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 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 S_g$S_g$. Two vertices co-bound an edge of the 1-skeleton, C¹(S_g)${\mathcal{C}}^1(S_g)$, if there are disjoint representatives in 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 C¹(S_g)${\mathcal{C}}^1(S_g)$. Thus, the distance between two vertices, d(v,w)$d(v,w)$, corresponds to the length of a geodesic—a shortest edge-path between v and w   in C¹(S_g)${\mathcal{C}}^1(S_g)$. In Birman et al. (2016), the authors introduced the concept of efficient geodesics   in 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 g=2$g=2$ that intersect 12 times, the minimal number of intersections needed for this distance and genus.