#### Volume 9, issue 3 (2009)

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 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Surgery presentations of coloured knots and of their covering links

### Andrew Kricker and Daniel Moskovich

Algebraic & Geometric Topology 9 (2009) 1341–1398
##### 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 $±1$–framed unknots in the kernel of the representation if and only if they have the same coloured untying invariant (a ${ℤ}_{n}$–valued algebraic invariant of ${D}_{2n}$–coloured knots).

##### Keywords
dihedral covering, covering space, covering linkage, Fox $n$–colouring, surgery presentation
Primary: 57M12
Secondary: 57M25