  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.
