Pub Date : 2020-04-01DOI: 10.1016/J.TOPOL.2021.107836
M. Elhamdadi, M. Saito, E. Zappala
{"title":"Skein theoretic approach to Yang-Baxter homology","authors":"M. Elhamdadi, M. Saito, E. Zappala","doi":"10.1016/J.TOPOL.2021.107836","DOIUrl":"https://doi.org/10.1016/J.TOPOL.2021.107836","url":null,"abstract":"","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89085618","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 article is a survey on the authors' proof of the isomorphism between Heegaard Floer homology and embedded contact homology appeared in This article is a survey on the authors' proof of the isomorphism between Heegaard Floer homology and embedded contact homology appeared in arXiv:1208.1074, arXiv:1208.1077 and arXiv:1208.1526.
{"title":"An exposition of the equivalence of Heegaard\u0000 Floer homology and embedded contact homology","authors":"P. Ghiggini, V. Colin, K. Honda","doi":"10.1090/conm/760/15286","DOIUrl":"https://doi.org/10.1090/conm/760/15286","url":null,"abstract":"This article is a survey on the authors' proof of the isomorphism between Heegaard Floer homology and embedded contact homology appeared in This article is a survey on the authors' proof of the isomorphism between Heegaard Floer homology and embedded contact homology appeared in arXiv:1208.1074, arXiv:1208.1077 and arXiv:1208.1526.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90887867","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}
Pub Date : 2020-03-30DOI: 10.1142/s021821652050090x
H. Abchir, Mohammed Sabak
We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all not alternating and quasi-alternating Montesinos links can be obtained in that way. We check that all the obtained quasi-alternating links satisfy Conjecture 3.1 of Qazaqzeh et al. (JKTR 22 (06), 2013), that is the crossing number of a quasi-alternating link is less than or equal to its determinant. We also prove that the converse of the Theorem 3.3 of Qazaqzeh et al. (JKTR 24 (01), 2015) is false.
我们从一个给定的表示拟交替连杆的几乎交替图或连通约简交替缠结图中构造了一个无限族的几乎交替和拟交替连杆。为了做到这一点，我们使用所谓的除电器扩展，即用一个合理的缠结来取代除电器。我们注意到所有非交替和拟交替的蒙特西诺斯链都可以用这种方法得到。我们检验得到的所有拟交变连杆都满足Qazaqzeh et al. (JKTR 22(06)， 2013)的猜想3.1，即拟交变连杆的交叉数小于等于其行列式。我们还证明了Qazaqzeh et al. (JKTR 24(01)， 2015)的定理3.3的逆为假。
{"title":"Generating links that are both quasi-alternating and almost alternating","authors":"H. Abchir, Mohammed Sabak","doi":"10.1142/s021821652050090x","DOIUrl":"https://doi.org/10.1142/s021821652050090x","url":null,"abstract":"We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all not alternating and quasi-alternating Montesinos links can be obtained in that way. We check that all the obtained quasi-alternating links satisfy Conjecture 3.1 of Qazaqzeh et al. (JKTR 22 (06), 2013), that is the crossing number of a quasi-alternating link is less than or equal to its determinant. We also prove that the converse of the Theorem 3.3 of Qazaqzeh et al. (JKTR 24 (01), 2015) is false.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"128 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85337719","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 show that the Seifert-Weber dodecahedral space $mathsf{SW}$ is an $L$-space. The proof builds on our work relating Floer homology and spectral geometry of hyperbolic three-manifolds. A direct application of our previous techniques runs into difficulties arising from the computational complexity of the problem. We overcome this by exploiting the large symmetry group and the arithmetic and tetrahedral group structure of $mathsf{SW}$ to prove that small eigenvalues on coexact $1$-forms must have large multiplicity.
{"title":"Monopole Floer Homology, Eigenform Multiplicities, and the Seifert–Weber Dodecahedral Space","authors":"Francesco Lin, Michael Lipnowski","doi":"10.1093/imrn/rnaa310","DOIUrl":"https://doi.org/10.1093/imrn/rnaa310","url":null,"abstract":"We show that the Seifert-Weber dodecahedral space $mathsf{SW}$ is an $L$-space. The proof builds on our work relating Floer homology and spectral geometry of hyperbolic three-manifolds. A direct application of our previous techniques runs into difficulties arising from the computational complexity of the problem. We overcome this by exploiting the large symmetry group and the arithmetic and tetrahedral group structure of $mathsf{SW}$ to prove that small eigenvalues on coexact $1$-forms must have large multiplicity.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84800679","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}
There is a compactification of the space of representations of a finitely generated group into the groups of isometries of all spaces with $Delta$-thin triangles. The ideal points are actions on $mathbb R$-trees. It is a geometric reformulation and extension of the Culler-Morgan-Shalen theory concerning limits of representations into $operatorname{SL}(2,{mathbb C})$ and more generally $operatorname{O}(n, 1)$. This paper was written and circulated in the early 90's, but never published.
{"title":"Degenerations of representations and thin\u0000 triangles","authors":"D. Cooper","doi":"10.1090/conm/760/15287","DOIUrl":"https://doi.org/10.1090/conm/760/15287","url":null,"abstract":"There is a compactification of the space of representations of a finitely generated group into the groups of isometries of all spaces with $Delta$-thin triangles. The ideal points are actions on $mathbb R$-trees. It is a geometric reformulation and extension of the Culler-Morgan-Shalen theory concerning limits of representations into $operatorname{SL}(2,{mathbb C})$ and more generally $operatorname{O}(n, 1)$. This paper was written and circulated in the early 90's, but never published.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86357993","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 book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers basic tools in hyperbolic geometry and geometric structures on 3-manifolds. The second part focuses on families of knots and links that have been amenable to study via hyperbolic geometry, particularly twist knots, 2-bridge knots, and alternating knots. It also develops geometric techniques used to study these families, such as angle structures and normal surfaces. The third part gives more detail on three important knot invariants that come directly from hyperbolic geometry, namely volume, canonical polyhedra, and the A-polynomial.
{"title":"Hyperbolic Knot Theory","authors":"J. Purcell","doi":"10.1090/gsm/209","DOIUrl":"https://doi.org/10.1090/gsm/209","url":null,"abstract":"This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers basic tools in hyperbolic geometry and geometric structures on 3-manifolds. The second part focuses on families of knots and links that have been amenable to study via hyperbolic geometry, particularly twist knots, 2-bridge knots, and alternating knots. It also develops geometric techniques used to study these families, such as angle structures and normal surfaces. The third part gives more detail on three important knot invariants that come directly from hyperbolic geometry, namely volume, canonical polyhedra, and the A-polynomial.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85063140","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 prove that each lower-dimensional face of a quasi-arithmetic Coxeter polytope, which happens to be itself a Coxeter polytope, is also quasi-arithmetic. We also provide a sufficient condition for a codimension $1$ face to be actually arithmetic, as well as a few computed examples.
{"title":"On Faces of Quasi-arithmetic Coxeter Polytopes","authors":"N. Bogachev, A. Kolpakov","doi":"10.1093/IMRN/RNAA278","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA278","url":null,"abstract":"We prove that each lower-dimensional face of a quasi-arithmetic Coxeter polytope, which happens to be itself a Coxeter polytope, is also quasi-arithmetic. We also provide a sufficient condition for a codimension $1$ face to be actually arithmetic, as well as a few computed examples.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80105608","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}
Pub Date : 2020-02-24DOI: 10.1142/s0218216520500571
D. Silver, Susan G. Williams
Laplacian matrices of weighted graphs in surfaces $S$ are used to define module and polynomial invariants of $Z/2$-homologically trivial links in $S times [0,1]$. Information about virtual genus is obtained.
{"title":"Links in surfaces and Laplacian modules","authors":"D. Silver, Susan G. Williams","doi":"10.1142/s0218216520500571","DOIUrl":"https://doi.org/10.1142/s0218216520500571","url":null,"abstract":"Laplacian matrices of weighted graphs in surfaces $S$ are used to define module and polynomial invariants of $Z/2$-homologically trivial links in $S times [0,1]$. Information about virtual genus is obtained.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84526581","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 homomorphism from a knot group to a fixed group, we introduce an element of a $K_1$-group, which is a generalization of (twisted) Alexander polynomials. We compare this $K_1$-class with other Alexander polynomials. In terms of semi-local rings, we compute the $K_1$-classes of some knots and show their non-triviality. We also introduce metabelian Alexander polynomials.
{"title":"Twisted Alexander Invariants of Knot Group Representations","authors":"Takefumi Nosaka","doi":"10.3836/tjm/1502179346","DOIUrl":"https://doi.org/10.3836/tjm/1502179346","url":null,"abstract":"Given a homomorphism from a knot group to a fixed group, we introduce an element of a $K_1$-group, which is a generalization of (twisted) Alexander polynomials. We compare this $K_1$-class with other Alexander polynomials. In terms of semi-local rings, we compute the $K_1$-classes of some knots and show their non-triviality. We also introduce metabelian Alexander polynomials.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"156 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76089909","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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