Knot Colouring Polynomials

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 πK be the fundamental group of the knot complement S 3 rK, and let mK , lK ∈ πK be a meridian-longitude pair. Given a finite group G and an element x ∈ G we consider the set of representations ρ : πK → G with ρ(mK) = x and define the colouring polynomial Px G(K) := ∑ρ ρ(lK). The resulting invariant maps knots to the group ring ZG. 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 Px G 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.