On Jones' subgroup of R. Thompson group F

Abstract

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 F→ of F encodes all oriented knots and links. We answer several questions of Jones about F→. In particular we prove that the subgroup F→ is generated by x0x1, x1x2, x2x3 (where xi, 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 F→ coincides with its commensurator. Hence the linearization of the permutational representation of F on F/F→ 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.