Jones rational coincidences

TL;DR

This paper explores coincidences in Jones polynomials among rational knots, introduces moves on continued fractions that preserve the polynomial, and conjectures these generate all such coincidences, supported by computational evidence.

Contribution

It introduces new moves on continued fractions that preserve the Jones polynomial and conjectures they generate all rational knot coincidences, along with a simplified formula for the Jones polynomial.

Findings

Moves on continued fractions do not change the Jones polynomial.

Conjecture that these moves generate all Jones rational coincidences.

New, simpler formula for the Jones polynomial of rational knots.

Abstract

We investigate coincidences of the (one-variable) Jones polynomial amongst rational knots, what we call `Jones rational coincidences'. We provide moves on the continued fraction expansion of the associated rational which we prove do not change the Jones polynomial and conjecture (based on experimental evidence from all rational knots with determinant $<900$) that these moves are sufficient to generate all Jones rational coincidences. These coincidences are generically not mutants, as is verified by checking the HOMFLYPT polynomial. In the process we give a new formula for the Jones polynomial of a rational knot based on a continued fraction expansion of the associated rational, which has significantly fewer terms than other formulae known to us. The paper is based on the second author's Ph.D. thesis and gives an essentially self-contained account.