Surgery presentations of coloured knots and of their covering links

TL;DR

This paper explores the relationship between dihedral-coloured knots and their covering links, providing algorithms for surgery presentations and proving a conjecture linking surgeries to a specific algebraic invariant.

Contribution

It introduces effective algorithms for constructing surgery presentations of dihedral branched covers and proves a conjecture relating surgeries to the coloured untying invariant.

Findings

Algorithms for surgery presentations of dihedral branched covers

Proof of the conjecture linking surgeries and the coloured untying invariant

Establishment of the equivalence between certain surgeries and algebraic invariants

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).