On Jones' subgroup of R. Thompson group F
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- 作者：Gili Golan1 ; gili.golan@math.biu.ac.il" class="auth_mail" title="E-mail the corresponding author ; Mark Sapir ; 2 ; m.sapir@vanderbilt.edu" class="auth_mail" title="E-mail the corresponding author
- 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 R3, and a certain subgroup ac1dcc7">
of F encodes all oriented knots and links. We answer several questions of Jones about ac1dcc7">. In particular we prove that the subgroup ac1dcc7"> is generated by x0x1, acc61ad3f3bdacb24f5b33" title="Click to view the MathML source">x1x2, x2x3 (where xi, acd4b4278ed06a0943379" title="Click to view the MathML source">i∈N are the standard generators of F ) and is isomorphic to F3, the analog of F where all slopes are powers of 3 and break points are 3-adic rationals. We also show that ac1dcc7"> coincides with its commensurator. Hence the linearization of the permutational representation of F on 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.
of F encodes all oriented knots and links. We answer several questions of Jones about ac1dcc7">. In particular we prove that the subgroup ac1dcc7"> is generated by x0x1, acc61ad3f3bdacb24f5b33" title="Click to view the MathML source">x1x2, x2x3 (where xi, acd4b4278ed06a0943379" title="Click to view the MathML source">i∈N are the standard generators of F ) and is isomorphic to F3, the analog of F where all slopes are powers of 3 and break points are 3-adic rationals. We also show that ac1dcc7"> coincides with its commensurator. Hence the linearization of the permutational representation of F on 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.