We consider knots equipped with a representation of their knot groups onto a dihedral
group
(where
is odd). To each such knot there corresponds a closed
–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
–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
–coloured
knots are related by a sequence of surgeries along
–framed unknots
in the kernel of the representation if and only if they have the same coloured untying invariant (a
–valued algebraic
invariant of –coloured
knots).
Keywords
dihedral covering, covering space, covering linkage, Fox
$n$–colouring, surgery presentation
