Surgery description of colored knots

TL;DR

This paper investigates the classification of p-colored knots under surgery, establishing an upper bound of 2p equivalence classes, which improves previous bounds and connects to bordism invariants.

Contribution

It provides a new upper bound of 2p for the number of equivalence classes of p-colored knots under surgery, refining Moskovich's conjecture.

Findings

Upper bound of 2p equivalence classes for p-colored knots.

Connection between colored knot invariants and bordism invariants.

Improvement over previous bounds for p=3 and p=5.

Abstract

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.