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The search for alternating surgeries 寻求交替手术
arXiv - MATH - Geometric Topology Pub Date : 2024-09-15 DOI: arxiv-2409.09842
Kenneth L. Baker, Marc Kegel, Duncan McCoy
{"title":"The search for alternating surgeries","authors":"Kenneth L. Baker, Marc Kegel, Duncan McCoy","doi":"arxiv-2409.09842","DOIUrl":"https://doi.org/arxiv-2409.09842","url":null,"abstract":"Surgery on a knot in $S^3$ is said to be an alternating surgery if it yields\u0000the double branched cover of an alternating link. The main theoretical\u0000contribution is to show that the set of alternating surgery slopes is\u0000algorithmically computable and to establish several structural results.\u0000Furthermore, we calculate the set of alternating surgery slopes for many\u0000examples of knots, including all hyperbolic knots in the SnapPy census. These\u0000examples exhibit several interesting phenomena including strongly invertible\u0000knots with a unique alternating surgery and asymmetric knots with two\u0000alternating surgery slopes. We also establish upper bounds on the set of\u0000alternating surgeries, showing that an alternating surgery slope on a\u0000hyperbolic knot satisfies $|p/q| leq 3g(K)+4$. Notably, this bound applies to\u0000lens space surgeries, thereby strengthening the known genus bounds from the\u0000conjecture of Goda and Teragaito.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256734","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}
引用次数: 0
On the Kauffman bracket skein module of a class of small Seifert manifolds 论一类小塞弗流形的考夫曼括号矢模
arXiv - MATH - Geometric Topology Pub Date : 2024-09-14 DOI: arxiv-2409.09438
Minyi Liang, Shangjun Shi, Xiao Wang
{"title":"On the Kauffman bracket skein module of a class of small Seifert manifolds","authors":"Minyi Liang, Shangjun Shi, Xiao Wang","doi":"arxiv-2409.09438","DOIUrl":"https://doi.org/arxiv-2409.09438","url":null,"abstract":"In this paper, we provide a presentation of the Kauffman bracket skein module\u0000for each small Seifert manifold. As one application, we demonstrate how to get\u0000the Kauffman bracket skein module of lens spaces from our main theorem.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256736","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}
引用次数: 0
Existence of 5 minimal tori in 3-spheres of positive Ricci curvature 正利玛窦曲率的 3 球体中存在 5 个最小转矩
arXiv - MATH - Geometric Topology Pub Date : 2024-09-14 DOI: arxiv-2409.09315
Adrian Chun-Pong Chu, Yangyang Li
{"title":"Existence of 5 minimal tori in 3-spheres of positive Ricci curvature","authors":"Adrian Chun-Pong Chu, Yangyang Li","doi":"arxiv-2409.09315","DOIUrl":"https://doi.org/arxiv-2409.09315","url":null,"abstract":"In 1989, B. White conjectured that every Riemannian 3-sphere has at least 5\u0000embedded minimal tori. We confirm this conjecture for 3-spheres of positive\u0000Ricci curvature. While our proof uses min-max theory, the underlying heuristics\u0000are largely inspired by mean curvature flow.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256742","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}
引用次数: 0
The polyhedral decomposition of cusped hyperbolic $n$-manifolds with totally geodesic boundary 具有完全大地边界的尖顶双曲$n$网格的多面体分解
arXiv - MATH - Geometric Topology Pub Date : 2024-09-13 DOI: arxiv-2409.08923
Ge Huabin, Jia Longsong, Zhang Faze
{"title":"The polyhedral decomposition of cusped hyperbolic $n$-manifolds with totally geodesic boundary","authors":"Ge Huabin, Jia Longsong, Zhang Faze","doi":"arxiv-2409.08923","DOIUrl":"https://doi.org/arxiv-2409.08923","url":null,"abstract":"Let $M$ be a volume finite non-compact complete hyperbolic $n$-manifold with\u0000totally geodesic boundary. We show that there exists a polyhedral decomposition\u0000of $M$ such that each cell is either an ideal polyhedron or a partially\u0000truncated polyhedron with exactly one truncated face. This result parallels\u0000Epstein-Penner's ideal decomposition cite{EP} for cusped hyperbolic manifolds\u0000and Kojima's truncated polyhedron decomposition cite{Kojima} for compact\u0000hyperbolic manifolds with totally geodesic boundary. We take two different\u0000approaches to demonstrate the main result in this paper. We also show that the\u0000number of polyhedral decompositions of $M$ is finite.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256744","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}
引用次数: 0
Factor system for graphs and combinatorial HHS 图形和组合 HHS 的因子系统
arXiv - MATH - Geometric Topology Pub Date : 2024-09-13 DOI: arxiv-2409.08663
Jihoon Park
{"title":"Factor system for graphs and combinatorial HHS","authors":"Jihoon Park","doi":"arxiv-2409.08663","DOIUrl":"https://doi.org/arxiv-2409.08663","url":null,"abstract":"We relaxe the constraint on the domains of combinatorial HHS machinery so\u0000combinatorial HHS machinery works for most cubical curve graphs. As an\u0000application we extend the factor system machinery of the CAT(0) cube complex to\u0000the quasi-median graphs.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256743","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}
引用次数: 0
The unknotting number, hard unknot diagrams, and reinforcement learning 解结数、硬解结图和强化学习
arXiv - MATH - Geometric Topology Pub Date : 2024-09-13 DOI: arxiv-2409.09032
Taylor Applebaum, Sam Blackwell, Alex Davies, Thomas Edlich, András Juhász, Marc Lackenby, Nenad Tomašev, Daniel Zheng
{"title":"The unknotting number, hard unknot diagrams, and reinforcement learning","authors":"Taylor Applebaum, Sam Blackwell, Alex Davies, Thomas Edlich, András Juhász, Marc Lackenby, Nenad Tomašev, Daniel Zheng","doi":"arxiv-2409.09032","DOIUrl":"https://doi.org/arxiv-2409.09032","url":null,"abstract":"We have developed a reinforcement learning agent that often finds a minimal\u0000sequence of unknotting crossing changes for a knot diagram with up to 200\u0000crossings, hence giving an upper bound on the unknotting number. We have used\u0000this to determine the unknotting number of 57k knots. We took diagrams of\u0000connected sums of such knots with oppositely signed signatures, where the\u0000summands were overlaid. The agent has found examples where several of the\u0000crossing changes in an unknotting collection of crossings result in hyperbolic\u0000knots. Based on this, we have shown that, given knots $K$ and $K'$ that satisfy\u0000some mild assumptions, there is a diagram of their connected sum and $u(K) +\u0000u(K')$ unknotting crossings such that changing any one of them results in a\u0000prime knot. As a by-product, we have obtained a dataset of 2.6 million distinct\u0000hard unknot diagrams; most of them under 35 crossings. Assuming the additivity\u0000of the unknotting number, we have determined the unknotting number of 43 at\u0000most 12-crossing knots for which the unknotting number is unknown.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"117 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256741","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}
引用次数: 0
Counting subgroups via Mirzakhani's curve counting 通过米尔扎哈尼曲线计数法计数子群
arXiv - MATH - Geometric Topology Pub Date : 2024-09-12 DOI: arxiv-2409.08109
Dounnu Sasaki
{"title":"Counting subgroups via Mirzakhani's curve counting","authors":"Dounnu Sasaki","doi":"arxiv-2409.08109","DOIUrl":"https://doi.org/arxiv-2409.08109","url":null,"abstract":"Given a hyperbolic surface $Sigma$ of genus $g$ with $r$ cusps, Mirzakhani\u0000proved that the number of closed geodesics of length at most $L$ and of a given\u0000type is asymptotic to $cL^{6g-6+2r}$ for some $c>0$. Since a closed geodesic\u0000corresponds to a conjugacy class of the fundamental group $pi_1(Sigma )$, we\u0000extend this to the counting problem of conjugacy classes of finitely generated\u0000subgroups of $pi_1(Sigma )$. Using `half the sum of the lengths of the\u0000boundaries of the convex core of a subgroup' instead of the length of a closed\u0000geodesic, we prove that the number of such conjugacy classes is similarly\u0000asymptotic to $cL^{6g-6+2r}$ for some $c>0$. Furthermore, we see that this\u0000measurement for subgroups is `natural' within the framework of subset currents,\u0000which serve as a completion of weighted conjugacy classes of finitely generated\u0000subgroups of $pi_1(Sigma )$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192235","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}
引用次数: 0
One-cusped complex hyperbolic 2-manifolds 单凹凸复双曲2-漫场
arXiv - MATH - Geometric Topology Pub Date : 2024-09-12 DOI: arxiv-2409.08028
Martin Deraux, Matthew Stover
{"title":"One-cusped complex hyperbolic 2-manifolds","authors":"Martin Deraux, Matthew Stover","doi":"arxiv-2409.08028","DOIUrl":"https://doi.org/arxiv-2409.08028","url":null,"abstract":"This paper builds one-cusped complex hyperbolic $2$-manifolds by an explicit\u0000geometric construction. Specifically, for each odd $d ge 1$ there is a smooth\u0000projective surface $Z_d$ with $c_1^2(Z_d) = c_2(Z_d) = 6d$ and a smooth\u0000irreducible curve $E_d$ on $Z_d$ of genus one so that $Z_d smallsetminus E_d$\u0000admits a finite volume uniformization by the unit ball $mathbb{B}^2$ in\u0000$mathbb{C}^2$. This produces one-cusped complex hyperbolic $2$-manifolds of\u0000arbitrarily large volume. As a consequence, the $3$-dimensional nilmanifold of\u0000Euler number $12d$ bounds geometrically for all odd $d ge 1$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192237","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}
引用次数: 0
A short proof of the classification of higher rank invariant subvarieties in genus three 三属中高阶不变子变量分类的简短证明
arXiv - MATH - Geometric Topology Pub Date : 2024-09-11 DOI: arxiv-2409.07603
Paul Apisa
{"title":"A short proof of the classification of higher rank invariant subvarieties in genus three","authors":"Paul Apisa","doi":"arxiv-2409.07603","DOIUrl":"https://doi.org/arxiv-2409.07603","url":null,"abstract":"We give a new short proof of the classification of rank at least two\u0000invariant subvarieties in genus three, which is due to Aulicino, Nguyen, and\u0000Wright. The proof uses techniques developed in recent work of Apisa and Wright.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192236","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}
引用次数: 0
Exotic diffeomorphisms on $4$-manifolds with $b_2^+ = 2$ 具有 $b_2^+ = 2$ 的 $4$-manifolds 上的奇异衍射
arXiv - MATH - Geometric Topology Pub Date : 2024-09-11 DOI: arxiv-2409.07009
Haochen Qiu
{"title":"Exotic diffeomorphisms on $4$-manifolds with $b_2^+ = 2$","authors":"Haochen Qiu","doi":"arxiv-2409.07009","DOIUrl":"https://doi.org/arxiv-2409.07009","url":null,"abstract":"While the exotic diffeomorphisms turned out to be very rich, we know much\u0000less about the $b^+_2 =2$ case, as parameterized gauge-theoretic invariants are\u0000not well defined. In this paper we present a method (that is, comparing the\u0000winding number of parameter families) to find exotic diffeomorphisms on\u0000simply-connected smooth closed $4$-manifolds with $b^+_2 =2$, and as a result\u0000we obtain that $2mathbb{C}mathbb{P}^2 # 10 (-{mathbb{C}mathbb{P}^2})$\u0000admits exotic diffeomorphisms. This is currently the smallest known example of\u0000a closed $4$-manifold that supports exotic diffeomorphisms.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"137 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192016","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}
引用次数: 0
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