Mutation invariance for the zeroth coefficients of the colored HOMFLY polynomial

TL;DR

This paper proves that the zeroth coefficient of colored HOMFLY polynomials remains unchanged under mutation, highlighting a limitation in using this invariant to distinguish mutant knots, unlike the full polynomial.

Contribution

The study demonstrates that the zeroth coefficient of cables of the HOMFLY polynomial is mutation-invariant, contrasting with the full polynomial's ability to distinguish mutants.

Findings

Zeroth coefficient of colored HOMFLY polynomial is mutation-invariant.

3-cables of the HOMFLY polynomial can distinguish mutants.

Zeroth coefficient does not distinguish mutant knots.

Abstract

We show that the zeroth coefficient of the cables of the HOMFLY polynomial (colored HOMFLY polynomials) does not distinguish mutants. This makes a sharp contrast with the total HOMFLY polynomial whose 3-cables can distinguish mutants.