{"title":"Augmented Legendrian cobordism in $J^1S^1$","authors":"Yu Pan, Dan Rutherford","doi":"10.4171/qt/195","DOIUrl":null,"url":null,"abstract":"We consider Legendrian links and tangles in $J^1S^1$ and $J^1\\[0,1]$ equipped with Morse complex families over a field $\\mathbb{F}$ and classify them up to Legendrian cobordism. When the coefficient field is $\\mathbb{F}\\_2$, this provides a cobordism classification for Legendrians equipped with augmentations of the Legendrian contact homology DG-algebras. A complete set of invariants, for which arbitrary values may be obtained, is provided by the fiber cohomology, a graded monodromy matrix, and a mod $2$ spin number. We apply the classification to construct augmented Legendrian surfaces in $J^1M$ with $\\mathrm{dim} M = 2$ realizing any prescribed monodromy representation, $\\Phi:\\pi\\_1(M,x\\_0) \\to \\mathrm{GL}(\\mathbf{n}, \\mathbb{F})$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"16 1","pages":"0"},"PeriodicalIF":1.0000,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quantum Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4171/qt/195","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider Legendrian links and tangles in $J^1S^1$ and $J^1\[0,1]$ equipped with Morse complex families over a field $\mathbb{F}$ and classify them up to Legendrian cobordism. When the coefficient field is $\mathbb{F}\_2$, this provides a cobordism classification for Legendrians equipped with augmentations of the Legendrian contact homology DG-algebras. A complete set of invariants, for which arbitrary values may be obtained, is provided by the fiber cohomology, a graded monodromy matrix, and a mod $2$ spin number. We apply the classification to construct augmented Legendrian surfaces in $J^1M$ with $\mathrm{dim} M = 2$ realizing any prescribed monodromy representation, $\Phi:\pi\_1(M,x\_0) \to \mathrm{GL}(\mathbf{n}, \mathbb{F})$.
期刊介绍:
Quantum Topology is a peer reviewed journal dedicated to publishing original research articles, short communications, and surveys in quantum topology and related areas of mathematics. Topics covered include in particular:
Low-dimensional Topology
Knot Theory
Jones Polynomial and Khovanov Homology
Topological Quantum Field Theory
Quantum Groups and Hopf Algebras
Mapping Class Groups and Teichmüller space
Categorification
Braid Groups and Braided Categories
Fusion Categories
Subfactors and Planar Algebras
Contact and Symplectic Topology
Topological Methods in Physics.