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

From skein theory to presentations for Thompson group