{"title":"结调和不变量的组合描述","authors":"Subhankar Dey, Hakan Doga","doi":"10.1142/S021821652150036X","DOIUrl":null,"url":null,"abstract":"In this paper, we give a combinatorial description of the concordance invariant $\\varepsilon$ defined by Hom in \\cite{hom2011knot}, prove some properties of this invariant using grid homology techniques. We also compute $\\varepsilon$ of $(p,q)$ torus knots and prove that $\\varepsilon(\\mathbb{G}_+)=1$ if $\\mathbb{G}_+$ is a grid diagram for a positive braid. Furthermore, we show how $\\varepsilon$ behaves under $(p,q)$-cabling of negative torus knots.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"76 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Combinatorial Description of the Knot Concordance Invariant Epsilon\",\"authors\":\"Subhankar Dey, Hakan Doga\",\"doi\":\"10.1142/S021821652150036X\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we give a combinatorial description of the concordance invariant $\\\\varepsilon$ defined by Hom in \\\\cite{hom2011knot}, prove some properties of this invariant using grid homology techniques. We also compute $\\\\varepsilon$ of $(p,q)$ torus knots and prove that $\\\\varepsilon(\\\\mathbb{G}_+)=1$ if $\\\\mathbb{G}_+$ is a grid diagram for a positive braid. Furthermore, we show how $\\\\varepsilon$ behaves under $(p,q)$-cabling of negative torus knots.\",\"PeriodicalId\":8454,\"journal\":{\"name\":\"arXiv: Geometric Topology\",\"volume\":\"76 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-10-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Geometric Topology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1142/S021821652150036X\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Geometric Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/S021821652150036X","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A Combinatorial Description of the Knot Concordance Invariant Epsilon
In this paper, we give a combinatorial description of the concordance invariant $\varepsilon$ defined by Hom in \cite{hom2011knot}, prove some properties of this invariant using grid homology techniques. We also compute $\varepsilon$ of $(p,q)$ torus knots and prove that $\varepsilon(\mathbb{G}_+)=1$ if $\mathbb{G}_+$ is a grid diagram for a positive braid. Furthermore, we show how $\varepsilon$ behaves under $(p,q)$-cabling of negative torus knots.