{"title":"Simplified SFT moduli spaces for Legendrian links","authors":"Russell Avdek","doi":"10.4310/jsg.2023.v21.n2.a2","DOIUrl":null,"url":null,"abstract":"We study moduli spaces $\\mathcal{M}$ of holomorphic maps $U$ to $\\mathbb{R}^4$ with boundaries on the Lagrangian cylinder over a Legendrian link $\\Lambda \\subset (\\mathbb{R}^3, \\xi_{std})$. We allow our domains, $\\dot{\\Sigma}$ , to have non-trivial topology in which case $\\mathcal{M}$ is the zero locus of an obstruction function $\\mathcal{O}$, sending a moduli space of holomorphic maps in $\\mathbb{C}$ to $H^1 (\\dot{\\Sigma})$. In general, $\\mathcal{O}^{-1} (0)$ is not combinatorially computable. However after a Legendrian isotopy $\\Lambda$ can be made <i>left-right-simple</i>, implying that any $U$ 1) of index $1$ is a disk with one or two positive punctures for which $\\pi_\\mathbb{C} \\circ U$ is an embedding. 2) of index $2$ is either a disk or an annulus with $\\pi_\\mathbb{C} \\circ U$ simply covered and without interior critical points. Therefore any SFT invariant of $\\Lambda$ is combinatorially computable using only disks with $\\leq 2$ positive punctures.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Symplectic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/jsg.2023.v21.n2.a2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ to $\mathbb{R}^4$ with boundaries on the Lagrangian cylinder over a Legendrian link $\Lambda \subset (\mathbb{R}^3, \xi_{std})$. We allow our domains, $\dot{\Sigma}$ , to have non-trivial topology in which case $\mathcal{M}$ is the zero locus of an obstruction function $\mathcal{O}$, sending a moduli space of holomorphic maps in $\mathbb{C}$ to $H^1 (\dot{\Sigma})$. In general, $\mathcal{O}^{-1} (0)$ is not combinatorially computable. However after a Legendrian isotopy $\Lambda$ can be made left-right-simple, implying that any $U$ 1) of index $1$ is a disk with one or two positive punctures for which $\pi_\mathbb{C} \circ U$ is an embedding. 2) of index $2$ is either a disk or an annulus with $\pi_\mathbb{C} \circ U$ simply covered and without interior critical points. Therefore any SFT invariant of $\Lambda$ is combinatorially computable using only disks with $\leq 2$ positive punctures.
期刊介绍:
Publishes high quality papers on all aspects of symplectic geometry, with its deep roots in mathematics, going back to Huygens’ study of optics and to the Hamilton Jacobi formulation of mechanics. Nearly all branches of mathematics are treated, including many parts of dynamical systems, representation theory, combinatorics, packing problems, algebraic geometry, and differential topology.