Volume 9, issue 3 (2009)

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Surgery presentations of coloured knots and of their covering links

Andrew Kricker and Daniel Moskovich

Algebraic & Geometric Topology 9 (2009) 1341–1398

We consider knots equipped with a representation of their knot groups onto a dihedral group D2n (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 D2n–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 D2n–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 n–valued algebraic invariant of D2n–coloured knots).

dihedral covering, covering space, covering linkage, Fox $n$–colouring, surgery presentation
Mathematical Subject Classification 2000
Primary: 57M12
Secondary: 57M25
Received: 22 August 2008
Revised: 1 June 2009
Accepted: 1 June 2009
Published: 5 July 2009
Andrew Kricker
School of Physical & Mathematical Sciences
Nanyang Technological University
21 Nanyang Link
Singapore 637371
Daniel Moskovich
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502