Algebraic & Geometric Topology 9 (2009) 1341–1398

DOI: 10.2140/agt.2009.9.1341

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

Mathematical Subject Classification

Primary: 57M12

Secondary: 57M25

References

Publication

Received: 22 August 2008
Revised: 1 June 2009
Accepted: 1 June 2009
Published: 5 July 2009

Authors