Surgery untying of coloured knots

A p–colouring of a knot K is a surjective homomorphism ρ from its knot group G := π1(S − K) to D2p := {t, s|t2 = sp = 1, tst = sp−1} the dihedral group of order 2p, when p is any odd integer. The pair (K, ρ) is called a p–coloured knot. It is a well-known fact that we can encode ρ as a colouring of arcs of a knot diagram by elements of Zp (the cyclic group of order p), subject to the ‘colouring rule’ that at least two colours are used, and that at each crossing half the sum of the labels of the under-crossing arcs equals the label of the over-crossing arc modulo p. By labeling an arc by an element n ∈ Zp , we are indicating that ρ maps its meridian to the element tsn ∈ D2p . If a knot K admits such a colouring the knot is said to be p–colourable, and ρ is said to be a p–colouring of K . These definitions may be extended to links and to tangles. A necessary and sufficient condition for a knot to be p–colourable is that its determinant be divisible by p. Whether or not a knot is p–colourable is the simplest non-trivial invariant which detects non-commutativity of the knot group. For more about p–colourability we refer the reader to Fox’s original paper [4].