Using Chebyshev polynomials, C. Frohman and R. Gelca introduced 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.
Copyright © 2015 World Scientific Publishing.
The definitive version is available at: http://www.worldscientific.com/doi/abs/10.1142/S0218216515500236?journalCode=jktr
Russell, Heather M., and Hoel Queffelec. "Chebyshev Polynomials and the Frohman-Gelca Formula." Journal of Knot Theory and Its Ramifications 24, no. 4 (2015): 1-13. doi:10.1142/S0218216515500236.
Russell, Heather M. and Queffelec, Hoel, "Chebyshev Polynomials and the Frohman-Gelca Formula" (2015). Math and Computer Science Faculty Publications. 125.