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 hmlsrc">hImg" 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)hContainer hidden">hCode">h altimg="si1.gif" overflow="scroll">hvariant="script">Chy="false">(w>Sw>w>gw>hy="false">)h> of a closed orientable surface of genus hmlsrc">hImg" 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≥2hContainer hidden">hCode">h altimg="si2.gif" overflow="scroll">g≥2h> is the simplicial complex whose vertices, hmlsrc">hImg" 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)hContainer hidden">hCode">h altimg="si3.gif" overflow="scroll">w>hvariant="script">Cw>w>0w>hy="false">(w>Sw>w>gw>hy="false">)h>, are isotopy classes of essential simple closed curves in hmlsrc">hImg" 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_ghContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">w>Sw>w>gw>h>. Two vertices co-bound an edge of the 1-skeleton, hmlsrc">hImg" 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)hContainer hidden">hCode">h altimg="si5.gif" overflow="scroll">w>hvariant="script">Cw>w>1w>hy="false">(w>Sw>w>gw>hy="false">)h>, if there are disjoint representatives in hmlsrc">hImg" 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_ghContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">w>Sw>w>gw>h>. A metric is obtained on hmlsrc">hImg" 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)hContainer hidden">hCode">h altimg="si3.gif" overflow="scroll">w>hvariant="script">Cw>w>0w>hy="false">(w>Sw>w>gw>hy="false">)h> by assigning unit length to each edge of hmlsrc">hImg" 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)hContainer hidden">hCode">h altimg="si5.gif" overflow="scroll">w>hvariant="script">Cw>w>1w>hy="false">(w>Sw>w>gw>hy="false">)h>. Thus, the distance between two vertices, hmlsrc">hImg" 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)hContainer hidden">hCode">h altimg="si6.gif" overflow="scroll">dhy="false">(v,why="false">)h>, corresponds to the length of a geodesic—a shortest edge-path between v and w   in hmlsrc">hImg" 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)hContainer hidden">hCode">h altimg="si5.gif" overflow="scroll">w>hvariant="script">Cw>w>1w>hy="false">(w>Sw>w>gw>hy="false">)h>. In Birman et al. (2016), the authors introduced the concept of efficient geodesics   in hmlsrc">hImg" 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)hContainer hidden">hCode">h altimg="si5.gif" overflow="scroll">w>hvariant="script">Cw>w>1w>hy="false">(w>Sw>w>gw>hy="false">)h> 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 hmlsrc">hImg" 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=2hContainer hidden">hCode">h altimg="si7.gif" overflow="scroll">g=2h> that intersect 12 times, the minimal number of intersections needed for this distance and genus.