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is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.14417">arXiv:2411.14417</a> <span> [<a href="https://arxiv.org/pdf/2411.14417">pdf</a>, <a href="https://arxiv.org/ps/2411.14417">ps</a>, <a href="https://arxiv.org/format/2411.14417">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Construction of Lie algebra weight system kernel via Vogel algebra </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Khudoteplov%2C+D">Dmitry Khudoteplov</a>, <a href="/search/math?searchtype=author&query=Lanina%2C+E">Elena Lanina</a>, <a href="/search/math?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.14417v1-abstract-short" style="display: inline;"> We develop a method of constructing a kernel of Lie algebra weight system. A main tool we use in the analysis is Vogel's $螞$ algebra and the surrounding framework. As an example of a developed technique we explicitly provide all Jacobi diagrams lying in the kernel of $\mathfrak{sl}_N$ weight system at low orders. We also discuss consequences of the presence of the kernel in Lie algebra weight syst… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14417v1-abstract-full').style.display = 'inline'; document.getElementById('2411.14417v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.14417v1-abstract-full" style="display: none;"> We develop a method of constructing a kernel of Lie algebra weight system. A main tool we use in the analysis is Vogel's $螞$ algebra and the surrounding framework. As an example of a developed technique we explicitly provide all Jacobi diagrams lying in the kernel of $\mathfrak{sl}_N$ weight system at low orders. We also discuss consequences of the presence of the kernel in Lie algebra weight systems for detection of correlators in the 3D Chern-Simons topological field theory and for distinguishing of knots by the corresponding quantum knot invariants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14417v1-abstract-full').style.display = 'none'; document.getElementById('2411.14417v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.18525">arXiv:2410.18525</a> <span> [<a href="https://arxiv.org/pdf/2410.18525">pdf</a>, <a href="https://arxiv.org/format/2410.18525">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Planar decomposition of bipartite HOMFLY polynomials in symmetric representations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Anokhina%2C+A">A. Anokhina</a>, <a href="/search/math?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/math?searchtype=author&query=Morozov%2C+A">A. Morozov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.18525v1-abstract-short" style="display: inline;"> We generalize the recently discovered planar decomposition (Kauffman bracket) for the HOMFLY polynomials of bipartite knot/link diagrams to (anti)symmetrically colored HOMFLY polynomials. Cabling destroys planarity, but it is restored after projection to (anti)symmetric representations. This allows to go beyond arborescent calculus, which so far produced the majority of results for colored polynom… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.18525v1-abstract-full').style.display = 'inline'; document.getElementById('2410.18525v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.18525v1-abstract-full" style="display: none;"> We generalize the recently discovered planar decomposition (Kauffman bracket) for the HOMFLY polynomials of bipartite knot/link diagrams to (anti)symmetrically colored HOMFLY polynomials. Cabling destroys planarity, but it is restored after projection to (anti)symmetric representations. This allows to go beyond arborescent calculus, which so far produced the majority of results for colored polynomials. Technicalities include combinations of projectors, and these can be handled rigorously, without any guess-work -- what can be also useful for other considerations, where reliable quantization was so far unavailable. We explicitly provide simple examples of calculation of the HOMFLY polynomials in symmetric representations with the use of our planar technique. These examples reveal what we call the bipartite evolution and the bipartite decomposition of squares of $\mathcal{R}$-matrices eigenvalues in the antiparallel channel. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.18525v1-abstract-full').style.display = 'none'; document.getElementById('2410.18525v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.08724">arXiv:2407.08724</a> <span> [<a href="https://arxiv.org/pdf/2407.08724">pdf</a>, <a href="https://arxiv.org/format/2407.08724">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1140/epjc/s10052-024-13309-0">10.1140/epjc/s10052-024-13309-0 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Planar decomposition of the HOMFLY polynomial for bipartite knots and links </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Anokhina%2C+A">A. Anokhina</a>, <a href="/search/math?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/math?searchtype=author&query=Morozov%2C+A">A. Morozov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.08724v2-abstract-short" style="display: inline;"> The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but for a special class of bipartite diagrams made entirely from the anitparallel lock tangle. Many amusing and important knots and links can be described in this wa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.08724v2-abstract-full').style.display = 'inline'; document.getElementById('2407.08724v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.08724v2-abstract-full" style="display: none;"> The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but for a special class of bipartite diagrams made entirely from the anitparallel lock tangle. Many amusing and important knots and links can be described in this way, from twist and double braid knots to the celebrated Kanenobu knots for even parameters -- and for all of them the entire HOMFLY polynomials possess planar decomposition. This provides an approach to evaluation of HOMFLY polynomials, which is complementary to the arborescent calculus, and this opens a new direction to homological techniques, parallel to Khovanov-Rozansky generalisations of the Kauffman calculus. Moreover, this planar calculus is also applicable to other symmetric representations beyond the fundamental one, and to links which are not fully bipartite what is illustrated by examples of Kanenobu-like links. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.08724v2-abstract-full').style.display = 'none'; document.getElementById('2407.08724v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages, published version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> The European Physical Journal C 84 (2024) 990 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.02553">arXiv:2402.02553</a> <span> [<a href="https://arxiv.org/pdf/2402.02553">pdf</a>, <a href="https://arxiv.org/format/2402.02553">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Closed 4-braids and the Jones unknot conjecture </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Korzun%2C+D">Dmitriy Korzun</a>, <a href="/search/math?searchtype=author&query=Lanina%2C+E">Elena Lanina</a>, <a href="/search/math?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.02553v2-abstract-short" style="display: inline;"> The Jones problem is a question whether there is a non-trivial knot with the trivial Jones polynomial in one variable $q$. The answer to this fundamental question is still unknown despite numerous attempts to explore it. In braid presentation the case of 4-strand braids is already open. S. Bigelow showed in 2000 that if the Burau representation for four-strand braids is unfaithful, then there is a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02553v2-abstract-full').style.display = 'inline'; document.getElementById('2402.02553v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.02553v2-abstract-full" style="display: none;"> The Jones problem is a question whether there is a non-trivial knot with the trivial Jones polynomial in one variable $q$. The answer to this fundamental question is still unknown despite numerous attempts to explore it. In braid presentation the case of 4-strand braids is already open. S. Bigelow showed in 2000 that if the Burau representation for four-strand braids is unfaithful, then there is an infinite number of non-trivial knots with the trivial two-variable HOMFLY-PT polynomial and hence, with the trivial Jones polynomial, since it is obtained from the HOMFLY-PT polynomial by the specialisation of one of the variables $A=q^2$. In this paper, we study four-strand braids and ask whether there are non-trivial knots with the trivial Jones polynomial but a non-trivial HOMFLY-PT polynomial. We have discovered that there is a whole 1-parameter family, parameterised by the writhe number, of 2-variable polynomials that can be HOMFLY-PT polynomials of some knots. We explore various properties of the obtained hypothetical HOMFLY-PT polynomials and suggest several checks to test these formulas. A generalisation is also proposed for the case of a large number of strands. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02553v2-abstract-full').style.display = 'none'; document.getElementById('2402.02553v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.13095">arXiv:2308.13095</a> <span> [<a href="https://arxiv.org/pdf/2308.13095">pdf</a>, <a href="https://arxiv.org/ps/2308.13095">ps</a>, <a href="https://arxiv.org/format/2308.13095">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.nuclphysb.2023.116403">10.1016/j.nuclphysb.2023.116403 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Towards tangle calculus for Khovanov polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Anokhina%2C+A">A. Anokhina</a>, <a href="/search/math?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/math?searchtype=author&query=Morozov%2C+A">A. Morozov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2308.13095v1-abstract-short" style="display: inline;"> We provide new evidence that the tangle calculus and "evolution" are applicable to the Khovanov polynomials for families of long braids inside the knot diagram. We show that jumps in evolution, peculiar for superpolynomials, are much less abundant than it was originally expected. Namely, for torus and twist satellites of a fixed companion knot, the main (most complicated) contribution does not jum… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.13095v1-abstract-full').style.display = 'inline'; document.getElementById('2308.13095v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.13095v1-abstract-full" style="display: none;"> We provide new evidence that the tangle calculus and "evolution" are applicable to the Khovanov polynomials for families of long braids inside the knot diagram. We show that jumps in evolution, peculiar for superpolynomials, are much less abundant than it was originally expected. Namely, for torus and twist satellites of a fixed companion knot, the main (most complicated) contribution does not jump, all jumps are concentrated in the torus and twist part correspondingly, where these jumps are necessary to make the Khovanov polynomial positive. Among other things, this opens a way to define a jump-free part of the colored Khovanov polynomials, which differs from the naive colored polynomial just "infinitesimally". The separation between jumping and smooth parts involves a combination of Rasmussen index and a new knot invariant, which we call "Thickness". <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.13095v1-abstract-full').style.display = 'none'; document.getElementById('2308.13095v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Nuclear Physics B 998 (2023) 116403 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2210.07874">arXiv:2210.07874</a> <span> [<a href="https://arxiv.org/pdf/2210.07874">pdf</a>, <a href="https://arxiv.org/ps/2210.07874">ps</a>, <a href="https://arxiv.org/format/2210.07874">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2023.138138">10.1016/j.physletb.2023.138138 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Tug-the-hook symmetry for quantum 6j-symbols </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/math?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2210.07874v2-abstract-short" style="display: inline;"> We introduce a novel symmetry for quantum 6j-symbols, which we call the tug-the-hook symmetry. Unlike other known symmetries, it is applicable for any representations, including ones with multiplicities. We provide several evidences in favour of the tug-the-hook symmetry. First, this symmetry follows from the eigenvalue conjecture. Second, it is shown by several new examples of explicit coincidenc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.07874v2-abstract-full').style.display = 'inline'; document.getElementById('2210.07874v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2210.07874v2-abstract-full" style="display: none;"> We introduce a novel symmetry for quantum 6j-symbols, which we call the tug-the-hook symmetry. Unlike other known symmetries, it is applicable for any representations, including ones with multiplicities. We provide several evidences in favour of the tug-the-hook symmetry. First, this symmetry follows from the eigenvalue conjecture. Second, it is shown by several new examples of explicit coincidence of 6j-symbols with multiplicities. Third, the tug-the-hook symmetry for Wilson loops for knots in the 3d Chern-Simons theory implies the tug-the-hook symmetry for quantum 6j-symbols. An important implication of the analysis is the generalization of the tug-the-hook symmetry for the Chern-Simons Wilson loops to the case of links. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.07874v2-abstract-full').style.display = 'none'; document.getElementById('2210.07874v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">12 pages, published version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Physics Letters B 845 (2023) 138138 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.01585">arXiv:2208.01585</a> <span> [<a href="https://arxiv.org/pdf/2208.01585">pdf</a>, <a href="https://arxiv.org/format/2208.01585">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1140/epjc/s10052-022-10969-8">10.1140/epjc/s10052-022-10969-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Defect and degree of the Alexander polynomial </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/math?searchtype=author&query=Morozov%2C+A">A. Morozov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.01585v1-abstract-short" style="display: inline;"> Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in $q^{\pm 2}$ of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.01585v1-abstract-full').style.display = 'inline'; document.getElementById('2208.01585v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.01585v1-abstract-full" style="display: none;"> Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in $q^{\pm 2}$ of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of $C$-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.01585v1-abstract-full').style.display = 'none'; document.getElementById('2208.01585v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> The European Physical Journal C 82 (2022) 1022 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2207.02284">arXiv:2207.02284</a> <span> [<a href="https://arxiv.org/pdf/2207.02284">pdf</a>, <a href="https://arxiv.org/format/2207.02284">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1134/S0040577923070024">10.1134/S0040577923070024 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On an alternative stratification of knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/math?searchtype=author&query=Popolitov%2C+A">A. Popolitov</a>, <a href="/search/math?searchtype=author&query=Tselousov%2C+N">N. Tselousov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2207.02284v1-abstract-short" style="display: inline;"> We introduce an alternative stratification of knots: by the size of lattice on which a knot can be first met. Using this classification, we find ratio of unknots and knots with more than 10 minimal crossings inside different lattices and answer the question which knots can be realized inside $3\times 3$ and $5\times 5$ lattices. In accordance with previous research, the ratio of unknots decreases… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.02284v1-abstract-full').style.display = 'inline'; document.getElementById('2207.02284v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2207.02284v1-abstract-full" style="display: none;"> We introduce an alternative stratification of knots: by the size of lattice on which a knot can be first met. Using this classification, we find ratio of unknots and knots with more than 10 minimal crossings inside different lattices and answer the question which knots can be realized inside $3\times 3$ and $5\times 5$ lattices. In accordance with previous research, the ratio of unknots decreases exponentially with the growth of the lattice size. Our computational results are approved with theoretical estimates for amounts of knots with fixed crossing number lying inside lattices of given size. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.02284v1-abstract-full').style.display = 'none'; document.getElementById('2207.02284v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">12 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Theoretical and Mathematical Physics 216 (2023) 924-937 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2111.11751">arXiv:2111.11751</a> <span> [<a href="https://arxiv.org/pdf/2111.11751">pdf</a>, <a href="https://arxiv.org/format/2111.11751">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.nuclphysb.2021.115644">10.1016/j.nuclphysb.2021.115644 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/math?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/math?searchtype=author&query=Tselousov%2C+N">N. Tselousov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2111.11751v1-abstract-short" style="display: inline;"> We have recently proposed arXiv:2105.11565 a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with $SU(N)$ gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values. First, we discuss the computation of Vassiliev i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.11751v1-abstract-full').style.display = 'inline'; document.getElementById('2111.11751v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2111.11751v1-abstract-full" style="display: none;"> We have recently proposed arXiv:2105.11565 a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with $SU(N)$ gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values. First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots $T[2,2k+1]$. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank $N$ and the representation $R$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.11751v1-abstract-full').style.display = 'none'; document.getElementById('2111.11751v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Nuclear Physics B 974 (2022) 115644 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.11565">arXiv:2105.11565</a> <span> [<a href="https://arxiv.org/pdf/2105.11565">pdf</a>, <a href="https://arxiv.org/ps/2105.11565">ps</a>, <a href="https://arxiv.org/format/2105.11565">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2021.136727">10.1016/j.physletb.2021.136727 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Chern-Simons perturbative series revisited </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/math?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/math?searchtype=author&query=Tselousov%2C+N">N. Tselousov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.11565v2-abstract-short" style="display: inline;"> A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with $SU(N)$ gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra $ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods ha… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.11565v2-abstract-full').style.display = 'inline'; document.getElementById('2105.11565v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.11565v2-abstract-full" style="display: none;"> A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with $SU(N)$ gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra $ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.11565v2-abstract-full').style.display = 'none'; document.getElementById('2105.11565v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Physics Letters B 823 (2021) 136727 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <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 class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 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