{"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":"15 1","pages":"0"},"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}
{"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":"93 1","pages":"0"},"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}
{"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":"79 1","pages":"0"},"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}
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.
{"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":"190 1","pages":"0"},"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}
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.
Vladimir Turaev在量子拓扑学的早期发现，模范畴的概念是构建包含3-流形(如Witten-Reshetikhin-Turaev流形)中链路不变量的三维拓扑量子场论(tqft，简称tqft)的合适结构。近年来，模范畴的广义概念放宽了对半简单性的要求，成功地将Turaev的构造推广到各种非半简单的情况。我们报道了这些领域的最新发展，展示了弗拉基米尔血统的丰富性。
{"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":"20 1","pages":"0"},"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}
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.
{"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":"17 1","pages":"0"},"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}
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.
{"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":"38 1","pages":"0"},"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}
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.
{"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":"96 1","pages":"0"},"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}
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.
{"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":"24 1","pages":"0"},"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}
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.
{"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":"13 1","pages":"0"},"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}
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