The Turaev cobracket, a loop operation introduced by V. Turaev, which measures self-intersection of a loop on a surface, is a modification of a path operation introduced earlier by Turaev himself, as well as a counterpart of the Goldman bracket. In this survey based on the author's joint works with A. Alekseev, Y. Kuno and F. Naef, we review some algebraic aspects of the cobracket and its framed variants including their formal description, an application to the mapping class group of the surface and a relation to the (higher genus) Kashiwara-Vergne problem. In addition, we review a homological description of the cobracket after R. Hain.
Turaev协括号是由V. Turaev引入的一种循环运算，用于测量表面上环路的自交，它是对Turaev自己之前引入的路径运算的改进，也是高盛括号的对应。本文基于作者与a . Alekseev, Y. Kuno和F. Naef的合著，回顾了协括号及其框架变体的代数方面，包括它们的形式描述，在曲面的映射类群中的应用以及与(高属)Kashiwara-Vergne问题的关系。此外，我们回顾了R. Hain之后的一种对协括号的同源描述。
{"title":"Some algebraic aspects of the Turaev cobracket","authors":"Nariya Kawazumi","doi":"10.4171/irma/33-1/17","DOIUrl":"https://doi.org/10.4171/irma/33-1/17","url":null,"abstract":"The Turaev cobracket, a loop operation introduced by V. Turaev, which measures self-intersection of a loop on a surface, is a modification of a path operation introduced earlier by Turaev himself, as well as a counterpart of the Goldman bracket. In this survey based on the author's joint works with A. Alekseev, Y. Kuno and F. Naef, we review some algebraic aspects of the cobracket and its framed variants including their formal description, an application to the mapping class group of the surface and a relation to the (higher genus) Kashiwara-Vergne problem. In addition, we review a homological description of the cobracket after R. Hain.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134408173","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}
Every knot can be unknotted with two generalized twists; this was first proved by Ohyama. Here we prove that any knot of genus g can be unknotted with 2g null-homologous twists and that there exist genus g knots that cannot be unknotted with fewer than 2g null-homologous twists.
{"title":"Null-homologous unknottings","authors":"C. Livingston","doi":"10.4171/irma/33-1/3","DOIUrl":"https://doi.org/10.4171/irma/33-1/3","url":null,"abstract":"Every knot can be unknotted with two generalized twists; this was first proved by Ohyama. Here we prove that any knot of genus g can be unknotted with 2g null-homologous twists and that there exist genus g knots that cannot be unknotted with fewer than 2g null-homologous twists.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131331418","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 provide an (almost) self-contained construction of the Witten-Reshetikhin-Turaev representations of the mapping class group. We describe its properties including its Hermitian structure, irreducibility and integrality (at prime level). The construction of these notes relies only on skein theory (Kauffman Bracket) and does not use surgery techniques. We hope that they will be accessible to non-specialists.
{"title":"Introduction to quantum representations of mapping class groups","authors":"Julien March'e","doi":"10.4171/irma/33-1/7","DOIUrl":"https://doi.org/10.4171/irma/33-1/7","url":null,"abstract":"We provide an (almost) self-contained construction of the Witten-Reshetikhin-Turaev representations of the mapping class group. We describe its properties including its Hermitian structure, irreducibility and integrality (at prime level). The construction of these notes relies only on skein theory (Kauffman Bracket) and does not use surgery techniques. We hope that they will be accessible to non-specialists.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125398033","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 existence of essential clsoed surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.
{"title":"Essential closed surfaces and finite coverings of negatively curved cusped 3-manifolds","authors":"C. Charitos","doi":"10.4171/IRMA/33-1/21","DOIUrl":"https://doi.org/10.4171/IRMA/33-1/21","url":null,"abstract":"The existence of essential clsoed surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115915179","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 introduce a framework in noncommutative geometry consisting of a $*$-algebra $mathcal A$, a bimodule $Omega^1$ endowed with a derivation $mathcal Ato Omega^1$ and with a Hermitian structure $Omega^1otimes bar{Omega}^1to mathcal A$ (a "noncommutative Kahler form"), and a cyclic 1-cochain $mathcal Ato mathbb C$ whose coboundary is determined by the previous structures. These data give moment map equations on the space of connections on an arbitrary finitely-generated projective $mathcal A$-module. As particular cases, we obtain a large class of equations in algebra (King's equations for representations of quivers, including ADHM equations), in classical gauge theory (Hermitian Yang-Mills equations, Hitchin equations, Bogomolny and Nahm equations, etc.), as well as in noncommutative gauge theory by Connes, Douglas and Schwarz. We also discuss Nekrasov's beautiful proposal for re-interpreting noncommutative instantons on $mathbb{C}^nsimeq mathbb{R}^{2n}$ as infinite-dimensional solutions of King's equation $$sum_{i=1}^n [T_i^dagger, T_i]=hbarcdot ncdotmathrm{Id}_{mathcal H}$$ where $mathcal H$ is a Hilbert space completion of a finitely-generated $mathbb C[T_1,dots,T_n]$-module (e.g. an ideal of finite codimension).
{"title":"A generalization of King’s equation via noncommutative geometry","authors":"Gourab Bhattacharya, M. Kontsevich","doi":"10.4171/irma/33-1/23","DOIUrl":"https://doi.org/10.4171/irma/33-1/23","url":null,"abstract":"We introduce a framework in noncommutative geometry consisting of a $*$-algebra $mathcal A$, a bimodule $Omega^1$ endowed with a derivation $mathcal Ato Omega^1$ and with a Hermitian structure $Omega^1otimes bar{Omega}^1to mathcal A$ (a \"noncommutative Kahler form\"), and a cyclic 1-cochain $mathcal Ato mathbb C$ whose coboundary is determined by the previous structures. These data give moment map equations on the space of connections on an arbitrary finitely-generated projective $mathcal A$-module. As particular cases, we obtain a large class of equations in algebra (King's equations for representations of quivers, including ADHM equations), in classical gauge theory (Hermitian Yang-Mills equations, Hitchin equations, Bogomolny and Nahm equations, etc.), as well as in noncommutative gauge theory by Connes, Douglas and Schwarz. \u0000We also discuss Nekrasov's beautiful proposal for re-interpreting noncommutative instantons on $mathbb{C}^nsimeq mathbb{R}^{2n}$ as infinite-dimensional solutions of King's equation $$sum_{i=1}^n [T_i^dagger, T_i]=hbarcdot ncdotmathrm{Id}_{mathcal H}$$ where $mathcal H$ is a Hilbert space completion of a finitely-generated $mathbb C[T_1,dots,T_n]$-module (e.g. an ideal of finite codimension).","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128066934","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}
Let $D$ be an oriented link diagram with the set of regions $operatorname{r}_{D}$. We define a symmetric map (or matrix) $operatorname{tau}_{D}colonoperatorname{r}_{D}times operatorname{r}_{D} to mathbb{Z}[x]$ that gives rise to an invariant of oriented links, based on a slightly modified $S$-equivalence of Trotter and Murasugi in the space of symmetric matrices. In particular, for real $x$, the negative signature of $operatorname{tau}_{D}$ corrected by the writhe is conjecturally twice the Tristram--Levine signature function, where $2x=sqrt{t}+frac1{sqrt{t}}$ with $t$ being the indeterminate of the Alexander polynomial.
设$D$为具有区域集$operatorname{r}_{D}$的定向链接图。基于Trotter和Murasugi在对称矩阵空间中的一个略微修改的$S$等价，我们定义了一个对称映射(或矩阵)$operatorname{tau}_{D}colonoperatorname{r}_{D}times operatorname{r}_{D} to mathbb{Z}[x]$，它产生了定向链接的不变量。特别是，对于实数$x$，由writhe修正的$operatorname{tau}_{D}$的负签名推测是Tristram- Levine签名函数的两倍，其中$2x=sqrt{t}+frac1{sqrt{t}}$与$t$是Alexander多项式的不定式。
{"title":"On symmetric matrices associated with oriented link diagrams","authors":"R. Kashaev","doi":"10.4171/irma/33-1/8","DOIUrl":"https://doi.org/10.4171/irma/33-1/8","url":null,"abstract":"Let $D$ be an oriented link diagram with the set of regions $operatorname{r}_{D}$. We define a symmetric map (or matrix) $operatorname{tau}_{D}colonoperatorname{r}_{D}times operatorname{r}_{D} to mathbb{Z}[x]$ that gives rise to an invariant of oriented links, based on a slightly modified $S$-equivalence of Trotter and Murasugi in the space of symmetric matrices. In particular, for real $x$, the negative signature of $operatorname{tau}_{D}$ corrected by the writhe is conjecturally twice the Tristram--Levine signature function, where $2x=sqrt{t}+frac1{sqrt{t}}$ with $t$ being the indeterminate of the Alexander polynomial.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122030047","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 expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they generate Brane Topological Field Theories. For general Hurwitz numbers corresponding algebra is an algebra of coverings of lesser dimension. For regular Hurwitz numbers the Frobenius algebra is an algebra of families of subgroups of finite groups.
{"title":"Brane Topological Field Theory and Hurwitz numbers for CW-complexes","authors":"S. Natanzon","doi":"10.4171/irma/33-1/13","DOIUrl":"https://doi.org/10.4171/irma/33-1/13","url":null,"abstract":"We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they generate Brane Topological Field Theories. For general Hurwitz numbers corresponding algebra is an algebra of coverings of lesser dimension. For regular Hurwitz numbers the Frobenius algebra is an algebra of families of subgroups of finite groups.","PeriodicalId":270093,"journal":{"name":"Topology and Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128925291","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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