Surgery Description of Colored Knots

The pair (K, ρ) consisting of a knotK ⊂ S and a surjective map ρ from the knot group onto a dihedral group is said to be a p-colored knot. In [Mos], D. Moskovich conjectures 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 an 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 [CGO], define invariants of the surgery equivalence class of a closed 3-manifold M in the context of bordism. By taking M to be 0-framed surgery of S 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.