From skein theory to presentations for Thompson group

TL;DR

This paper explores the connection between skein theory, subfactor planar algebras, and Thompson group representations, revealing new subgroup structures and addressing a question about 3-colorable subgroups.

Contribution

It establishes a link between skein theory and subgroup presentations in Thompson group representations, including a resolution of Jones's question on 3-colorable subgroups.

Findings

Identified relations between skein theory and subgroup presentations.

Characterized a large class of subgroups within Thompson group F.

Answered Jones's question on the 3-colorable subgroup.

Abstract

Jones introduced unitary representations of Thompson group $F$ starting from a given subfactor planar algebra, and all unoriented links arise as matrix coefficients of these representations. Moreover, all oriented links arise as matrix coefficients of a subgroup $\vec{F}$ which is the stabilizer of a certain vector. Later Golan and Sapir determined the subgroup $\vec{F}$ and showed many interesting properties. In this paper, we investigate into a large class of groups which arises as subgroups of Thompson group $F$ and reveal the relation between the skein theory of the subfactor planar algebra and the presentation of subgroup related to the corresponding unitary representation. Specifically, we answer a question by Jones about the 3-colorable subgroup.