The Jones polynomials of three-bridge knots via Chebyshev knots and billiard table diagrams

Abstract

This work presents formulas for the Kauffman bracket and Jones polynomials of three-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation expansion. The subject is introduced by considering the easier case of two-bridge knots, where some geometric interpretation is provided, as well, via combinatorial tiling problems.