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 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, x...

Full description

Saved in:
Bibliographic Details
Published in:Journal of algebra 2017-01, Vol.470, p.122-159
Main Authors: Golan, Gili, Sapir, Mark
Format: Article
Language:English
Subjects:
Diagram groups
Knots and links
R. Thompson group
Tree-diagrams
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2016.09.001