The Jones polynomials of 3-bridge knots via Chebyshev knots and billiard table diagrams

TL;DR

This paper derives simplified formulas for the Jones polynomials of 3-bridge knots using Chebyshev knots and billiard table diagrams, reducing computational complexity and providing geometric insights.

Contribution

It introduces a novel method to compute Jones polynomials of 3-bridge knots with fewer terms, leveraging Chebyshev knot structures and billiard table diagrams.

Findings

Fewer terms in polynomial formulas compared to Skein relation expansion

Geometric interpretation for 2-bridge knots via tiling problems

Explicit formulas for 3-bridge knot Jones polynomials

Abstract

This work presents formulas for the Kauffman bracket and Jones polynomials of 3-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 2-bridge knots, where some geometric interpretation is provided, as well, via combinatorial tiling problems.