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2009 Surgery presentations of coloured knots and of their covering links
Andrew Kricker, Daniel Moskovich
Algebr. Geom. Topol. 9(3): 1341-1398 (2009). DOI: 10.2140/agt.2009.9.1341

Abstract

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).

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Andrew Kricker. Daniel Moskovich. "Surgery presentations of coloured knots and of their covering links." Algebr. Geom. Topol. 9 (3) 1341 - 1398, 2009. https://doi.org/10.2140/agt.2009.9.1341

Information

Received: 22 August 2008; Revised: 1 June 2009; Accepted: 1 June 2009; Published: 2009
First available in Project Euclid: 20 December 2017

zbMATH: 1179.57003
MathSciNet: MR2520403
Digital Object Identifier: 10.2140/agt.2009.9.1341

Subjects:
Primary: 57M12
Secondary: 57M25

Rights: Copyright © 2009 Mathematical Sciences Publishers

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