Chebyshev polynomials and the Frohman-Gelca formula

TL;DR

This paper provides a diagrammatic proof of the Frohman-Gelca formula for the Kauffman bracket skein module of the torus, highlighting the role of Chebyshev polynomials in simplifying the Jones-Kauffman product.

Contribution

It offers a new diagrammatic proof that clarifies how Chebyshev polynomials facilitate the Product-to-Sum formula in skein modules.

Findings

Diagrammatic proof of Frohman-Gelca formula

Simplification of Jones-Kauffman product via Chebyshev polynomials

Enhanced understanding of skein module basis on the torus

Abstract

Using Chebyshev polynomials, C. Frohman and R. Gelca introduce a basis of the Kauffman bracket skein module of the torus. This basis is especially useful because the Jones-Kauffman product can be described via a very simple Product-to-Sum formula. Presented in this work is a diagrammatic proof of this formula, which emphasizes and demystifies the role played by Chebyshev polynomials.