">$R_{\xi _5 \rightarrow \xi _2} \cong \Delta ^3_{\sigma _2 \rightarrow \sigma _1} \times \big ({{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _5)\big ) \times {\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _2)$</span>, hence </p><div id="Equ88" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(R_{\xi _5 \rightarrow \xi _2})&=e( \Delta ^3_{\sigma _2 \rightarrow \sigma _1}) e\big ({{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _5)\big ) e({\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _2)) \\&=(3n-3) (q^3-1)(q^3+q^2)(q-1)q^3. \end{aligned}$$</span></div></div> </li> <li> <p><span class="mathjax-tex">$R_{\xi _4 \rightarrow \xi _1} \cong \Delta ^3_{\sigma _2 \rightarrow \sigma _1} \times \big ({{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _4)\big ) \times {\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _1)$</span>, therefore </p><div id="Equ89" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(R_{\xi _4 \rightarrow \xi _1})&=e( \Delta ^3_{\sigma _2 \rightarrow \sigma _1}) e\big ({{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _4)\big ) e({\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _1)) \\&=(3n-3) (q^2+q+1)q^2(q^3-1)(q^2-1)q^3. \end{aligned}$$</span></div></div> </li> <li> <p><span class="mathjax-tex">$R_{\xi _6 \rightarrow \xi _1} \cong {\tilde{R}}_{\xi _6 \rightarrow \xi _1}/S_3$</span>, where <span class="mathjax-tex">${\tilde{R}}_{\xi _6 \rightarrow \xi _1} \cong \Delta ^3_{\sigma _3 \rightarrow \sigma _1} \times \big ({{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _6)\big ) \times {\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _1) = \Delta ^3_{\sigma _3 \rightarrow \sigma _1} \times \big ({{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/ {\mathcal {D}}\big ) \times {{\,\textrm{SL}\,}}_3({\mathbb {C}})$</span>. By (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ15">15</a>), (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ20">20</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ22">22</a>), we have </p><div id="Equ90" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}{} & {} e_{S_3}({\tilde{R}}_{\xi _6 \rightarrow \xi _1}) = \left( \frac{n^2 - 3n + 2}{2} T + \frac{n^2 - 3n + 2}{2}S + (n^2 - 3n + 2)D \right) \\{} & {} \qquad \qquad \quad \qquad \otimes ( q^6 T+q^3 S + (q^5+q^4)D ) \otimes (q^8 - q^6 - q^5 + q^3)T . \end{aligned}$$</span></div></div><p> Taking the <i>T</i>-component, </p><div id="Equ91" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(R_{\xi _6 \rightarrow \xi _1})&= \frac{n^2-3n+2}{2} (q^3-1)^2 (q+1)^2 q^6 . \end{aligned}$$</span></div></div> </li> <li> <p><span class="mathjax-tex">$R_{\xi _6 \rightarrow \xi _4} \cong {\tilde{R}}_{\xi _6 \rightarrow \xi _4}/S_2$</span>, where <span class="mathjax-tex">$R_{\xi _6 \rightarrow \xi _4} \cong \Delta ^3_{\sigma _3 \rightarrow \sigma _2} \times \big ({{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/ {{\,\textrm{Stab}\,}}(\xi _6)\big ) \times {\widetilde{{{\,\textrm{Stab}\,}}}}(\xi _4)$</span>. By (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ16">16</a>), (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ21">21</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ22">22</a>) we have </p><div id="Equ92" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned}e_{S_2}({\tilde{R}}_{\xi _6 \rightarrow \xi _4})&= \left( \left( \left\lfloor \frac{n}{2} \right\rfloor (q-1) - \frac{3n(n-1)}{2} \right) T\right. \\ {}&\left. \qquad + \left( \left\lfloor \frac{n-1}{2} \right\rfloor (q-1) - \frac{3n(n-1)}{2} \right) N\right) \\ {}&\qquad \otimes \left( (q^6+q^5+q^4) T+ (q^5 + q^4 + q^3)N\right) \otimes (q^4-q^3-q^2+q) T. \end{aligned} \end{aligned}$$</span></div></div><p> Taking the <i>T</i>-component, </p><div id="Equ93" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(R_{\xi _6 \rightarrow \xi _4}) =&\, q^4 (q^3-1)(q^2-1) \Big ( \left\lfloor \frac{n}{2} \right\rfloor (q-1)^2\\&-\frac{3n^2-5n+2}{2} (q-1) -3n(n-1)\Big ). \end{aligned}$$</span></div></div> </li> <li> <p><span class="mathjax-tex">$R_{\xi _6 \rightarrow \xi _6} ={\tilde{R}}_{\xi _6 \rightarrow \xi _6}/S_3$</span>, where <span class="mathjax-tex">${\tilde{R}}_{\xi _6\rightarrow \xi _6} \cong \Delta ^3_{\sigma _3 \rightarrow \sigma _3} \times \big ({{\,\textrm{PGL}\,}}_{3}({\mathbb {C}})/ {\mathcal {D}}\big ) \times {\mathcal {D}}$</span>. By (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ17">17</a>), (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ20">20</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ19">19</a>) we have </p><div id="Equ94" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&e_{S_3}( {\tilde{R}}_{\xi _6 \rightarrow \xi _6}) = \Big ( \left( q^{2} - q - \left\lfloor \frac{n}{2} \right\rfloor (q-1) + n^2 \right) T\\&\qquad - \left( \left\lfloor \frac{n-1}{2} \right\rfloor (q-1)- n^2 \right) S - \left( (n+1) (q-1) - 2n^2 \right) D \Big )\\&\qquad \otimes \left( q^6 T+q^3 S + (q^5+q^4)D\right) \otimes \left( q^2 T+ S -q D\right) . \end{aligned}$$</span></div></div><p> Taking the <i>T</i>-component, </p><div id="Equ95" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(R_{\xi _6 \rightarrow \xi _6})&= \left( q^{2} - q - \left\lfloor \frac{n}{2} \right\rfloor (q-1) + n^2 \right) (q^8-q^6-q^5+q^3) \end{aligned}$$</span></div></div> </li> </ul><p>Adding up all the contributions, we finally get</p><div id="Equ96" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&e\big (R(H_n, {{\,\textrm{SL}\,}}_{3}({\mathbb {C}}))\big ) =\, (q^3-1)(q^2-1)q^2 \Big ( \left\lfloor \frac{n}{2} \right\rfloor (q^2-q)(q^2-q-1) \\&\qquad +\frac{1}{2}n^2 (q^7+2q^6+2q^5+q^4-3q^3-3q^2+2q)\\&\qquad -\frac{1}{2}n (3q^7+6q^6-3q^4 -17q^3 \\&\qquad -q^2+12q) + q^7+2q^6-q^5-2q^4-6q^3+2q^2+13q\Big ). \end{aligned}$$</span></div></div></div></div></section><section data-title="Rank 3 Character Variety of the Twisted Hopf Link"><div class="c-article-section" id="Sec18-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec18"><span class="c-article-section__title-number">9 </span>Rank 3 Character Variety of the Twisted Hopf Link</h2><div class="c-article-section__content" id="Sec18-content"><p>We end up with the computation of <span class="mathjax-tex">$e({\mathfrak {M}}(H_n,G))$</span>, for <span class="mathjax-tex">$G={{\,\textrm{SL}\,}}_3({\mathbb {C}})$</span>. First, we deal with reducible representations (<i>A</i>, <i>B</i>). The ones of type (1, 1, 1) are the direct sums of three one-dimensional representations. This means that</p><div id="Equ97" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} (A,B)= \left( \left( \begin{array}{ccc} \lambda _1 &{}0 &{} 0 \\ 0 &{} \lambda _2 &{}0 \\ 0 &{}0&{}\lambda _3 \end{array} \right) , \left( \begin{array}{ccc} \mu _1 &{}0 &{} 0 \\ 0 &{} \mu _2 &{}0 \\ 0 &{}0&{}\mu _3 \end{array} \right) \right) , \end{aligned}$$</span></div></div><p>which is parametrized by <span class="mathjax-tex">$({\mathcal {D}})^2/S_3$</span>. Using (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ19">19</a>), we take the <i>T</i>-component of <span class="mathjax-tex">$e_{S_3}(({\mathcal {D}})^2)=e_{S_3}({\mathcal {D}})^2=(q^2T+S-qD)^2$</span>, which is</p><div id="Equ98" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e({\mathfrak {M}}^{\textrm{red}}_{1,1,1}(H_n,G))= e(({\mathcal {D}})^2/S_3)= q^4+ q^2 + 1\,. \end{aligned}$$</span></div></div><p>Next, we consider the reducible representations of type (2, 1). Then</p><div id="Equ99" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} (A,B)= \left( \left( \begin{array}{c|c} A_1 &{}0 \\ \hline 0 &{} \lambda _1 \end{array} \right) , \left( \begin{array}{c|c} B_1 &{}0 \\ \hline 0 &{} \mu _1 \end{array} \right) \right) , \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\lambda _1=(\det A_1)^{-1}$</span>, <span class="mathjax-tex">$\mu _1=(\det B_1)^{-1}$</span>, and <span class="mathjax-tex">$(A_1,B_1)$</span> is an irreducible <span class="mathjax-tex">${{\,\textrm{GL}\,}}_2({\mathbb {C}})$</span>-representation. The computation is similar to the case of <span class="mathjax-tex">${{\,\textrm{SL}\,}}_2({\mathbb {C}})$</span> in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s00009-023-02300-w#Sec12">6</a>. Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar18">6.1</a> applies here, and we only have to see at the reductions <span class="mathjax-tex">$\xi _3\rightarrow \xi _1$</span>. Therefore <span class="mathjax-tex">$(A_1,B_1)$</span> can be put on the form (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00009-023-02300-w#Equ10">10</a>), where <span class="mathjax-tex">$\lambda \in {\mathbb {C}}^*$</span>, <span class="mathjax-tex">$\varepsilon \in \mu _n^*$</span>, <span class="mathjax-tex">$bc\ne 0$</span>, <span class="mathjax-tex">$ad-bc\ne 0$</span>. Here, we find two options. </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>If <span class="mathjax-tex">$\varepsilon \ne -1$</span> (which happens always if <i>n</i> is odd), then the <span class="mathjax-tex">$S_2$</span>-action sends <span class="mathjax-tex">$\varepsilon \mapsto \varepsilon ^{-1}$</span>. The quotient of one of such sets is parametrized by <span class="mathjax-tex">$({{\,\textrm{GL}\,}}_2({\mathbb {C}})-\{bc=0\})/{\mathbb {C}}^*$</span>, whose <i>E</i>-polynomial is <span class="mathjax-tex">$(q^2-q+1)(q-1)=q^3-2q^2+2q-1$</span>.</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>If <span class="mathjax-tex">$\varepsilon =-1$</span> then we have to quotient by the swap of the eigenvalues, which yields the space <span class="mathjax-tex">$(({{\,\textrm{GL}\,}}_2({\mathbb {C}})-\{bc=0\})/{\mathbb {C}}^*)/S_2$</span>. Let us consider the fibration </p><div id="Equ100" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F=\{(a,d) \,|\, ad \ne bc+1\} \rightarrow {{\,\textrm{GL}\,}}_2({\mathbb {C}})-\{bc=0\} \rightarrow B=\{(b,c) \,|\, bc\ne 0\}. \end{aligned}$$</span></div></div><p> Note that the action of <span class="mathjax-tex">${\mathbb {C}}^*$</span> is only on <i>B</i>, <span class="mathjax-tex">$(b,c)\mapsto (b\varpi ^2, c\varpi ^{-2})$</span>. The quotient is <span class="mathjax-tex">$B/{\mathbb {C}}^* \cong {\mathbb {C}}^*$</span> with <span class="mathjax-tex">$S_2$</span>-action <span class="mathjax-tex">$\alpha \mapsto \alpha ^{-1}$</span>, whence <span class="mathjax-tex">$e_{S^2}(B/{\mathbb {C}}^*)=qT-N$</span>. For the fiber, <span class="mathjax-tex">$S_2$</span> swaps (<i>a</i>, <i>d</i>), hence the quotient is parametrized by <span class="mathjax-tex">$s=a+d$</span>, <span class="mathjax-tex">$p=ad \ne bc+1$</span>, so <span class="mathjax-tex">$e(F/S_2)=q(q-1)$</span>, and thus <span class="mathjax-tex">$e_{S^2}(F)=(q^2-q)T+N$</span>. Therefore we get </p><div id="Equ101" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e_{S_2}(({{\,\textrm{GL}\,}}_2({\mathbb {C}})-\{bc=0\})/{\mathbb {C}}^*)&= e_{S_2}(B/{\mathbb {C}}^*) e_{S_2}(F) \\&=(q^3-q^2-1)T+ (2q-q^2)N. \end{aligned}$$</span></div></div> </li> </ol><p>All together, we have</p><div id="Equ102" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e({\mathfrak {M}}^{\textrm{red}}_{2,1}(H_n,G))&= \left\lfloor \frac{n-1}{2} \right\rfloor (q^3-2q^2+2q-1)\\&\quad + \Big (n-1-2\left\lfloor \frac{n-1}{2} \right\rfloor \Big )(q^3-q^2-1)\\&= (n-1)(q^3-q^2-1) - \left\lfloor \frac{n-1}{2} \right\rfloor (q^3-2q-1)\,. \end{aligned}$$</span></div></div><p>Now we move to <span class="mathjax-tex">${\mathfrak {M}}^{\textrm{irr}}(H_n,G)$</span>. For the irreducible representations, Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00009-023-02300-w#FPar18">6.1</a> implies that the only non-empty strata are <span class="mathjax-tex">$R^{\textrm{irr}}_{\xi _4\rightarrow \xi _1}$</span> and <span class="mathjax-tex">$R^{\textrm{irr}}_{\xi _6\rightarrow \xi _1}$</span>. We start with <span class="mathjax-tex">$(A,B)\in R^{\textrm{irr}}_{\xi _6\rightarrow \xi _1}$</span>. Choosing a suitable basis,</p><div id="Equ103" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} (A,B)= \left( \left( \begin{array}{ccc} \lambda _1 &{}0 &{} 0 \\ 0 &{} \lambda _2 &{}0 \\ 0 &{}0&{}\lambda _3 \end{array} \right) , \left( \begin{array}{ccc} a &{} b &{} c \\ d &{} e &{} f \\ g &{} h &{} i \end{array} \right) \right) , \end{aligned}$$</span></div></div><p>modulo the action of <span class="mathjax-tex">${\mathcal {D}}=({\mathbb {C}}^*)^2$</span>. As <span class="mathjax-tex">$\lambda _1^n=\lambda _2^n=\lambda _3^n=\varpi $</span>, <span class="mathjax-tex">$\varpi ^3=1$</span>, <span class="mathjax-tex">$\lambda _i\ne \lambda _j$</span> for <span class="mathjax-tex">$i\ne j$</span>, the count of matrices is given by <span class="mathjax-tex">$\Delta ^3_{\sigma _3\rightarrow \sigma _1}/S_3$</span>, i.e. <span class="mathjax-tex">$\frac{1}{2} (n^2-3n+2)$</span> points. In order for (<i>A</i>, <i>B</i>) to be irreducible, they cannot leave invariant a line (that is, no column of <i>B</i> is the coordinate vector) or a plane (that is, no row of <i>B</i> is the coordinate vector). We count the contribution:</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">$d\ne 0$</span> and <span class="mathjax-tex">$g\ne 0$</span>. Using the action of <span class="mathjax-tex">${\mathcal {D}}$</span>, we arrange <span class="mathjax-tex">$d=1, g=1$</span>. The space of such matrices in <span class="mathjax-tex">${{\,\textrm{SL}\,}}_3({\mathbb {C}})$</span> has <i>E</i>-polynomial <span class="mathjax-tex">$q(q^3-q)q^2$</span>. Now we have to remove <span class="mathjax-tex">$U_1=\{b=c=0\}$</span>, <span class="mathjax-tex">$U_2=\{b=h=0\}$</span>, <span class="mathjax-tex">$U_3=\{c=f=0\}$</span>. Denote <span class="mathjax-tex">$U_{ij}=U_i\cap U_j$</span>. Note that <span class="mathjax-tex">$U_{23}=U_{123}$</span>. The contribution is: </p><div id="Equ104" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(U_1\cup U_2 \cup U_3 )&= e(U_1)+ e(U_2)+ e(U_3 ) - e(U_{12})- e(U_{13}) \\&= (q^2-1)(q^2-q) + q^2(q^2-q) + q^2(q^2-q) - q(q-1)^2- q(q-1)^2 \\&= 3 q^{4} - 5 q^{3} + 3 q^{2} - q. \end{aligned}$$</span></div></div><p> Hence the <i>E</i>-polynomial of this stratum is <span class="mathjax-tex">$q^{6} - 4 q^{4} + 5 q^{3} - 3 q^{2} + q$</span>.</p> </li> <li> <p><span class="mathjax-tex">$d\ne 0$</span> and <span class="mathjax-tex">$g=0$</span>. It must be <span class="mathjax-tex">$h\ne 0$</span>. We can arrange <span class="mathjax-tex">$d=1$</span>, <span class="mathjax-tex">$h=1$</span>. Then <i>B</i> has determinant <span class="mathjax-tex">$c+aei-bi-af=1$</span>, so <i>c</i> is fixed. Therefore the space of such matrices in <span class="mathjax-tex">${{\,\textrm{SL}\,}}_3({\mathbb {C}})$</span> has <i>E</i>-polynomial <span class="mathjax-tex">$q^5$</span>. Now we remove <span class="mathjax-tex">$U_1=\{b=c=0\}$</span>, <span class="mathjax-tex">$U_2=\{c=0,f=0\}$</span>. The contribution is: </p><div id="Equ105" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e(U_1\cup U_2 )&= e(U_1)+ e(U_2) - e(U_{12}) \\ {}&= q(q^2-q) + q(q^2-q)-(q-1)^2 \\&=2q^3-3q^2+2q-1. \end{aligned}$$</span></div></div><p> Thus the <i>E</i>-polynomial of this stratum is <span class="mathjax-tex">$q^5-2q^3+3q^2-2q+1$</span>.</p> </li> <li> <p><span class="mathjax-tex">$d=0$</span>, <span class="mathjax-tex">$g\ne 0$</span>. This is analogous to the previous one. It has <i>E</i>-polynomial <span class="mathjax-tex">$q^5-2q^3+3q^2-2q+1$</span>.</p> </li> </ul><p>Adding up,</p><div id="Equ106" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e({\mathfrak {M}}^{\textrm{irr}}_{\xi _6\rightarrow \xi _1} )= \frac{1}{2} (n^2-3n+2)(q^{6} + 2 q^{5} - 4 q^{4} + q^{3} + 3 q^{2} - 3 q + 2). \end{aligned}$$</span></div></div><p>We end up with <span class="mathjax-tex">$(A,B)\in R^{\textrm{irr}}_{\xi _4\rightarrow \xi _1}$</span>. Choosing a suitable basis,</p><div id="Equ107" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} (A,B)= \left( \left( \begin{array}{ccc} \lambda &{}0 &{} 0 \\ 0 &{} \lambda &{}0 \\ 0 &{}0&{}\lambda \varepsilon \end{array} \right) , \left( \begin{array}{c|c} B_1 &{} \begin{array}{c} c \\ f \end{array} \\ g \,\,\, h &{} i \end{array} \right) \right) , \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\varepsilon =\lambda ^{-3}$</span>, <span class="mathjax-tex">$\lambda \in \mu _{3n}- \mu _3$</span>. This space is modulo the action of <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_2({\mathbb {C}})\times {\mathbb {C}}^*$</span>. The action of <span class="mathjax-tex">${{\,\textrm{PGL}\,}}_2({\mathbb {C}})$</span> conjugates <span class="mathjax-tex">$B_1$</span>, therefore we can put it in Jordan form. There are two options:</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">$B_1$</span> is diagonalizable. Therefore we can put </p><div id="Equ108" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} B=\left( \begin{array}{ccc} a &{} 0 &{} c \\ 0 &{} e &{} f \\ g &{} h &{} i \end{array} \right) . \end{aligned}$$</span></div></div><p> It must be <span class="mathjax-tex">$g\ne 0$</span>, <span class="mathjax-tex">$h\ne 0$</span>, <span class="mathjax-tex">$c\ne 0$</span> and <span class="mathjax-tex">$f\ne 0$</span>. With the residual action of <span class="mathjax-tex">${\mathcal {D}}={\mathbb {C}}^*\times {\mathbb {C}}^*$</span>, we can arrange <span class="mathjax-tex">$g=1$</span>, <span class="mathjax-tex">$h=1$</span>. No eigenvector of <i>A</i> of the form (<i>x</i>, <i>y</i>, 0) should be eigenvector of <i>B</i>, which translates into <span class="mathjax-tex">$a\ne e$</span>. Also, no invariant plane of the form <span class="mathjax-tex">$\langle (x,y,0),(0,0,1)\rangle $</span> should be invariant for <i>B</i>, which also means <span class="mathjax-tex">$a\ne e$</span>. Now we distinguish two cases: </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p><span class="mathjax-tex">$a,e\ne 0$</span>. Then the condition <span class="mathjax-tex">$\det B=1$</span> says that <i>i</i> is given in terms of <span class="mathjax-tex">$(a,e,c,f) \in (({\mathbb {C}}^*)^2-\Delta )\times ({\mathbb {C}}^*)^2$</span>. There is an action of <span class="mathjax-tex">$S_2$</span> swapping eigenvalues of <span class="mathjax-tex">$B_1$</span>, that is <span class="mathjax-tex">$(a,e,c,f) \mapsto (e,a, f,c)$</span>. The equivariant <i>E</i>-polynomials are: <span class="mathjax-tex">$e_{S_2}(({\mathbb {C}}^*)^2-\Delta ) =(q^2-2q+1)T-(q-1)N$</span>, <span class="mathjax-tex">$e_{S_2}(({\mathbb {C}}^*)^2=(q^2-q)T-(q-1)N$</span>. This gives the final <i>E</i>-polynomial <span class="mathjax-tex">$(q^2-q)(q^2-2q+1)+(q-1)^2= q^4-3q^3+4q^2-3q+1$</span>.</p> </li> <li> <span class="u-custom-list-number">2.</span> <p><span class="mathjax-tex">$a=0$</span>, <span class="mathjax-tex">$e\ne 0$</span> (and there is no swapping of eigenvalues now). Then the parameters are <span class="mathjax-tex">$(c,f,i)\in ({\mathbb {C}}^*)^2\times {\mathbb {C}}$</span>, and <span class="mathjax-tex">$e=c^{-1}$</span>. The <i>E</i>-polynomial is <span class="mathjax-tex">$(q-1)^2q$</span>.</p> </li> </ol> </li> <li> <p><span class="mathjax-tex">$B_1$</span> is not diagonalizable. Therefore we can put </p><div id="Equ109" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} B=\left( \begin{array}{ccc} a &{} 0 &{} c \\ 1 &{} a &{} f \\ g &{} h &{} i \end{array} \right) . \end{aligned}$$</span></div></div><p> There is a residual action of <span class="mathjax-tex">$\tiny \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ x &{} 1 &{} 0 \\ 0 &{} 0 &{} y \end{array} \right) $</span>. As it must be <span class="mathjax-tex">$h\ne 0$</span>, we can arrange <span class="mathjax-tex">$h=1$</span>, and also <span class="mathjax-tex">$g=0$</span>. The irreducibility means that (0, 1, 0), (0, 0, 1) are not eigenvectors of <i>B</i>, and <span class="mathjax-tex">$\langle (1,0,0),(0,1,0)\rangle $</span> and <span class="mathjax-tex">$\langle (0,1,0), (0,0,1)\rangle $</span> are not invariant planes of <i>B</i>. This translates into <span class="mathjax-tex">$c\ne 0$</span>. The determinant condition is <span class="mathjax-tex">$\det B= a^2 i+c-af =1$</span>, so <i>c</i> is determined, and the space is <span class="mathjax-tex">$\{(a,i,f) | \, a^2i-af\ne 1\}$</span>. For <span class="mathjax-tex">$a\ne 0$</span>, this is <span class="mathjax-tex">${\mathbb {C}}^2- {\mathbb {C}}$</span>; and for <span class="mathjax-tex">$a=0$</span>, it is <span class="mathjax-tex">${\mathbb {C}}^2$</span>. So the <i>E</i>-polynomial is <span class="mathjax-tex">$(q-1)(q^2-q)+q^2=q^3-q^2+q$</span>.</p> </li> </ul><p>Adding up, this amounts to <span class="mathjax-tex">$q^4-q^3+q^2-q+1$</span>, and taking into account the possible values of <span class="mathjax-tex">$\lambda $</span> we get</p><div id="Equ110" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e({\mathfrak {M}}^{\textrm{irr}}_{\xi _4\rightarrow \xi _1} )=(3n-3) (q^4-q^3+q^2-q+1). \end{aligned}$$</span></div></div><p>Putting all together we finally get</p><div id="Equ111" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} e({\mathfrak {M}}(H_n, {{\,\textrm{SL}\,}}_3({\mathbb {C}})))&= q^{4} + \frac{1}{2} {\left( q^{6} + 2 q^{5} - 4 q^{4} + q^{3} + 3 q^{2} - 3 q + 2\right) } {\left( n^{2} - 3 n + 2\right) } \\&\quad - {\left( q^{3} - 2 q - 1\right) } \left\lfloor \frac{n-1}{2} \right\rfloor + 3 {\left( q^{4} - q^{3} + q^{2} - q + 1\right) } {\left( n - 1\right) } \\&\quad - (q^{2} + 1){\left( n - 2\right) } + q^{3}(n -1). \end{aligned}$$</span></div></div></div></div></section> </div> <section data-title="Data Availability Statement"><div class="c-article-section" id="data-availability-statement-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="data-availability-statement">Data Availability Statement</h2><div class="c-article-section__content" id="data-availability-statement-content"> <p>This work has no associated data.</p> </div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Cavazos, S., Lawton, S.: E-polynomial of <span class="mathjax-tex">$SL_2({\mathbb{C} })$</span>-character varieties of free groups. 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The first author is partially supported by Madrid Government (Comunidad de Madrid - Spain) under the Multiannual Agreement with the Universidad Complutense de Madrid in the line Research Incentive for Young PhDs, in the context of the V PRICIT (Regional Programme of Research and Technological Innovation) through the project PR27/21-029 and by Project MCI (Spain) PID2019-106493RB-I00. The second author is partially supported by Project MCI (Spain) PID2020-118452GB-I00.</p></div></div></section><section data-title="Funding"><div class="c-article-section" id="Fun-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Fun">Funding</h2><div class="c-article-section__content" id="Fun-content"><p>Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040, Madrid, Spain</p><p class="c-article-author-affiliation__authors-list">Ángel González-Prieto</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13-15, 28049, Madrid, Spain</p><p class="c-article-author-affiliation__authors-list">Ángel González-Prieto & Vicente Muñoz</p></li><li id="Aff3"><p class="c-article-author-affiliation__address">Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos S/N, 29071, Málaga, Spain</p><p class="c-article-author-affiliation__authors-list">Vicente Muñoz</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-_ngel-Gonz_lez_Prieto-Aff1-Aff2"><span class="c-article-authors-search__title u-h3 js-search-name">Ángel González-Prieto</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?dc.creator=%C3%81ngel%20Gonz%C3%A1lez-Prieto" class="c-article-button" data-track="click" data-track-action="author link - 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">González-Prieto, Á., Muñoz, V. Representation Varieties of Twisted Hopf Links. <i>Mediterr. J. Math.</i> <b>20</b>, 89 (2023). https://doi.org/10.1007/s00009-023-02300-w</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s00009-023-02300-w?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2022-02-22">22 February 2022</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Revised<span 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