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Continuous and discontinuous functions on deformation spaces of Kleinian groups Kleinian群变形空间上的连续与不连续函数
Topology and Geometry Pub Date : 2021-07-15 DOI: 10.4171/IRMA/33-1/22
Ken'ichi Ohshika
{"title":"Continuous and discontinuous functions on deformation spaces of Kleinian groups","authors":"Ken'ichi Ohshika","doi":"10.4171/IRMA/33-1/22","DOIUrl":"https://doi.org/10.4171/IRMA/33-1/22","url":null,"abstract":"","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133993898","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}
引用次数: 0
Fundamental groups in projective knot theory 射影结理论中的基本群
Topology and Geometry Pub Date : 2021-07-15 DOI: 10.4171/IRMA/33-1/5
Julia Viro, O. Viro
{"title":"Fundamental groups in projective knot theory","authors":"Julia Viro, O. Viro","doi":"10.4171/IRMA/33-1/5","DOIUrl":"https://doi.org/10.4171/IRMA/33-1/5","url":null,"abstract":"","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117287493","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}
引用次数: 0
A question of Turaev about triple higher Milnor linking numbers of divide links 图拉耶夫关于三倍高米尔诺分链数的问题
Topology and Geometry Pub Date : 2021-07-15 DOI: 10.4171/IRMA/33-1/4
N. A'campo
{"title":"A question of Turaev about triple higher Milnor linking numbers of divide links","authors":"N. A'campo","doi":"10.4171/IRMA/33-1/4","DOIUrl":"https://doi.org/10.4171/IRMA/33-1/4","url":null,"abstract":"","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134279163","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}
引用次数: 0
The Affine Index Polynomial and the Sawollek Polynomial 仿射指数多项式与Sawollek多项式
Topology and Geometry Pub Date : 2020-12-03 DOI: 10.4171/irma/33-1/6
L. Kauffman
{"title":"The Affine Index Polynomial and the Sawollek Polynomial","authors":"L. Kauffman","doi":"10.4171/irma/33-1/6","DOIUrl":"https://doi.org/10.4171/irma/33-1/6","url":null,"abstract":"The purpose of this paper is to give a new basis for examining the relationships of the Affine Index Polynomial and the Sawollek Polynomial. Blake Mellor has written a pioneering paper showing how the Affine Index Polynomial may be extracted from the Sawollek Polynomial. The Affine Index Polynomial is an elementary combinatorial invariant of virtual knots. The Sawollek polynomial is a relative of the classical Alexander polynomial and is defined in terms of a generalization of the Alexander module to virtual knots that derives from the so-called Alexander Biquandle. The present paper constructs the groundwork for a new approach to this relationship, and gives a concise proof of the basic Theorem of Mellor extracting the Affine Index Polynomial from the Sawollek Polynomial.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126031826","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}
引用次数: 3
Modular categories and TQFTs beyond semisimplicity 模类别和tqft超越了半简单性
Topology and Geometry Pub Date : 2020-11-25 DOI: 10.4171/IRMA/33-1/11
C. Blanchet, M. Renzi
{"title":"Modular categories and TQFTs beyond semisimplicity","authors":"C. Blanchet, M. Renzi","doi":"10.4171/IRMA/33-1/11","DOIUrl":"https://doi.org/10.4171/IRMA/33-1/11","url":null,"abstract":"Vladimir Turaev discovered in the early years of quantum topology that the notion of modular category was an appropriate structure for building 3-dimensional Topological Quantum Field Theories (TQFTs for short) containing invariants of links in 3-manifolds such as Witten-Reshetikhin-Turaev ones. In recent years, generalized notions of modular categories, which relax the semisimplicity requirement, have been successfully used to extend Turaev's construction to various non-semisimple settings. We report on these recent developments in the domain, showing the richness of Vladimir's lineage.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115149775","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}
引用次数: 6
Non-semisimple invariants and Habiro’s series 非半单不变量和Habiro级数
Topology and Geometry Pub Date : 2020-09-28 DOI: 10.4171/irma/33-1/10
A. Beliakova, K. Hikami
{"title":"Non-semisimple invariants and Habiro’s series","authors":"A. Beliakova, K. Hikami","doi":"10.4171/irma/33-1/10","DOIUrl":"https://doi.org/10.4171/irma/33-1/10","url":null,"abstract":"In this paper we establish an explicit relationship between Habiro's cyclotomic expansion of the colored Jones polynomial (evaluated at a p-th root of unity) and the Akutsu-Deguchi-Ohtsuki (ADO) invariants of the double twist knots. This allows us to compare the Witten-Reshetikhin-Turaev (WRT) and Costantino-Geer-Patureau (CGP) invariants of 3-manifolds obtained by 0-surgery on these knots. The difference between them is determined by the p-1 coefficient of the Habiro series. We expect these to hold for all Seifert genus 1 knots.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115345240","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}
引用次数: 7
On mapping class group quotients by powers of Dehn twists and their representations Dehn捻幂映射类群商及其表示
Topology and Geometry Pub Date : 2020-09-13 DOI: 10.4171/irma/33-1/15
L. Funar
{"title":"On mapping class group quotients by powers of Dehn twists and their representations","authors":"L. Funar","doi":"10.4171/irma/33-1/15","DOIUrl":"https://doi.org/10.4171/irma/33-1/15","url":null,"abstract":"The aim of this paper is to survey some known results about mapping class group quotients by powers of Dehn twists, related to their finite dimensional representations and to state some open questions. One can construct finite quotients of them, out of representations with Zariski dense images into semisimple Lie groups. We show that, in genus 2, the Fibonacci TQFT representation is actually a specialization of the Jones representation. Eventually, we explain a method of Long and Moody which provides large families of mapping class group representations.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134005798","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}
引用次数: 1
Resurgence of Faddeev’s quantum dilogarithm Faddeev量子二对数的复兴
Topology and Geometry Pub Date : 2020-08-28 DOI: 10.4171/irma/33-1/14
S. Garoufalidis, R. Kashaev
{"title":"Resurgence of Faddeev’s quantum dilogarithm","authors":"S. Garoufalidis, R. Kashaev","doi":"10.4171/irma/33-1/14","DOIUrl":"https://doi.org/10.4171/irma/33-1/14","url":null,"abstract":"The quantum dilogarithm function of Faddeev is a special function that plays a key role as the building block of quantum invariants of knots and 3-manifolds, of quantum Teichmuller theory and of complex Chern-Simons theory. Motivated by conjectures on resurgence and recent interest in wall-crossing phenomena, we prove that the Borel summation of a formal power series solution of a linear difference equation produces Faddeev's quantum dilogarithm. Along the way, we give an explicit formula for the meromorphic function in Borel plane, locate its poles and residues, and describe the Stokes phenomenon of its Laplace transforms along the Stokes rays.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122105181","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}
引用次数: 8
Generalized Dehn twists in low-dimensional topology 低维拓扑中的广义Dehn扭转
Topology and Geometry Pub Date : 2019-09-20 DOI: 10.4171/irma/33-1/18
Y. Kuno, G. Massuyeau, Shunsuke Tsuji
{"title":"Generalized Dehn twists in low-dimensional topology","authors":"Y. Kuno, G. Massuyeau, Shunsuke Tsuji","doi":"10.4171/irma/33-1/18","DOIUrl":"https://doi.org/10.4171/irma/33-1/18","url":null,"abstract":"The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-intersection, it is induced from the usual Dehn twist along the curve. In this expository article, after explaining their definition, we review several results about generalized Dehn twists such as their realizability as diffeomorphisms of the surface, their diagrammatic description in terms of decorated trees and the Hopf-algebraic framework underlying their construction. Going to the dimension three, we also overview the relation between generalized Dehn twists and $3$-dimensional homology cobordisms, and we survey the variants of generalized Dehn twists for skein algebras of the surface.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121579631","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}
引用次数: 1
State sums for some super quantum link invariants 一些超量子链路不变量的状态和
Topology and Geometry Pub Date : 2019-09-05 DOI: 10.4171/irma/33-1/12
Louis-Hadrien Robert, E. Wagner
{"title":"State sums for some super quantum link invariants","authors":"Louis-Hadrien Robert, E. Wagner","doi":"10.4171/irma/33-1/12","DOIUrl":"https://doi.org/10.4171/irma/33-1/12","url":null,"abstract":"We present state sums for quantum link invariants arising from the representation theory of $U_q(mathfrak{gl}_{N|M})$. We investigate the case of the $N$-th exterior power of the standard representation of $U_q(mathfrak{gl}_{N|1})$ and explicit the relation with Kashaev invariants.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130474850","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}
引用次数: 0
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