MICC: A tool for computing short distances in the curve complex

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• 作者：Paul Glenn^a ; ^{paulglen@berkeley.edu" class="auth_mail" title="E-mail the corresponding author} ; William W. Menasco^b ; ^{menasco@buffalo.edu" class="auth_mail" title="E-mail the corresponding author} ; Kayla Morrell^c ; ^{morrelke01@mail.buffalostate.edu" class="auth_mail" title="E-mail the corresponding author} ; Matthew J. Morse^d ; ^{mmorse@cs.nyu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Mapping class group ; Curve complex ; Distance ; Geodesic
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：January-February 2017
• 年：2017
• 卷：78
• 期：Complete
• 页码：115-132
• 全文大小：575 K

文摘

The complex of curves C(S_g)$\mathcal{C}(S_g)$ of a closed orientable surface of genus 8e3146c5778dd7d0358fd5e" title="Click to view the MathML source">g≥2$g\ge 2$ is the simplicial complex whose vertices, C⁰(S_g)${\mathcal{C}}^0(S_g)$, are isotopy classes of essential simple closed curves in af1e835" title="Click to view the MathML source">S_g$S_g$. Two vertices co-bound an edge of the 1-skeleton, e4cd957ee79b7eba422aa52a442" title="Click to view the MathML source">C¹(S_g)${\mathcal{C}}^1(S_g)$, if there are disjoint representatives in af1e835" title="Click to view the MathML source">S_g$S_g$. A metric is obtained on C⁰(S_g)${\mathcal{C}}^0(S_g)$ by assigning unit length to each edge of e4cd957ee79b7eba422aa52a442" title="Click to view the MathML source">C¹(S_g)${\mathcal{C}}^1(S_g)$. Thus, the distance between two vertices, e4c024e58259ba618e5dc628449d53" title="Click to view the MathML source">d(v,w)$d(v,w)$, corresponds to the length of a geodesic—a shortest edge-path between v and w   in e4cd957ee79b7eba422aa52a442" title="Click to view the MathML source">C¹(S_g)${\mathcal{C}}^1(S_g)$. In Birman et al. (2016), the authors introduced the concept of efficient geodesics   in e4cd957ee79b7eba422aa52a442" title="Click to view the MathML source">C¹(S_g)${\mathcal{C}}^1(S_g)$ and used them to give a new algorithm for computing the distance between vertices. In this note, we introduce the software package MICC (Metric in the Curve Complex  ), a partial implementation of the efficient geodesic algorithm. We discuss the mathematics underlying MICC and give applications. In particular, up to an action of an element of the mapping class group, we give a calculation which produces all distance 4 vertex pairs for 9da991fe6c3deb11c5fc86aa" title="Click to view the MathML source">g=2$g=2$ that intersect 12 times, the minimal number of intersections needed for this distance and genus.