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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si1.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=9529b76f6212ac0d6bbb9747da168676" title="Click to view the MathML source">C(S_g)class="mathContainer hidden">class="mathCode">$\mathcal{C}(S_g)$ of a closed orientable surface of genus class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si2.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=79bd4cdc28e3146c5778dd7d0358fd5e" title="Click to view the MathML source">g≥2class="mathContainer hidden">class="mathCode">$g\ge 2$ is the simplicial complex whose vertices, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si3.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=f99230f00cff508873e397927f102261" title="Click to view the MathML source">C⁰(S_g)class="mathContainer hidden">class="mathCode">${\mathcal{C}}^0(S_g)$, are isotopy classes of essential simple closed curves in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si4.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=00839a11bcdd049e2002ea813af1e835" title="Click to view the MathML source">S_gclass="mathContainer hidden">class="mathCode">$S_g$. Two vertices co-bound an edge of the 1-skeleton, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si5.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=50400e4cd957ee79b7eba422aa52a442" title="Click to view the MathML source">C¹(S_g)class="mathContainer hidden">class="mathCode">${\mathcal{C}}^1(S_g)$, if there are disjoint representatives in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si4.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=00839a11bcdd049e2002ea813af1e835" title="Click to view the MathML source">S_gclass="mathContainer hidden">class="mathCode">$S_g$. A metric is obtained on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si3.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=f99230f00cff508873e397927f102261" title="Click to view the MathML source">C⁰(S_g)class="mathContainer hidden">class="mathCode">${\mathcal{C}}^0(S_g)$ by assigning unit length to each edge of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si5.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=50400e4cd957ee79b7eba422aa52a442" title="Click to view the MathML source">C¹(S_g)class="mathContainer hidden">class="mathCode">${\mathcal{C}}^1(S_g)$. Thus, the distance between two vertices, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si6.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=5ee4c024e58259ba618e5dc628449d53" title="Click to view the MathML source">d(v,w)class="mathContainer hidden">class="mathCode">$d(v,w)$, corresponds to the length of a geodesic—a shortest edge-path between v and w   in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si5.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=50400e4cd957ee79b7eba422aa52a442" title="Click to view the MathML source">C¹(S_g)class="mathContainer hidden">class="mathCode">${\mathcal{C}}^1(S_g)$. In Birman et al. (2016), the authors introduced the concept of efficient geodesics   in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si5.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=50400e4cd957ee79b7eba422aa52a442" title="Click to view the MathML source">C¹(S_g)class="mathContainer hidden">class="mathCode">${\mathcal{C}}^1(S_g)$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300116&_mathId=si7.gif&_user=111111111&_pii=S0747717116300116&_rdoc=1&_issn=07477171&md5=e3f8fc6e9da991fe6c3deb11c5fc86aa" title="Click to view the MathML source">g=2class="mathContainer hidden">class="mathCode">$g=2$ that intersect 12 times, the minimal number of intersections needed for this distance and genus.