From skein theory to presentations for Thompson group

Jones introduced some unitary representations of Thompson group F constructed 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 F→ which is the stabilizer of a certain vector. Later Golan and Sapir determined the subgroup 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 presentations of subgroups related to the corresponding unitary representation. Specifically, we answer a question by Jones about the 3-colorable subgroup.