  The pair (K,r) consisting of a knot K and a surjective map r from the knot
group onto a dihedral group is said to be a p-colored knot. D. Moskovich
conjectured that for any odd prime p there are exactly p equivalence classes of
p-colored knots up to surgery along unknots in the kernel of the coloring. We
show that there are at most 2p equivalence classes. This is a vast improvement
upon the previous results by Moskovich for p=3, and 5, with no upper bound
given in general. T. Cochran, A. Gerges, and K. Orr, in "Dehn surgery
equivalence relations of 3-manifolds", define invariants of the surgery
equivalence class of a closed 3-manifold M in the context of bordisms. By
taking M to be 0-framed surgery of the 3-sphere along K we may define
Moskovich's colored untying invariant in the same way as the Cochran-Gerges-Orr
invariants. This bordism definition of the colored untying invariant will be
then used to establish the upper bound.

Geometric Topology