Bilinear Forms on Skein Modules and Steps in Dyck Paths

TL;DR

This paper constructs bases for a skein module using Jones-Wenzl idempotents, analyzes a bilinear form on it, and relates its determinant to combinatorial objects like Dyck paths and semi-meanders.

Contribution

It introduces a new basis for the skein module and connects the bilinear form's determinant to combinatorial path counting and semi-meander determinants.

Findings

Calculated the determinant of the bilinear form matrix.

Reduced the determinant computation to counting steps in generalized Dyck paths.

Established a relation between the determinant and semi-meander structures.

Abstract

We use Jones-Wenzl idempotents to construct bases for the relative Kauffman bracket skein module of a square with n points colored 1 and one point colored h. We consider a natural bilinear form on this skein module. We calculate the determinant of the matrix for this form with respect to the natural basis. We reduce the computation to count some steps in generalized Dyck paths. Moreover, we relate our determinant to a determinant on semi-meanders.