Mutant knots with symmetry

TL;DR

This paper investigates the properties of mutant knots with symmetry, showing that under certain conditions their m-string satellites share the same Homfly polynomial for small m, with exceptions demonstrated via quantum invariants.

Contribution

It establishes conditions under which mutant knots with symmetry have identical Homfly polynomials for m-string satellites, extending known results for 2-string satellites.

Findings

Mutant knots with symmetry share the same Homfly polynomial for m<6 under certain conditions.

All m-string satellites share the same Homfly polynomial when based on a cable knot pattern.

Counterexamples are provided where mutants differ in their 6-string satellite invariants.

Abstract

Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m>2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directed m-string satellites of mutants share the same Homfly polynomial for m<6 in general, and for all choices of m when the satellite is based on a cable knot pattern. We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantum sl(3) invariants.