Knot colouring polynomials

TL;DR

This paper introduces colouring polynomials as a new knot invariant that generalizes existing invariants, relates to Yang-Baxter invariants and quandle 2-cocycle invariants, and can distinguish complex knot symmetries.

Contribution

It defines colouring polynomials for knots, links them to Yang-Baxter and quandle invariants, and demonstrates their effectiveness in distinguishing knots and symmetries.

Findings

Colouring polynomials distinguish knots where other invariants fail.

They can differentiate mutants, reverses, and inverses of knots.

Every quandle 2-cocycle invariant is a specialization of a colouring polynomial.

Abstract

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longitude pair in \pi_K. Given a finite group G and an element x in G, we consider the set of representations \rho from \pi_K to G that map the meridian m_K to x, and define the colouring polynomial P(K) as the sum over all longitude images \rho(l_K). The resulting invariant maps knots to the group ring Z[G]. It is multiplicative with respect to connected sum and equivariant with respect to symmetry operations of knots. Examples are given to show that colouring polynomials distinguish knots for which other invariants fail, in particular they can distinguish knots from their mutants, obverses, inverses, or reverses. We prove that every quandle 2-cocycle state-sum invariant of knots is a specialization of some knot colouring polynomial. This provides a complete topological interpretation of these invariants in terms of the knot group and its peripheral system. Furthermore, we show that P can be presented as a Yang-Baxter invariant, i.e. as the trace of some linear braid group representation. This entails in particular that Yang-Baxter invariants can detect non-inversible and non-reversible knots.