Mathematics – Geometric Topology
Scientific paper
2007-09-10
Mathematics
Geometric Topology
41 pages, 23 figures (Version 3) Minor revisions and typos fixed. Proofs of Propositions 4.1 and 4.8 revised
Scientific paper
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.
Litherland Richard A.
Wallace Steven D.
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