Distinguishing Mutant Knots

TL;DR

This paper investigates methods to distinguish mutant knots using advanced polynomial invariants, calculating specific polynomials in complex representations to identify differences that simpler invariants cannot detect.

Contribution

It introduces calculations of mutant knot polynomials in higher and non-symmetric representations, expanding the tools for knot distinction beyond traditional invariants.

Findings

Mutant knots can be distinguished using polynomials in representations [3,1] and [4,2].

Differences in polynomial invariants reveal distinctions not captured by symmetric or antisymmetric representations.

Complex representations are essential for identifying certain mutant knots.

Abstract

Knot theory is actively studied both by physicists and mathematicians as it provides a connecting centerpiece for many physical and mathematical theories. One of the challenging problems in knot theory is distinguishing mutant knots. Mutant knots are not distinguished by colored HOMFLY-PT polynomials for knots colored by either symmetric and or antisymmetric representations of $SU(N)$. Some of the mutant knots can be distinguished by the simplest non-symmetric representation $[2,1]$. However there is a class of mutant knots which require more complex representations like $[4,2]$. In this paper we calculate polynomials and differences for the mutant knot polynomials in representations $[3,1]$ and $[4,2]$ and study their properties.