Pub Date : 2021-03-02DOI: 10.2140/agt.2022.22.3983
I. Sundberg, Jonah Swann
To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we define an invariant of its boundary-preserving isotopy class from the Khovanov homology of its boundary link. Previous work showed that when the boundary link is empty, this invariant is determined by the genus of the surface. We show that this relative invariant: can obstruct sliceness of knots; detects a pair of slices for $9_{46}$; is not hindered by detecting connected sums with knotted $2$-spheres.
{"title":"Relative Khovanov–Jacobsson classes","authors":"I. Sundberg, Jonah Swann","doi":"10.2140/agt.2022.22.3983","DOIUrl":"https://doi.org/10.2140/agt.2022.22.3983","url":null,"abstract":"To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we define an invariant of its boundary-preserving isotopy class from the Khovanov homology of its boundary link. Previous work showed that when the boundary link is empty, this invariant is determined by the genus of the surface. We show that this relative invariant: can obstruct sliceness of knots; detects a pair of slices for $9_{46}$; is not hindered by detecting connected sums with knotted $2$-spheres.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"115 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79457676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rub'en Blasco-Garc'ia, J. I. Cogolludo-Agust'in, Conchita Mart'inez-P'erez
. We explicitly compute the homology groups with coefficients in a field of characteristic zero of cocyclic subgroups or even Artin groups of FC-type. We also give some partial results in the case when the coefficients are taken in a field of prime characteristic.
{"title":"Homology of even Artin kernels","authors":"Rub'en Blasco-Garc'ia, J. I. Cogolludo-Agust'in, Conchita Mart'inez-P'erez","doi":"10.2140/agt.2022.22.349","DOIUrl":"https://doi.org/10.2140/agt.2022.22.349","url":null,"abstract":". We explicitly compute the homology groups with coefficients in a field of characteristic zero of cocyclic subgroups or even Artin groups of FC-type. We also give some partial results in the case when the coefficients are taken in a field of prime characteristic.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"19 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72490294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-01DOI: 10.2140/agt.2022.22.2997
Francisco Nicol'as
We prove that if a surface group embeds as a normal subgroup in a K¨ahler group and the conjugation action of the K¨ahler group on the surface group preserves the conjugacy class of a non-trivial element, then the K¨ahler group is virtually given by a direct product, where one factor is a surface group. Moreover we prove that if a one-ended hyperbolic group with infinite outer automorphism group embeds as a normal subgroup in a K¨ahler group then it is virtually a surface group. More generally we give restrictions on normal subgroups of K¨ahler groups which are amalgamated products or HNN extensions.
{"title":"On finitely generated normal subgroups of\u0000Kähler groups","authors":"Francisco Nicol'as","doi":"10.2140/agt.2022.22.2997","DOIUrl":"https://doi.org/10.2140/agt.2022.22.2997","url":null,"abstract":"We prove that if a surface group embeds as a normal subgroup in a K¨ahler group and the conjugation action of the K¨ahler group on the surface group preserves the conjugacy class of a non-trivial element, then the K¨ahler group is virtually given by a direct product, where one factor is a surface group. Moreover we prove that if a one-ended hyperbolic group with infinite outer automorphism group embeds as a normal subgroup in a K¨ahler group then it is virtually a surface group. More generally we give restrictions on normal subgroups of K¨ahler groups which are amalgamated products or HNN extensions.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"17 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88302235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-30DOI: 10.2140/agt.2023.23.2191
Kouyemon Iriye, D. Kishimoto
The notions Golodness and tightness for simplicial complexes come from algebra and geometry, respectively. We prove these two notions are equivalent for 3-manifold triangulations, through a topological characterization of a polyhedral product for a tight-neighborly manifold triangulation of dimension $ge 3$.
{"title":"Golod and tight 3–manifolds","authors":"Kouyemon Iriye, D. Kishimoto","doi":"10.2140/agt.2023.23.2191","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2191","url":null,"abstract":"The notions Golodness and tightness for simplicial complexes come from algebra and geometry, respectively. We prove these two notions are equivalent for 3-manifold triangulations, through a topological characterization of a polyhedral product for a tight-neighborly manifold triangulation of dimension $ge 3$.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"36 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82874850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-13DOI: 10.2140/agt.2023.23.1831
L. Eakins, Thomas Fleming, T. Mattman
A graph is maximal knotless if it is edge maximal for the property of knotless embedding in $R^3$. We show that such a graph has at least $frac74 |V|$ edges, and construct an infinite family of maximal knotless graphs with $|E|
{"title":"Maximal knotless graphs","authors":"L. Eakins, Thomas Fleming, T. Mattman","doi":"10.2140/agt.2023.23.1831","DOIUrl":"https://doi.org/10.2140/agt.2023.23.1831","url":null,"abstract":"A graph is maximal knotless if it is edge maximal for the property of knotless embedding in $R^3$. We show that such a graph has at least $frac74 |V|$ edges, and construct an infinite family of maximal knotless graphs with $|E|<frac52|V|$. With the exception of $|E| = 22$, we show that for any $|E| geq 20$ there exists a maximal knotless graph of size $|E|$. We classify the maximal knotless graphs through nine vertices and 20 edges. We determine which of these maxnik graphs are the clique sum of smaller graphs and construct an infinite family of maxnik graphs that are not clique sums.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"3 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88235075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-18DOI: 10.2140/agt.2022.22.3023
Eduardo Mart'inez-Pedroza, Luis Jorge S'anchez Saldana
Let $G$ and $H$ be quasi-isometric finitely generated groups and let $Pleq G$; is there a subgroup $Q$ (or a collection of subgroups) of $H$ whose left cosets coarsely reflect the geometry of the left cosets of $P$ in $G$? We explore sufficient conditions for a positive answer. The article consider pairs of the form $(G,mathcal{P})$ where $G$ is a finitely generated group and $mathcal{P}$ a finite collection of subgroups, there is a notion of quasi-isometry of pairs, and quasi-isometrically characteristic collection of subgroups. A subgroup is qi-characteristic if it belongs to a qi-characteristic collection. Distinct classes of qi-characteristic collections of subgroups have been studied in the literature on quasi-isometric rigidity, we list in the article some of them and provide other examples. The first part of the article proves: if $G$ and $H$ are finitely generated quasi-isometric groups and $mathcal{P}$ is a qi-characteristic collection of subgroups of $G$, then there is a collection of subgroups $mathcal{Q}$ of $H$ such that $ (G, mathcal{P})$ and $(H, mathcal{Q})$ are quasi-isometric pairs. The second part of the article studies the number of filtered ends $tilde e (G, P)$ of a pair of groups, a notion introduced by Bowditch, and provides an application of our main result: if $G$ and $H$ are quasi-isometric groups and $Pleq G$ is qi-characterstic, then there is $Qleq H$ such that $tilde e (G, P) = tilde e (H, Q)$.
设$G$和$H$是拟等距有限生成群，设$Pleq G$;是否存在一个$H$的子群$Q$(或子群的集合)，其左余集大致反映$G$中$P$的左余集的几何形状?我们探索一个正答案的充分条件。本文考虑$(G,mathcal{P})$形式的对，其中$G$是有限生成的群，$mathcal{P}$是有限子群的集合，存在对的拟等距概念，以及子群的拟等距特征集合。如果子群属于一个气特征集合，则子群是气特征的。关于准等距刚性的文献中已经研究了不同种类的子群的气特征集合，本文列举了其中的一些，并给出了其他的例子。本文第一部分证明:如果$G$和$H$是有限生成的拟等距群，$mathcal{P}$是$G$的子群的一个齐特征集合，那么存在$H$的子群$mathcal{Q}$的一个集合，使得$ (G, mathcal{P})$和$(H, mathcal{Q})$是拟等距对。文章的第二部分研究了Bowditch引入的一对群的过滤端个数$tilde e (G, P)$，并给出了我们的主要结果的一个应用:如果$G$和$H$是拟等距群，$Pleq G$是齐特征群，则存在$Qleq H$使得$tilde e (G, P) = tilde e (H, Q)$。
{"title":"Quasi-isometric rigidity of subgroups and filtered ends","authors":"Eduardo Mart'inez-Pedroza, Luis Jorge S'anchez Saldana","doi":"10.2140/agt.2022.22.3023","DOIUrl":"https://doi.org/10.2140/agt.2022.22.3023","url":null,"abstract":"Let $G$ and $H$ be quasi-isometric finitely generated groups and let $Pleq G$; is there a subgroup $Q$ (or a collection of subgroups) of $H$ whose left cosets coarsely reflect the geometry of the left cosets of $P$ in $G$? We explore sufficient conditions for a positive answer. The article consider pairs of the form $(G,mathcal{P})$ where $G$ is a finitely generated group and $mathcal{P}$ a finite collection of subgroups, there is a notion of quasi-isometry of pairs, and quasi-isometrically characteristic collection of subgroups. A subgroup is qi-characteristic if it belongs to a qi-characteristic collection. Distinct classes of qi-characteristic collections of subgroups have been studied in the literature on quasi-isometric rigidity, we list in the article some of them and provide other examples. The first part of the article proves: if $G$ and $H$ are finitely generated quasi-isometric groups and $mathcal{P}$ is a qi-characteristic collection of subgroups of $G$, then there is a collection of subgroups $mathcal{Q}$ of $H$ such that $ (G, mathcal{P})$ and $(H, mathcal{Q})$ are quasi-isometric pairs. The second part of the article studies the number of filtered ends $tilde e (G, P)$ of a pair of groups, a notion introduced by Bowditch, and provides an application of our main result: if $G$ and $H$ are quasi-isometric groups and $Pleq G$ is qi-characterstic, then there is $Qleq H$ such that $tilde e (G, P) = tilde e (H, Q)$.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"128 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83971741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-17DOI: 10.2140/agt.2022.22.3965
Thiago de Paiva
The twisted torus knots K(p, q; r, s) are obtained by performing a sequence of s full twists on r adjacent strands of (p, q)-torus knots. In this paper, we answer two questions related to essential surfaces in the exteriors of twisted torus knots. Namely, we show there are prime numbers r greater than 2 such that K(p, q; r, s) contain closed essential surface in their exterior, answering a question of Morimoto and Yamada. Additionally, Morimoto asked whether all twisted torus knots with essential tori in the exterior fit into one of two families. We find a new infinite family that was previously unknown.
{"title":"Unexpected essential surfaces among exteriors of twisted torus knots","authors":"Thiago de Paiva","doi":"10.2140/agt.2022.22.3965","DOIUrl":"https://doi.org/10.2140/agt.2022.22.3965","url":null,"abstract":"The twisted torus knots K(p, q; r, s) are obtained by performing a sequence of s full twists on r adjacent strands of (p, q)-torus knots. In this paper, we answer two questions related to essential surfaces in the exteriors of twisted torus knots. Namely, we show there are prime numbers r greater than 2 such that K(p, q; r, s) contain closed essential surface in their exterior, answering a question of Morimoto and Yamada. Additionally, Morimoto asked whether all twisted torus knots with essential tori in the exterior fit into one of two families. We find a new infinite family that was previously unknown.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"14 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77352572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-14DOI: 10.2140/agt.2023.23.1891
Joseph Boninger
We define a polynomial invariant $J_n^T$ of links in the thickened torus. We call $J^T_n$ the $n$th toroidal colored Jones polynomial, and show it satisfies many properties of the original colored Jones polynomial. Most significantly, $J_n^T$ exhibits volume conjecture behavior. We prove the volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions of $J_n^T,$ one using operator invariants and another using the Kauffman bracket skein module of the torus. In the process we generalize the theory of operator invariants to links in $T^2 times I$, defining what we call a pseudo-operator invariant.
{"title":"A quantum invariant of links in T2× I with\u0000volume conjecture behavior","authors":"Joseph Boninger","doi":"10.2140/agt.2023.23.1891","DOIUrl":"https://doi.org/10.2140/agt.2023.23.1891","url":null,"abstract":"We define a polynomial invariant $J_n^T$ of links in the thickened torus. We call $J^T_n$ the $n$th toroidal colored Jones polynomial, and show it satisfies many properties of the original colored Jones polynomial. Most significantly, $J_n^T$ exhibits volume conjecture behavior. We prove the volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions of $J_n^T,$ one using operator invariants and another using the Kauffman bracket skein module of the torus. In the process we generalize the theory of operator invariants to links in $T^2 times I$, defining what we call a pseudo-operator invariant.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"27 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90962289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-06DOI: 10.2140/agt.2023.23.1249
J. Korinman
We provide finite presentations for stated skein algebras and deduce that those algebras are Koszul and that they are isomorphic to the quantum moduli algebras appearing in lattice gauge field theory, generalizing previous results of Bullock, Frohman, Kania-Bartoszynska and Faitg.
{"title":"Finite presentations for stated skein algebras and lattice gauge field theory","authors":"J. Korinman","doi":"10.2140/agt.2023.23.1249","DOIUrl":"https://doi.org/10.2140/agt.2023.23.1249","url":null,"abstract":"We provide finite presentations for stated skein algebras and deduce that those algebras are Koszul and that they are isomorphic to the quantum moduli algebras appearing in lattice gauge field theory, generalizing previous results of Bullock, Frohman, Kania-Bartoszynska and Faitg.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"61 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84765190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-02DOI: 10.2140/agt.2021.21.3675
Daniel Allcock
Let $X$ be a surface, possibly with boundary. Suppose it has infinite genus or infinitely many punctures, or a closed subset which is a disk with a Cantor set removed from its interior. For example, $X$ could be any surface of infinite type with only finitely many boundary components. We prove that the mapping class group of $X$ does not satisfy the Tits Alternative. That is, Map$(X)$ contains a finitely generated subgroup that is not virtually solvable and contains no nonabelian free group.
{"title":"Most big mapping class groups fail the Tits alternative","authors":"Daniel Allcock","doi":"10.2140/agt.2021.21.3675","DOIUrl":"https://doi.org/10.2140/agt.2021.21.3675","url":null,"abstract":"Let $X$ be a surface, possibly with boundary. Suppose it has infinite genus or infinitely many punctures, or a closed subset which is a disk with a Cantor set removed from its interior. For example, $X$ could be any surface of infinite type with only finitely many boundary components. We prove that the mapping class group of $X$ does not satisfy the Tits Alternative. That is, Map$(X)$ contains a finitely generated subgroup that is not virtually solvable and contains no nonabelian free group.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"2 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80904413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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