On Jones' subgroup of R. Thompson group F

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• 作者：Gili Golanp>1p>p> ; p>p>gili.golan@math.biu.ac.il" class="auth_mail" title="E-mail the corresponding authorp> ; Mark Sapirp> ; p>p>2p>p> ; p>p>m.sapir@vanderbilt.edu" class="auth_mail" title="E-mail the corresponding authorp>
• 关键词：R. Thompson group ; Diagram groups ; Tree-diagrams ; Knots and links
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：470
• 期：Complete
• 页码：122-159
• 全文大小：1853 K

文摘

Recently Vaughan Jones showed that the R. Thompson group F   encodes in a natural way all knots and links in pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si1.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=7d61cdbf8523b68be316608182562497" title="Click to view the MathML source">Rp>3p>pan>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">p>R3p>pan>pan>pan>, and a certain subgroup pan id="mmlsi148" class="mathmlsrc">f&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=9600d3fdff0d917cea640675eac1dcc7">pxl.gif" data-inlimgeid="1-s2.0-S0021869316302976-si148.gif">pt>p://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869316302976-si148.gif">pt>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">F→pan>pan>pan> of F   encodes all oriented knots and links. We answer several questions of Jones about pan id="mmlsi148" class="mathmlsrc">f&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=9600d3fdff0d917cea640675eac1dcc7">pxl.gif" data-inlimgeid="1-s2.0-S0021869316302976-si148.gif">pt>p://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869316302976-si148.gif">pt>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">F→pan>pan>pan>. In particular we prove that the subgroup pan id="mmlsi148" class="mathmlsrc">f&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=9600d3fdff0d917cea640675eac1dcc7">pxl.gif" data-inlimgeid="1-s2.0-S0021869316302976-si148.gif">pt>p://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869316302976-si148.gif">pt>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">F→pan>pan>pan> is generated by pan id="mmlsi18" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si18.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=4030a76762529c76cdef2f96cba665e6" title="Click to view the MathML source">x₀x₁pan>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">x0x1pan>pan>pan>, pan id="mmlsi19" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si19.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=4ddd595c51acc61ad3f3bdacb24f5b33" title="Click to view the MathML source">x₁x₂pan>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">x1x2pan>pan>pan>, pan id="mmlsi20" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si20.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=21efd567dc7a00799a40f3f671eec573" title="Click to view the MathML source">x₂x₃pan>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">x2x3pan>pan>pan> (where pan id="mmlsi21" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si21.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=341d6a87b69736dc3469259e9ecfb8d2" title="Click to view the MathML source">x_ipan>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">xipan>pan>pan>, pan id="mmlsi22" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si22.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=4f7aeb42098acd4b4278ed06a0943379" title="Click to view the MathML source">i∈Npan>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">i∈Npan>pan>pan> are the standard generators of F  ) and is isomorphic to pan id="mmlsi23" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302976&_mathId=si23.gif&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=d2033ad416883e85f51b4d146c54f426" title="Click to view the MathML source">F₃pan>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">F3pan>pan>pan>, the analog of F   where all slopes are powers of 3 and break points are 3-adic rationals. We also show that pan id="mmlsi148" class="mathmlsrc">f&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=9600d3fdff0d917cea640675eac1dcc7">pxl.gif" data-inlimgeid="1-s2.0-S0021869316302976-si148.gif">pt>p://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869316302976-si148.gif">pt>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">F→pan>pan>pan> coincides with its commensurator. Hence the linearization of the permutational representation of F   on pan id="mmlsi9" class="mathmlsrc">f&_user=111111111&_pii=S0021869316302976&_rdoc=1&_issn=00218693&md5=e2e643d7b1a3688eef794be55cfaead5">pxl.gif" data-inlimgeid="1-s2.0-S0021869316302976-si9.gif">pt>p://origin-ars.els-cdn.com/content/image/1-s2.0-S0021869316302976-si9.gif">pt>pan class="mathContainer hidden">pan class="mathCode">flow="scroll">Ffalse">/F→pan>pan>pan> is irreducible. We show how to replace 3 in the above results by an arbitrary n, and to construct a series of irreducible representations of F defined in a similar way. Finally we analyze Jones' construction and deduce that the Thompson index of a link is linearly bounded in terms of the number of crossings in a link diagram.