Thompson's group, braid groups and link homology 38 views

The Jones polynomial was introduced by Vaughan Jones as a polynomial invariant for links in his seminal work of braid group representations via von Neumann algebras. At the beginning of the century, Mikhail Khovanov developed a homology theory of links categorifying the Jones polynomial. By extending this theory to tangles, Khovanov constructed a braid group action on the category of geometric bimodule chain complexes, as part of the efforts to explore 4-dimensional TQFTs. In more recent work, Jones introduced a unitary representation of Thompson's group F, which can give rise to links and Jones polynomials similarly to braid groups. Inspired by Jones' work, this thesis studies the connection of Thompson's group to link homology and contact geometry, and examines deeper connections between Thompson's group and braid groups. First, we extend Jones' link construction to tangles using Thompson's group, and we construct a lax Thompson's group action on Khovanov's category of geometric bimodule chain complexes. Second, we associate to each element in Thompson's group a canonical Legendrian link representative, and we provide lower and upper bounds for the maximal Thurston-Bennequin number using the Thompson index. Third, we introduce the dyadic braid group to unify Jones braid group representation with his Thompson's group representation.

PHD (Doctor of Philosophy)

English

2026-04-29