We describe a geometric compactification of the moduli stack of left invariant complex structures on a fixed real Lie group or a fixed quotient. The extra points are CR structures transverse to a real foliation.
我们描述了固定实李群或固定商上左不变复结构模数堆栈的几何压缩。外点是横跨实叶的 CR 结构。
{"title":"A Geometric Compactification Of The Moduli Stack Of Left Invariant Complex Structures On A Lie Group","authors":"Laurent Meersseman","doi":"arxiv-2408.16182","DOIUrl":"https://doi.org/arxiv-2408.16182","url":null,"abstract":"We describe a geometric compactification of the moduli stack of left\u0000invariant complex structures on a fixed real Lie group or a fixed quotient. The\u0000extra points are CR structures transverse to a real foliation.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The cactus group was introduced by Henriques and Kamnitzer as an analogue of the braid group. In this note, we provide an explicit description of the relationship between the pure cactus group of degree three and the configuration space of four points on the circle.
{"title":"A note on the pure cactus group of degree three and the configuration space of four points on the circle","authors":"Takatoshi Hama, Kazuhiro Ichihara","doi":"arxiv-2408.15478","DOIUrl":"https://doi.org/arxiv-2408.15478","url":null,"abstract":"The cactus group was introduced by Henriques and Kamnitzer as an analogue of\u0000the braid group. In this note, we provide an explicit description of the\u0000relationship between the pure cactus group of degree three and the\u0000configuration space of four points on the circle.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Benard-Conway invariant of links in the 3-sphere is a Casson-Lin type invariant defined by counting irreducible SU(2) representations of the link group with fixed meridional traces. For two-component links with linking number one, the invariant has been shown to equal a symmetrized multivariable link signature. We extend this result to all two-component links with non-zero linking number. A key ingredient in the proof is an explicit calculation of the Benard-Conway invariant for (2, 2l)-torus links with the help of the Chebyshev polynomials.
{"title":"The Benard-Conway invariant of two-component links","authors":"Zedan Liu, Nikolai Saveliev","doi":"arxiv-2408.16161","DOIUrl":"https://doi.org/arxiv-2408.16161","url":null,"abstract":"The Benard-Conway invariant of links in the 3-sphere is a Casson-Lin type\u0000invariant defined by counting irreducible SU(2) representations of the link\u0000group with fixed meridional traces. For two-component links with linking number\u0000one, the invariant has been shown to equal a symmetrized multivariable link\u0000signature. We extend this result to all two-component links with non-zero\u0000linking number. A key ingredient in the proof is an explicit calculation of the\u0000Benard-Conway invariant for (2, 2l)-torus links with the help of the Chebyshev\u0000polynomials.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe an efficient algorithm to compute finite type invariants of type $k$ by first creating, for a given knot $K$ with $n$ crossings, a look-up table for all subdiagrams of $K$ of size $lceil frac{k}{2}rceil$ indexed by dyadic intervals in $[0,2n-1]$. Using this algorithm, any such finite type invariant can be computed on an $n$-crossing knot in time $sim n^{lceil frac{k}{2}rceil}$, a lot faster than the previously best published bound of $sim n^k$.
{"title":"Computing Finite Type Invariants Efficiently","authors":"Dror Bar-Natan, Itai Bar-Natan, Iva Halacheva, Nancy Scherich","doi":"arxiv-2408.15942","DOIUrl":"https://doi.org/arxiv-2408.15942","url":null,"abstract":"We describe an efficient algorithm to compute finite type invariants of type\u0000$k$ by first creating, for a given knot $K$ with $n$ crossings, a look-up table\u0000for all subdiagrams of $K$ of size $lceil frac{k}{2}rceil$ indexed by dyadic\u0000intervals in $[0,2n-1]$. Using this algorithm, any such finite type invariant\u0000can be computed on an $n$-crossing knot in time $sim n^{lceil\u0000frac{k}{2}rceil}$, a lot faster than the previously best published bound of\u0000$sim n^k$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Peter Feller, Lukas Lewark, Miguel Orbegozo Rodriguez
We show that closures of homogeneous braids are visually prime, addressing a question of Cromwell. The key technical tool for the proof is the following criterion concerning primeness of open books, which we consider to be of independent interest. For open books of 3-manifolds the property of having no fixed essential arcs is preserved under essential Murasugi sums with a strictly right-veering open book, if the plumbing region of the original open book veers to the left. We also provide examples of open books in S^3 demonstrating that primeness is not necessarily preserved under essential Murasugi sum, in fact not even under stabilizations a.k.a. Hopf plumbings. Furthermore, we find that trefoil plumbings need not preserve primeness. In contrast, we establish that figure-eight knot plumbings do preserve primeness.
{"title":"Homogeneous braids are visually prime","authors":"Peter Feller, Lukas Lewark, Miguel Orbegozo Rodriguez","doi":"arxiv-2408.15730","DOIUrl":"https://doi.org/arxiv-2408.15730","url":null,"abstract":"We show that closures of homogeneous braids are visually prime, addressing a\u0000question of Cromwell. The key technical tool for the proof is the following\u0000criterion concerning primeness of open books, which we consider to be of\u0000independent interest. For open books of 3-manifolds the property of having no\u0000fixed essential arcs is preserved under essential Murasugi sums with a strictly\u0000right-veering open book, if the plumbing region of the original open book veers\u0000to the left. We also provide examples of open books in S^3 demonstrating that\u0000primeness is not necessarily preserved under essential Murasugi sum, in fact\u0000not even under stabilizations a.k.a. Hopf plumbings. Furthermore, we find that\u0000trefoil plumbings need not preserve primeness. In contrast, we establish that\u0000figure-eight knot plumbings do preserve primeness.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A weaving knot is an alternating knot whose minimal diagram is a closed braid of a lattice-like pattern. In this paper, upper bounds of the unknotting number and the region unknotting number for some families of weaving knots are given by diagrammatical and combinatorial examination of the warping degree of weaving knot diagrams.
{"title":"A note on the unknotting number and the region unknotting number of weaving knots","authors":"Ayaka Shimizu, Amrendra Gill, Sahil Joshi","doi":"arxiv-2408.14938","DOIUrl":"https://doi.org/arxiv-2408.14938","url":null,"abstract":"A weaving knot is an alternating knot whose minimal diagram is a closed braid\u0000of a lattice-like pattern. In this paper, upper bounds of the unknotting number\u0000and the region unknotting number for some families of weaving knots are given\u0000by diagrammatical and combinatorial examination of the warping degree of\u0000weaving knot diagrams.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"132 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $M$ be a non-compact hyperbolic $3$-manifold with finite volume and totally geodesic boundary components. By subdividing mixed ideal polyhedral decompositions of $M$, under some certain topological conditions, we prove that $M$ has an ideal triangulation which admits an angle structure.
{"title":"Angle structure on general hyperbolic 3-manifolds","authors":"Ge Huabin, Jia Longsong, Zhang Faze","doi":"arxiv-2408.14003","DOIUrl":"https://doi.org/arxiv-2408.14003","url":null,"abstract":"Let $M$ be a non-compact hyperbolic $3$-manifold with finite volume and\u0000totally geodesic boundary components. By subdividing mixed ideal polyhedral\u0000decompositions of $M$, under some certain topological conditions, we prove that\u0000$M$ has an ideal triangulation which admits an angle structure.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish two spectral sequences in knot Floer homology associated to a directed strongly invertible knot K: one from the knot Floer homology of K to a two dimensional vector space, and one from the singular knot Floer homology of a singular knot associated to K to the knot Floer homology quotient knot of K. The first of these spectral sequences is used to define a numerical invariant of strongly invertible knots.
我们在与定向强可逆结 K 相关的结浮子同源性中建立了两个谱序列：一个是从 K 的结浮子同源性到二维向量空间，另一个是从与 K 相关的奇异结的奇异结浮子同源性到 K 的结浮子同源性商结。
{"title":"Localization and the Floer homology of strongly invertible knots","authors":"Aakash Parikh","doi":"arxiv-2408.13892","DOIUrl":"https://doi.org/arxiv-2408.13892","url":null,"abstract":"We establish two spectral sequences in knot Floer homology associated to a\u0000directed strongly invertible knot K: one from the knot Floer homology of K to a\u0000two dimensional vector space, and one from the singular knot Floer homology of\u0000a singular knot associated to K to the knot Floer homology quotient knot of K.\u0000The first of these spectral sequences is used to define a numerical invariant\u0000of strongly invertible knots.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is known that any periodic map of order $n$ on a closed oriented surface of genus $g$ can be equivariantly embedded into $S^m$ for some $m$. In the orientable and smooth category, we determine the smallest possible $m$ when $ngeq 3g$. We show that for each integer $k>1$ there exist infinitely many periodic maps such that the smallest possible $m$ is equal to $k$.
{"title":"Embedding periodic maps of surfaces into those of spheres with minimal dimensions","authors":"Chao Wang, Shicheng Wang, Zhongzi Wang","doi":"arxiv-2408.13749","DOIUrl":"https://doi.org/arxiv-2408.13749","url":null,"abstract":"It is known that any periodic map of order $n$ on a closed oriented surface\u0000of genus $g$ can be equivariantly embedded into $S^m$ for some $m$. In the\u0000orientable and smooth category, we determine the smallest possible $m$ when\u0000$ngeq 3g$. We show that for each integer $k>1$ there exist infinitely many\u0000periodic maps such that the smallest possible $m$ is equal to $k$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study endperiodic maps of an infinite graph with finitely many ends. We prove that any such map is homotopic to an endperiodic relative train track map. Moreover, we show that the (largest) Perron-Frobenius eigenvalue of the transition matrix is a canonical quantity associated to the map.
{"title":"Relative train tracks and endperiodic graph maps","authors":"Yan Mary He, Chenxi Wu","doi":"arxiv-2408.13401","DOIUrl":"https://doi.org/arxiv-2408.13401","url":null,"abstract":"We study endperiodic maps of an infinite graph with finitely many ends. We\u0000prove that any such map is homotopic to an endperiodic relative train track\u0000map. Moreover, we show that the (largest) Perron-Frobenius eigenvalue of the\u0000transition matrix is a canonical quantity associated to the map.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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