We show that the smooth homotopy 4-sphere obtained by Gluck twisting the m-twist n-roll spin of any unknotting number one knot is diffeomorphic to the standard 4-sphere, for any pair of integers (m,n). It follows as a corollary that an infinite collection of twisted doubles of Gompf's infinite order corks are standard.
{"title":"Gluck twisting roll spun knots","authors":"Patrick Naylor, Hannah R. Schwartz","doi":"10.2140/agt.2022.22.973","DOIUrl":"https://doi.org/10.2140/agt.2022.22.973","url":null,"abstract":"We show that the smooth homotopy 4-sphere obtained by Gluck twisting the m-twist n-roll spin of any unknotting number one knot is diffeomorphic to the standard 4-sphere, for any pair of integers (m,n). It follows as a corollary that an infinite collection of twisted doubles of Gompf's infinite order corks are standard.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"55 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89420248","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}
We show that a finitely generated subgroup of the genus two handlebody group is stable if and only if the orbit map to the disk graph is a quasi-isometric embedding. To this end, we prove that the genus two handlebody group is a hierarchically hyperbolic group, and that the maximal hyperbolic space in the hierarchy is quasi-isometric to the disk graph of a genus two handlebody by appealing to a construction of Hamenstadt-Hensel. We then utilize the characterization of stable subgroups of hierarchically hyperbolic groups provided by Abbott-Behrstock-Durham. We also provide a counterexample for the higher genus analogue of the main theorem.
{"title":"Stable subgroups of the genus 2 handlebody\u0000group","authors":"Marissa Chesser","doi":"10.2140/agt.2022.22.919","DOIUrl":"https://doi.org/10.2140/agt.2022.22.919","url":null,"abstract":"We show that a finitely generated subgroup of the genus two handlebody group is stable if and only if the orbit map to the disk graph is a quasi-isometric embedding. To this end, we prove that the genus two handlebody group is a hierarchically hyperbolic group, and that the maximal hyperbolic space in the hierarchy is quasi-isometric to the disk graph of a genus two handlebody by appealing to a construction of Hamenstadt-Hensel. We then utilize the characterization of stable subgroups of hierarchically hyperbolic groups provided by Abbott-Behrstock-Durham. We also provide a counterexample for the higher genus analogue of the main theorem.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"200 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80099855","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}
Leighton's Theorem states that if there is a tree $T$ that covers two finite graphs $G_1$ and $G_2$, then there is a finite graph $hat G$ that is covered by $T$ and covers both $G_1$ and $G_2$. We prove that this result does not extend to regular covers by graphs other than trees. Nor does it extend to non-regular covers by a quasitree, even if the automorphism group of the quasitree contains a uniform lattice. But it does extend to regular coverings by quasitrees.
{"title":"Leighton’s theorem : Extensions, limitations and\u0000quasitrees","authors":"M. Bridson, Sam Shepherd","doi":"10.2140/agt.2022.22.881","DOIUrl":"https://doi.org/10.2140/agt.2022.22.881","url":null,"abstract":"Leighton's Theorem states that if there is a tree $T$ that covers two finite graphs $G_1$ and $G_2$, then there is a finite graph $hat G$ that is covered by $T$ and covers both $G_1$ and $G_2$. We prove that this result does not extend to regular covers by graphs other than trees. Nor does it extend to non-regular covers by a quasitree, even if the automorphism group of the quasitree contains a uniform lattice. But it does extend to regular coverings by quasitrees.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88683960","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 : 2020-09-07DOI: 10.2140/agt.2022.22.1905
J. Porti
We consider varieties of representations and characters of 2 and 3-dimensional orbifolds in semisimple Lie groups, and we focus on computing their dimension. For hyperbolic 3-orbifolds, we consider the component of the variety of characters that contains the holonomy composed with the principal representation, we show that its dimension equals half the dimension of the variety of characters of the boundary. We also show that this is a lower bound for the dimension of generic components. We furthermore provide tools for computing dimensions of varieties of characters of 2-orbifolds, including the Hitchin component. We apply this computation to the dimension growth of varieties of characters of some 3-dimensional manifolds in SL(n,C).
{"title":"Dimension of representation and character varieties for two- and three-orbifolds","authors":"J. Porti","doi":"10.2140/agt.2022.22.1905","DOIUrl":"https://doi.org/10.2140/agt.2022.22.1905","url":null,"abstract":"We consider varieties of representations and characters of 2 and 3-dimensional orbifolds in semisimple Lie groups, and we focus on computing their dimension. For hyperbolic 3-orbifolds, we consider the component of the variety of characters that contains the holonomy composed with the principal representation, we show that its dimension equals half the dimension of the variety of characters of the boundary. We also show that this is a lower bound for the dimension of generic components. We furthermore provide tools for computing dimensions of varieties of characters of 2-orbifolds, including the Hitchin component. We apply this computation to the dimension growth of varieties of characters of some 3-dimensional manifolds in SL(n,C).","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"244 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76947701","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 : 2020-08-31DOI: 10.2140/agt.2023.23.1787
John H. Johnson
The residual torsion-free nilpotence of the commutator subgroup of a knot group has played a key role in studying the bi-orderability of knot groups. A technique developed by Mayland provides a sufficient condition for the commutator subgroup of a knot group to be residually-torsion-free nilpotent using work of Baumslag. In this paper, we apply Mayland's technique to several genus one pretzel knots and a family of pretzel knots with arbitrarily high genus. As a result, we obtain a large number of new examples of knots with bi-orderable knot groups. These are the first examples of bi-orderable knot groups for knots which are not fibered or alternating.
{"title":"Residual torsion-free nilpotence, biorderability and pretzel knots","authors":"John H. Johnson","doi":"10.2140/agt.2023.23.1787","DOIUrl":"https://doi.org/10.2140/agt.2023.23.1787","url":null,"abstract":"The residual torsion-free nilpotence of the commutator subgroup of a knot group has played a key role in studying the bi-orderability of knot groups. A technique developed by Mayland provides a sufficient condition for the commutator subgroup of a knot group to be residually-torsion-free nilpotent using work of Baumslag. In this paper, we apply Mayland's technique to several genus one pretzel knots and a family of pretzel knots with arbitrarily high genus. As a result, we obtain a large number of new examples of knots with bi-orderable knot groups. These are the first examples of bi-orderable knot groups for knots which are not fibered or alternating.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"189 ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72544029","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 : 2020-08-27DOI: 10.2140/agt.2022.22.3459
Shijie Gu
It is shown that there exist $mathcal{Z}$-compactifiable manifolds with noncompact boundary which fail to be pseudo-collarable.
证明了具有非紧边界的$数学{Z}$-可紧流形不具有伪可紧性。
{"title":"𝒵–compactifiable manifolds which are not\u0000pseudocollarable","authors":"Shijie Gu","doi":"10.2140/agt.2022.22.3459","DOIUrl":"https://doi.org/10.2140/agt.2022.22.3459","url":null,"abstract":"It is shown that there exist $mathcal{Z}$-compactifiable manifolds with noncompact boundary which fail to be pseudo-collarable.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"76 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86099652","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 : 2020-08-26DOI: 10.2140/agt.2022.22.1461
Sara Maloni, M. B. Pozzetti
We study geometric limits of convex-cocompact cyclic subgroups of the rank 1 groups SO_0(1, k+1) and SU(1, k+1). We construct examples of sequences of subgroups of such groups G that converge algebraically and whose geometric limit strictly contains the algebraic limit, thus generalizing the example first described by Jorgensen for subgroups of SO_0(1,3). We also give necessary and sufficient conditions for a subgroup of SO_0(1, k+1) to arise as geometric limit of a sequence of cyclic subgroups. We then discuss generalizations of such examples to sequence of representations of free groups, and applications of our constructions in that setting.
{"title":"Geometric limits of cyclic subgroups of\u0000SO0(1,k + 1) and SU(1,k + 1)","authors":"Sara Maloni, M. B. Pozzetti","doi":"10.2140/agt.2022.22.1461","DOIUrl":"https://doi.org/10.2140/agt.2022.22.1461","url":null,"abstract":"We study geometric limits of convex-cocompact cyclic subgroups of the rank 1 groups SO_0(1, k+1) and SU(1, k+1). We construct examples of sequences of subgroups of such groups G that converge algebraically and whose geometric limit strictly contains the algebraic limit, thus generalizing the example first described by Jorgensen for subgroups of SO_0(1,3). We also give necessary and sufficient conditions for a subgroup of SO_0(1, k+1) to arise as geometric limit of a sequence of cyclic subgroups. We then discuss generalizations of such examples to sequence of representations of free groups, and applications of our constructions in that setting.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"20 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74106036","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 : 2020-08-25DOI: 10.2140/agt.2022.22.1325
Andr'es Pedroza
We show that for any positive integer $k$ there exists a closed symplectic $4$-manifold, such that the rank of the fundamental group of the group of Hamiltonian diffeomorphisms is at least $k.$
证明了对于任意正整数k存在一个闭辛流形，使得哈密顿微分同态群的基群的秩至少为k
{"title":"On the rank of π1(Ham)","authors":"Andr'es Pedroza","doi":"10.2140/agt.2022.22.1325","DOIUrl":"https://doi.org/10.2140/agt.2022.22.1325","url":null,"abstract":"We show that for any positive integer $k$ there exists a closed symplectic $4$-manifold, such that the rank of the fundamental group of the group of Hamiltonian diffeomorphisms is at least $k.$","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"21 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82724396","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 : 2020-08-22DOI: 10.2140/agt.2023.23.2271
H. Boden, H. Karimi, Adam S. Sikora
The Kauffman bracket of classical links extends to an invariant of links in an arbitrary oriented 3-manifold $M$ with values in the skein module of $M$. In this paper, we consider the skein bracket in case $M$ is a thickened surface. We develop a theory of adequacy for link diagrams on surfaces and show that any alternating link diagram on a surface is skein adequate. We apply our theory to establish the first and second Tait conjectures for adequate link diagrams on surfaces. These are the statements that any adequate link diagram has minimal crossing number, and any two adequate diagrams of the same link have the same writhe. Given a link diagram $D$ on a surface $Sigma$, we use $[D]_Sigma$ to denote its skein bracket. If $D$ has minimal genus, we show that $${rm span}([D]_Sigma) leq 4c(D) + 4 |D|-4g(Sigma),$$ where $|D|$ is the number of connected components of $D$, $c(D)$ is the number of crossings, and $g(Sigma)$ is the genus of $Sigma.$ This extends a classical result proved by Kauffman, Murasugi, and Thistlethwaite. We further show that the above inequality is an equality if and only if $D$ is weakly alternating, namely if $D$ is the connected sum of an alternating link diagram on $Sigma$ with one or more alternating link diagrams on $S^2$. This last statement is a generalization of a well-known result for classical links due to Thistlethwaite, and it implies that the skein bracket detects the crossing number for weakly alternating links. As an application, we show that the crossing number is additive under connected sum for adequate links in thickened surfaces.
{"title":"Adequate links in thickened surfaces and the generalized Tait conjectures","authors":"H. Boden, H. Karimi, Adam S. Sikora","doi":"10.2140/agt.2023.23.2271","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2271","url":null,"abstract":"The Kauffman bracket of classical links extends to an invariant of links in an arbitrary oriented 3-manifold $M$ with values in the skein module of $M$. In this paper, we consider the skein bracket in case $M$ is a thickened surface. We develop a theory of adequacy for link diagrams on surfaces and show that any alternating link diagram on a surface is skein adequate. We apply our theory to establish the first and second Tait conjectures for adequate link diagrams on surfaces. These are the statements that any adequate link diagram has minimal crossing number, and any two adequate diagrams of the same link have the same writhe. \u0000Given a link diagram $D$ on a surface $Sigma$, we use $[D]_Sigma$ to denote its skein bracket. If $D$ has minimal genus, we show that $${rm span}([D]_Sigma) leq 4c(D) + 4 |D|-4g(Sigma),$$ where $|D|$ is the number of connected components of $D$, $c(D)$ is the number of crossings, and $g(Sigma)$ is the genus of $Sigma.$ This extends a classical result proved by Kauffman, Murasugi, and Thistlethwaite. We further show that the above inequality is an equality if and only if $D$ is weakly alternating, namely if $D$ is the connected sum of an alternating link diagram on $Sigma$ with one or more alternating link diagrams on $S^2$. This last statement is a generalization of a well-known result for classical links due to Thistlethwaite, and it implies that the skein bracket detects the crossing number for weakly alternating links. As an application, we show that the crossing number is additive under connected sum for adequate links in thickened surfaces.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"54 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90554750","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}
Let $A$ be the quotient of a graded polynomial ring $mathbb{Z}[x_1,cdots,x_m]otimesLambda[y_1,cdots,y_n]$ by an ideal generated by monomials with leading coefficients 1. Then we constructed a space~$X_A$ such that $A$ is isomorphic to $H^*(X_A)$ modulo torsion elements.
{"title":"Realization of graded monomial ideal rings modulo torsion","authors":"Tseleung So, Donald Stanley","doi":"10.2140/agt.2023.23.733","DOIUrl":"https://doi.org/10.2140/agt.2023.23.733","url":null,"abstract":"Let $A$ be the quotient of a graded polynomial ring $mathbb{Z}[x_1,cdots,x_m]otimesLambda[y_1,cdots,y_n]$ by an ideal generated by monomials with leading coefficients 1. Then we constructed a space~$X_A$ such that $A$ is isomorphic to $H^*(X_A)$ modulo torsion elements.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"99 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86725521","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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