Surgery on a knot in $S^3$ is said to be an alternating surgery if it yields the double branched cover of an alternating link. The main theoretical contribution is to show that the set of alternating surgery slopes is algorithmically computable and to establish several structural results. Furthermore, we calculate the set of alternating surgery slopes for many examples of knots, including all hyperbolic knots in the SnapPy census. These examples exhibit several interesting phenomena including strongly invertible knots with a unique alternating surgery and asymmetric knots with two alternating surgery slopes. We also establish upper bounds on the set of alternating surgeries, showing that an alternating surgery slope on a hyperbolic knot satisfies $|p/q| leq 3g(K)+4$. Notably, this bound applies to lens space surgeries, thereby strengthening the known genus bounds from the conjecture of Goda and Teragaito.
如果对$S^3$中的一个结进行的手术产生了交替链接的双支盖，那么这个结就被称为交替手术。我们的主要理论贡献是证明交替手术斜率集是可以算出的，并建立了几个结构性结果。此外，我们还计算了许多结的交替手术斜率集，包括 SnapPy 普查中的所有双曲结。这些例子展示了几个有趣的现象，包括具有唯一交替手术的强可逆结和具有两个交替手术斜率的不对称结。我们还建立了交替手术集的上限，表明双曲结上的交替手术斜率满足 $|p/q| leq 3g(K)+4$。值得注意的是，这一约束适用于lens空间手术，从而加强了来自 Goda 和 Teragaito 的猜想的已知种属约束。
{"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}
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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