We introduce a natural extension of the colouring numbers of knots, called colouring
polynomials, and study 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
${\mathbb{S}}^{3}\backslash K$, and
let
${m}_{K},{l}_{K}\in {\pi}_{K}$
be a meridianlongitude pair. Given a finite group
$G$ and an element
$x\in G$ we consider the set
of representations
$\rho :{\pi}_{K}\to G$
with
$\rho \left({m}_{K}\right)=x$ and define the
colouring polynomial as
${\sum}_{\rho}\rho \left({l}_{K}\right)$.
The resulting invariant maps knots to the group ring
$\mathbb{Z}G$. 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
statesum 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 the colouring
polynomial can be presented as a Yang–Baxter invariant, i.e. as the trace of some
linear braid group representation. This entails that Yang–Baxter invariants
can
detect noninversible and nonreversible knots.
