Surgery presentations of coloured knots and of their covering links

Kricker, Andrew; Moskovich, Daniel

doi:10.2140/agt.2009.9.1341

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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

DOI: 10.2140/agt.2009.9.1341

Abstract

We consider knots equipped with a representation of their knot groups onto a dihedral group $D_{2 n}$ (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 $D_{2 n}$ –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 $D_{2 n}$ –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 $ℤ_{n}$ –valued algebraic invariant of $D_{2 n}$ –coloured knots).

Keywords

dihedral covering, covering space, covering linkage, Fox $n$–colouring, surgery presentation

Mathematical Subject Classification 2000

Primary: 57M12

Secondary: 57M25

References

Publication

Received: 22 August 2008

Revised: 1 June 2009

Accepted: 1 June 2009

Published: 5 July 2009

Authors

Andrew Kricker
School of Physical & Mathematical Sciences Nanyang Technological University SPMS-04-01 21 Nanyang Link Singapore 637371
http://www.spms.ntu.edu.sg/MAS/
Daniel Moskovich
Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan
http://www.sumamathematica.com/

Volume 9, issue 3 (2009)