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${\mathbb{R}}^3$, 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$x_0{x}_1$, acc61ad3f3bdacb24f5b33" title="Click to view the MathML source">x₁x₂$x_1{x}_2$, x₂x₃$x_2{x}_3$ (where x_i$x_i$, aeb42098acd4b4278ed06a0943379" title="Click to view the MathML source">i∈N$i\in \mathbb{N}$ are the standard generators of F  ) and is isomorphic to F₃$F_3$, 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 aead5"> 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.