In this article, we explore phenomena relating to quasi-alternating surgeries on knots, where a quasi-alternating surgery on a knot is a Dehn surgery yielding the double branched cover of a quasi-alternating link. Since the double branched cover of a quasi-alternating link is an L-space, quasi-alternating surgeries are special examples of L-space surgeries. We show that all SnapPy census L-space knots admit quasi-alternating surgeries except for the knots t09847 and o9_30634 which both do not have any quasi-alternating surgeries. In particular, this finishes Dunfield's classification of the L-space knots among all SnapPy census knots. In addition, we show that all asymmetric census L-space knots have exactly two quasi-alternating slopes that are consecutive integers. Similar behavior is observed for some of the Baker-Luecke asymmetric L-space knots. We also classify the quasi-alternating surgeries on torus knots and explore briefly the notion of formal L-space surgeries. This allows us to give examples of asymmetric formal L-spaces.
本文探讨了与结上的准交替手术有关的现象，其中结上的准交替手术是产生准交替链接双支盖的 Dehn 手术。由于准交替链接的双支盖是一个 L 空间，因此准交替手术是 L 空间手术的特例。我们证明，除了 t09847 和 o9_30634 这两个节点没有准交替手术之外，所有 SnapPy 普查 L 空间节点都有准交替手术。特别是，这完成了邓菲尔德对所有 SnapPy 普查结中 L 空间结的分类。此外，我们还证明了所有非对称普查 L 空间结都有两个连续整数的准交替斜率。一些贝克-吕克非对称 L 空间结也有类似行为。我们还对环状结上的准交替手术进行了分类，并简要探讨了形式 L 空间手术的概念。这使我们能够给出不对称形式 L 空间的例子。
{"title":"Quasi-alternating surgeries","authors":"Kenneth L. Baker, Marc Kegel, Duncan McCoy","doi":"arxiv-2409.09839","DOIUrl":"https://doi.org/arxiv-2409.09839","url":null,"abstract":"In this article, we explore phenomena relating to quasi-alternating surgeries\u0000on knots, where a quasi-alternating surgery on a knot is a Dehn surgery\u0000yielding the double branched cover of a quasi-alternating link. Since the\u0000double branched cover of a quasi-alternating link is an L-space,\u0000quasi-alternating surgeries are special examples of L-space surgeries. We show that all SnapPy census L-space knots admit quasi-alternating\u0000surgeries except for the knots t09847 and o9_30634 which both do not have any\u0000quasi-alternating surgeries. In particular, this finishes Dunfield's\u0000classification of the L-space knots among all SnapPy census knots. In addition,\u0000we show that all asymmetric census L-space knots have exactly two\u0000quasi-alternating slopes that are consecutive integers. Similar behavior is\u0000observed for some of the Baker-Luecke asymmetric L-space knots. We also classify the quasi-alternating surgeries on torus knots and explore\u0000briefly the notion of formal L-space surgeries. This allows us to give examples\u0000of asymmetric formal L-spaces.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256738","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}
In this paper, we provide a presentation of the Kauffman bracket skein module for each small Seifert manifold. As one application, we demonstrate how to get the Kauffman bracket skein module of lens spaces from our main theorem.
{"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}
In 1989, B. White conjectured that every Riemannian 3-sphere has at least 5 embedded minimal tori. We confirm this conjecture for 3-spheres of positive Ricci curvature. While our proof uses min-max theory, the underlying heuristics are largely inspired by mean curvature flow.
{"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}
Let $M$ be a volume finite non-compact complete hyperbolic $n$-manifold with totally geodesic boundary. We show that there exists a polyhedral decomposition of $M$ such that each cell is either an ideal polyhedron or a partially truncated polyhedron with exactly one truncated face. This result parallels Epstein-Penner's ideal decomposition cite{EP} for cusped hyperbolic manifolds and Kojima's truncated polyhedron decomposition cite{Kojima} for compact hyperbolic manifolds with totally geodesic boundary. We take two different approaches to demonstrate the main result in this paper. We also show that the number of polyhedral decompositions of $M$ is finite.
{"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}
We relaxe the constraint on the domains of combinatorial HHS machinery so combinatorial HHS machinery works for most cubical curve graphs. As an application we extend the factor system machinery of the CAT(0) cube complex to the quasi-median graphs.
{"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}
Taylor Applebaum, Sam Blackwell, Alex Davies, Thomas Edlich, András Juhász, Marc Lackenby, Nenad Tomašev, Daniel Zheng
We have developed a reinforcement learning agent that often finds a minimal sequence of unknotting crossing changes for a knot diagram with up to 200 crossings, hence giving an upper bound on the unknotting number. We have used this to determine the unknotting number of 57k knots. We took diagrams of connected sums of such knots with oppositely signed signatures, where the summands were overlaid. The agent has found examples where several of the crossing changes in an unknotting collection of crossings result in hyperbolic knots. Based on this, we have shown that, given knots $K$ and $K'$ that satisfy some mild assumptions, there is a diagram of their connected sum and $u(K) + u(K')$ unknotting crossings such that changing any one of them results in a prime knot. As a by-product, we have obtained a dataset of 2.6 million distinct hard unknot diagrams; most of them under 35 crossings. Assuming the additivity of the unknotting number, we have determined the unknotting number of 43 at most 12-crossing knots for which the unknotting number is unknown.
{"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}
Given a hyperbolic surface $Sigma$ of genus $g$ with $r$ cusps, Mirzakhani proved that the number of closed geodesics of length at most $L$ and of a given type is asymptotic to $cL^{6g-6+2r}$ for some $c>0$. Since a closed geodesic corresponds to a conjugacy class of the fundamental group $pi_1(Sigma )$, we extend this to the counting problem of conjugacy classes of finitely generated subgroups of $pi_1(Sigma )$. Using `half the sum of the lengths of the boundaries of the convex core of a subgroup' instead of the length of a closed geodesic, we prove that the number of such conjugacy classes is similarly asymptotic to $cL^{6g-6+2r}$ for some $c>0$. Furthermore, we see that this measurement for subgroups is `natural' within the framework of subset currents, which serve as a completion of weighted conjugacy classes of finitely generated subgroups of $pi_1(Sigma )$.
{"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}
This paper builds one-cusped complex hyperbolic $2$-manifolds by an explicit geometric construction. Specifically, for each odd $d ge 1$ there is a smooth projective surface $Z_d$ with $c_1^2(Z_d) = c_2(Z_d) = 6d$ and a smooth irreducible curve $E_d$ on $Z_d$ of genus one so that $Z_d smallsetminus E_d$ admits a finite volume uniformization by the unit ball $mathbb{B}^2$ in $mathbb{C}^2$. This produces one-cusped complex hyperbolic $2$-manifolds of arbitrarily large volume. As a consequence, the $3$-dimensional nilmanifold of Euler number $12d$ bounds geometrically for all odd $d ge 1$.
本文通过显式几何构造建立了单弦复双曲$2$-manifolds。具体地说，对于每个奇数 $d ge 1$，都有一个光滑的投影面 $Z_d$，其上有$c_1^2(Z_d) = c_2(Z_d) = 6d$和一条光滑的可还原曲线 $E_d$ on $Z_d$ of genus one，这样 $Z_d smallsetminus E_d$ 就满足了单位球 $mathbb{B}^2$ inmathbb{C}^2$ 的有限体积均匀化。这就产生了任意大体积的单瓣复双曲$2$-manifolds。因此，对于所有奇数$d ge 1$，欧拉数$12d$的$3$维零芒福德在几何上都是有边界的。
{"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}
We give a new short proof of the classification of rank at least two invariant subvarieties in genus three, which is due to Aulicino, Nguyen, and Wright. The proof uses techniques developed in recent work of Apisa and Wright.
{"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}
While the exotic diffeomorphisms turned out to be very rich, we know much less about the $b^+_2 =2$ case, as parameterized gauge-theoretic invariants are not well defined. In this paper we present a method (that is, comparing the winding number of parameter families) to find exotic diffeomorphisms on simply-connected smooth closed $4$-manifolds with $b^+_2 =2$, and as a result we obtain that $2mathbb{C}mathbb{P}^2 # 10 (-{mathbb{C}mathbb{P}^2})$ admits exotic diffeomorphisms. This is currently the smallest known example of a closed $4$-manifold that supports exotic diffeomorphisms.
{"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}
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