Geometric Topology

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href=/list/math.GT/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> <dl id='articles'> <h3>New submissions (showing 6 of 6 entries)</h3> <dt> <a name='item1'>[1]</a> <a href ="/abs/2504.01254" title="Abstract" id="2504.01254"> arXiv:2504.01254 </a> [<a href="/pdf/2504.01254" title="Download PDF" id="pdf-2504.01254" aria-labelledby="pdf-2504.01254">pdf</a>, <a href="https://arxiv.org/html/2504.01254v1" title="View HTML" id="html-2504.01254" aria-labelledby="html-2504.01254" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2504.01254" title="Other formats" id="oth-2504.01254" aria-labelledby="oth-2504.01254">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A robot that unknots knots </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Hui,+C+O+Y">Connie On Yu Hui</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Ibarra,+D">Dionne Ibarra</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Kauffman,+L+H">Louis H. Kauffman</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=McQuire,+E+N">Emma N. McQuire</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Montoya-Vega,+G">Gabriel Montoya-Vega</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Mukherjee,+S">Sujoy Mukherjee</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Reid,+C">Corbin Reid</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 30 pages, 28 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Geometric Topology (math.GT)</span>; Combinatorics (math.CO) </div> <p class='mathjax'> Consider a robot that walks along a knot once on a knot diagram and switches every undercrossing it meets, stopping when it comes back to the starting position. We show that such a robot always unknots the knot. In fact, we prove that the robot produces an ascending diagram, and we provide a purely combinatorial proof that every ascending or descending knot diagram with C crossings can be transformed into the zero-crossing unknot diagram using at most (7C+1)C Reidemeister moves. Moreover, we show that an ascending or descending knot diagram can always be transformed into a zero-crossing unknot diagram using Reidemeister moves that do not increase the number of crossings. </p> </div> </dd> <dt> <a name='item2'>[2]</a> <a href ="/abs/2504.01436" title="Abstract" id="2504.01436"> arXiv:2504.01436 </a> [<a href="/pdf/2504.01436" title="Download PDF" id="pdf-2504.01436" aria-labelledby="pdf-2504.01436">pdf</a>, <a href="https://arxiv.org/html/2504.01436v1" title="View HTML" id="html-2504.01436" aria-labelledby="html-2504.01436" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2504.01436" title="Other formats" id="oth-2504.01436" aria-labelledby="oth-2504.01436">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> New perspectives on a classical embedding theorem </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Crabb,+M+C">M. C. Crabb</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Geometric Topology (math.GT)</span>; Algebraic Topology (math.AT) </div> <p class='mathjax'> In this expository note, recent results of Kishimoto and Matsushita on triangulated manifolds are linked to the classical criterion on the normal Stiefel-Whitney classes for existence of an embedding of a smooth closed manifold into Euclidean space of given dimension. We also look back at Atiyah's K-theoretic condition for the existence of a smooth embedding. </p> </div> </dd> <dt> <a name='item3'>[3]</a> <a href ="/abs/2504.01494" title="Abstract" id="2504.01494"> arXiv:2504.01494 </a> [<a href="/pdf/2504.01494" title="Download PDF" id="pdf-2504.01494" aria-labelledby="pdf-2504.01494">pdf</a>, <a href="https://arxiv.org/html/2504.01494v1" title="View HTML" id="html-2504.01494" aria-labelledby="html-2504.01494" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2504.01494" title="Other formats" id="oth-2504.01494" aria-labelledby="oth-2504.01494">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Zariski-Closures of Linear Reflection Groups </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Audibert,+J">Jacques Audibert</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Douba,+S">Sami Douba</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Lee,+G">Gye-Seon Lee</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Marquis,+L">Ludovic Marquis</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 34 pages, 4 figures. Comments welcome </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Geometric Topology (math.GT)</span>; Group Theory (math.GR) </div> <p class='mathjax'> We give necessary and sufficient conditions for a linear reflection group in the sense of Vinberg to be Zariski-dense in the ambient projective general linear group. As an application, we show that every irreducible right-angled Coxeter group of rank $N \geq 3$ virtually embeds Zariski-densely in $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq N$. This allows us to settle the existence of Zariski-dense surface subgroups of $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq 3$. Among the other applications are examples of Zariski-dense one-ended finitely generated subgroups of $\mathrm{SL}_n(\mathbb{Z})$ that are not finitely presented for all $n \geq 6$. </p> </div> </dd> <dt> <a name='item4'>[4]</a> <a href ="/abs/2504.01701" title="Abstract" id="2504.01701"> arXiv:2504.01701 </a> [<a href="/pdf/2504.01701" title="Download PDF" id="pdf-2504.01701" aria-labelledby="pdf-2504.01701">pdf</a>, <a href="https://arxiv.org/html/2504.01701v1" title="View HTML" id="html-2504.01701" aria-labelledby="html-2504.01701" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2504.01701" title="Other formats" id="oth-2504.01701" aria-labelledby="oth-2504.01701">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Twisted local G-wild mapping class groups </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Dou%C3%A7ot,+J">Jean Dou莽ot</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Rembado,+G">Gabriele Rembado</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Yamakawa,+D">Daisuke Yamakawa</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 52 pages: comments welcome! </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Geometric Topology (math.GT)</span>; Algebraic Geometry (math.AG); Exactly Solvable and Integrable Systems (nlin.SI) </div> <p class='mathjax'> We consider the (universal) local isomonodromic deformations of irregular-singular connections defined on principal bundles over complex curves: for any complex reductive structure group G, any pole order, and allowing for twisted/ramified formal normal forms at each pole. This covers the general case, and we particularly study the fundamental groups of the spaces of admissible deformations of irregular types/classes, in the viewpoint of (twisted/nonsplit) reflections cosets. </p> </div> </dd> <dt> <a name='item5'>[5]</a> <a href ="/abs/2504.01714" title="Abstract" id="2504.01714"> arXiv:2504.01714 </a> [<a href="/pdf/2504.01714" title="Download PDF" id="pdf-2504.01714" aria-labelledby="pdf-2504.01714">pdf</a>, <a href="https://arxiv.org/html/2504.01714v1" title="View HTML" id="html-2504.01714" aria-labelledby="html-2504.01714" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2504.01714" title="Other formats" id="oth-2504.01714" aria-labelledby="oth-2504.01714">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On Thompson knot theory and conjugacy classes of Thompson's group $F$ </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Bao,+Y">Yuanyuan Bao</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Sheng,+X">Xiaobing Sheng</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Geometric Topology (math.GT)</span> </div> <p class='mathjax'> Jones introduced a method to produce unoriented links from elements of the Thompson's group $F$, and proved that any link can be produced by this construction. In this paper, we attempt to investigate the relations between conjugacy classes of the group $F$ and the links being constructed. For each unoriented link $L$, we find a sequence of elements of $F$ from distinct conjugacy classes which yield $L$ via Jones's construction. We also show that a sequence of $2$-bridge links can be constructed from elements in the conjugacy class of $x_0$ (resp. $x_1$). </p> </div> </dd> <dt> <a name='item6'>[6]</a> <a href ="/abs/2504.01747" title="Abstract" id="2504.01747"> arXiv:2504.01747 </a> [<a href="/pdf/2504.01747" title="Download PDF" id="pdf-2504.01747" aria-labelledby="pdf-2504.01747">pdf</a>, <a href="/format/2504.01747" title="Other formats" id="oth-2504.01747" aria-labelledby="oth-2504.01747">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> The untangling number of 3-periodic tangles </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Andriamanalina,+T">Toky Andriamanalina</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Mahmoudi,+S">Sonia Mahmoudi</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Evans,+M+E">Myfanwy E. Evans</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Geometric Topology (math.GT)</span> </div> <p class='mathjax'> The entanglement of curves within a 3-periodic box provides a model for complicated space-filling entangled structures occurring in biological materials and structural chemistry. Quantifying the complexity of the entanglement within these models enhances the characterisation of these structures. In this paper, we introduce a new measure of entanglement complexity through the untangling number, reminiscent of the unknotting number in knot theory. The untangling number quantifies the minimum distance between a given 3-periodic structure and its least tangled version, called ground state, through a sequence of operations in a diagrammatic representation of the structure. For entanglements that consist of only infinite open curves, we show that the generic ground states of these structures are crystallographic rod packings, well-known in structural chemistry. </p> </div> </dd> </dl> <dl id='articles'> <h3>Replacement submissions (showing 5 of 5 entries)</h3> <dt> <a name='item7'>[7]</a> <a href ="/abs/2205.08802" title="Abstract" id="2205.08802"> arXiv:2205.08802 </a> (replaced) [<a href="/pdf/2205.08802" title="Download PDF" id="pdf-2205.08802" aria-labelledby="pdf-2205.08802">pdf</a>, <a href="https://arxiv.org/html/2205.08802v3" title="View HTML" id="html-2205.08802" aria-labelledby="html-2205.08802" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2205.08802" title="Other formats" id="oth-2205.08802" aria-labelledby="oth-2205.08802">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Recognising elliptic manifolds </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Lackenby,+M">Marc Lackenby</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Schleimer,+S">Saul Schleimer</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 36 pages, 4 figures, v3 - changes made to reflect referee comments </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Geometric Topology (math.GT)</span> </div> <p class='mathjax'> We show that the problem of deciding whether a closed three-manifold admits an elliptic structure lies in NP. Furthermore, determining the homeomorphism type of an elliptic manifold lies in the complexity class FNP. These are both consequences of the following result. Suppose that $M$ is a lens space which is neither $\mathbb{RP}^3$ nor a prism manifold. Suppose that $\mathcal{T}$ is a triangulation of $M$. Then there is a loop, in the one-skeleton of the 86th iterated barycentric subdivision of $\mathcal{T}$, whose simplicial neighbourhood is a Heegaard solid torus for $M$. </p> </div> </dd> <dt> <a name='item8'>[8]</a> <a href ="/abs/2312.05823" title="Abstract" id="2312.05823"> arXiv:2312.05823 </a> (replaced) [<a href="/pdf/2312.05823" title="Download PDF" id="pdf-2312.05823" aria-labelledby="pdf-2312.05823">pdf</a>, <a href="https://arxiv.org/html/2312.05823v2" title="View HTML" id="html-2312.05823" aria-labelledby="html-2312.05823" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2312.05823" title="Other formats" id="oth-2312.05823" aria-labelledby="oth-2312.05823">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Tight Contact Structures on Contact Mapping Tori and their Folded Sums </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Arikan,+M+F">M. Firat Arikan</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 14 pages, 2 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Geometric Topology (math.GT)</span>; Symplectic Geometry (math.SG) </div> <p class='mathjax'> It is known that the folded sum of two contact mapping tori whose fibers are compact exact symplectic manifolds having a common convex boundary (called the ``fold'') admits a cooriented contact structure compatible with the obvious fibration map onto the circle. Here we first provide an alternative bundle-theoretical construction of such a ``folded'' contact structure based on a gluing process near the fold. Moreover, we prove that in any odd dimension $2n+1\geq 7$ a folded contact structure on a folded sum of two contact mapping tori is tight if the induced contact form on the (common) contact fold admits no contractible Reeb orbit. In particular, any contact mapping torus of an odd dimension $2n+1\geq 7$ is tight if the induced contact form on the convex boundary of a fiber admits no contractible Reeb orbit. </p> </div> </dd> <dt> <a name='item9'>[9]</a> <a href ="/abs/2412.06572" title="Abstract" id="2412.06572"> arXiv:2412.06572 </a> (replaced) [<a href="/pdf/2412.06572" title="Download PDF" id="pdf-2412.06572" aria-labelledby="pdf-2412.06572">pdf</a>, <a href="https://arxiv.org/html/2412.06572v2" title="View HTML" id="html-2412.06572" aria-labelledby="html-2412.06572" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2412.06572" title="Other formats" id="oth-2412.06572" aria-labelledby="oth-2412.06572">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Quaternionic spinors and horospheres in 4-dimensional hyperbolic geometry </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Mathews,+D+V">Daniel V. Mathews</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Varsha">Varsha</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 56 pages, 3 figures. v2: updated references </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Geometric Topology (math.GT)</span>; Differential Geometry (math.DG) </div> <p class='mathjax'> We give explicit bijective correspondences between three families of objects: certain pairs of quaternions, which we regard as spinors; certain flags in (1+4)-dimensional Minkowski space; and horospheres in 4-dimensional hyperbolic space decorated with certain pairs of spinorial directions. These correspondences generalise previous work of the first author, Penrose--Rindler, and Penner in lower dimensions, and use the description of 4-dimensional hyperbolic isometries via Clifford matrices studied by Ahlfors and others. <br>We show that lambda lengths generalise to 4 dimensions, where they take quaternionic values, and are given by a certain bilinear form on quaternionic spinors. They satisfy a non-commutative Ptolemy equation, arising from quasi-Pl眉cker relations in the Gel'fand--Retakh theory of noncommutative determinants. We also study various structures of geometric and topological interest that arise in the process. </p> </div> </dd> <dt> <a name='item10'>[10]</a> <a href ="/abs/2212.11014" title="Abstract" id="2212.11014"> arXiv:2212.11014 </a> (replaced) [<a href="/pdf/2212.11014" title="Download PDF" id="pdf-2212.11014" aria-labelledby="pdf-2212.11014">pdf</a>, <a href="https://arxiv.org/html/2212.11014v3" title="View HTML" id="html-2212.11014" aria-labelledby="html-2212.11014" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2212.11014" title="Other formats" id="oth-2212.11014" aria-labelledby="oth-2212.11014">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Rigidity of mapping class groups mod powers of twists </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Mangioni,+G">Giorgio Mangioni</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Sisto,+A">Alessandro Sisto</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 68 pages, 30 figures. V3: rearranged to match published version (Proc. R. Soc. Edinb., Sect. A, Math.). The proof of quasi-isometric rigidity of pants graphs is now an Appendix. Comments are encouraged! </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Group Theory (math.GR)</span>; Geometric Topology (math.GT) </div> <p class='mathjax'> We study quotients of mapping class groups of punctured spheres by suitable large powers of Dehn twists, showing an analogue of Ivanov's theorem for the automorphisms of the corresponding quotients of curve graphs. Then we use this result to prove quasi-isometric rigidity of these quotients, answering a question of Behrstock, Hagen, Martin, and Sisto in the case of punctured spheres. Finally, we show that the automorphism groups of our quotients of mapping class groups are "small", as are their abstract commensurators. This is again an analogue of a theorem of Ivanov about the automorphism group of the mapping class group. In the process we develop techniques to extract combinatorial data from a quasi-isometry of a hierarchically hyperbolic space, and use them to give a different proof of a result of Bowditch about quasi-isometric rigidity of pants graphs of punctured spheres. </p> </div> </dd> <dt> <a name='item11'>[11]</a> <a href ="/abs/2503.16267" title="Abstract" id="2503.16267"> arXiv:2503.16267 </a> (replaced) [<a href="/pdf/2503.16267" title="Download PDF" id="pdf-2503.16267" aria-labelledby="pdf-2503.16267">pdf</a>, <a href="/format/2503.16267" title="Other formats" id="oth-2503.16267" aria-labelledby="oth-2503.16267">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Enumerating Smooth Structures on $\mathbb{C}P^3\times\mathbb{S}^k$ </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Basu,+S">Samik Basu</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Kasilingam,+R">Ramesh Kasilingam</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Sarkar,+A">Ankur Sarkar</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 34 pages. The Abstract and title have been changed. Comments are welcome </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Topology (math.AT)</span>; Geometric Topology (math.GT) </div> <p class='mathjax'> In this paper, we compute the concordance inertia group of the product $M \times \mathbb{S}^k$, where $M$ is a simply connected, closed, smooth 6-manifold, for $1 \leq k \leq 10$, using known low-dimensional computations of the stable homotopy groups of spheres. Specifically, for $M = \mathbb{C}P^3$, we determine the inertia group of $\mathbb{C}P^3 \times \mathbb{S}^k$ for $2 \leq k \leq 7, k \neq 6$, and establish a diffeomorphism classification of all smooth manifolds homeomorphic to $\mathbb{C}P^3 \times \mathbb{S}^k$ for $1 \leq k \leq 7$. </p> </div> </dd> </dl> <div class='paging'>Total of 11 entries </div> <div class='morefewer'>Showing up to 2000 entries per page: <a href=/list/math.GT/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> </div> </div> </div> </main> <footer style="clear: both;"> <div class="columns is-desktop" role="navigation" aria-label="Secondary" style="margin: -0.75em -0.75em 0.75em -0.75em">  <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul style="list-style: none; 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