On Jones' subgroup of R. Thompson group F

Recently Vaughan Jones showed that the R. Thompson group F encodes in a natural way all knots and links in R-3, and a certain subgroup (F) over right arrow of F encodes all oriented knots and links. We answer several questions of Jones about (F) over right arrow. In particular we prove that the subgroup (F) over right arrow is generated by x(0)x(1), x(1)x(2)) x(2)x(3) (where xi, i is an element of N are the standard generators of F) and is isomorphic to F-3, the analog of F where all slopes are powers of 3 and break points are 3-adic rationals. We also show that (F) over right arrow coincides with its commensurator. Hence the linearization of the permutational representation of F on F/(F) over right arrow 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. (C) 2016 Elsevier Inc. All rights reserved.