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arXiv · 0805.2307

Surgery presentations of coloured knots and of their covering links

Andrew Kricker·Daniel Moskovich
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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 unit-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).

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BibTeXRIS

Andrew Kricker, Daniel Moskovich. 2008-05-13. Surgery presentations of coloured knots and of their covering links. https://doi.org/10.2140/agt.2009.9.1341

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