Vol. 231, No. 2, 2007

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ISSN: 0030-8730
Knot colouring polynomials

Michael Eisermann

Vol. 231 (2007), No. 2, 305–336

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 πK be the fundamental group of the knot complement S3 \ K, 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 as ρρ(lK). The resulting invariant maps knots to the group ring 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 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.

fundamental group of a knot, peripheral system, knot group homomorphism, quandle 2-cocycle state-sum invariant, Yang–Baxter invariant
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27
Received: 24 July 2004
Revised: 25 July 2007
Accepted: 30 July 2007
Published: 1 June 2007
Michael Eisermann
Institut Fourier
Université Grenoble I
100 rue des Maths, BP 74
38402 St Martin d’Hères