Surgery presentations of coloured knots and of their covering links

Abstract

We consider knots equipped with a representation of their knot groups onto a dihedral group $D_{2n}$ (where $n$ is odd). To each such knot there corresponds a closed $3$–manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relating to the problem of passing from the initial data of a $D_{2n}$–coloured knot to a surgery presentation of the corresponding branched covering space and covering link. In particular, we describe effective algorithms for constructing such presentations. A by-product of these investigations is a proof of the conjecture that two $D_{2n}$–coloured knots are related by a sequence of surgeries along $\pm 1$–framed unknots in the kernel of the representation if and only if they have the same coloured untying invariant (a ${\mathbb{Z}}_n$–valued algebraic invariant of $D_{2n}$–coloured knots).