Surgery description of colored knots
The pair (K,ρ) consisting of a knot K S3 and a surjective map ρ from the knot group onto a dihedral group of order 2p for p an odd integer is said to be a p-colored knot. In [Algebr. Geom. Topol. 6(2006) 673-697] D Moskovich conjectures that there are exactly p equivalence classes of p-colored knots up to surgery along unknots in the kernel of the coloring. He shows that for p = 3 and 5 the conjecture holds and that for any odd p there are at least p distinct classes, but gives no general upper bound. We show that there are at most 2p equivalence classes for any odd p. In [Math. Proc. Cambridge Philos. Soc. 131 (2001) 97-127] T Cochran, A Gerges and K Orr, 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 S3 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 the nused to establish the upper bound as well as to obtain a complete invariant of p-colored knot surgery equivalence. © 2008 Mathematical Sciences Publishers.
Publication Source (Journal or Book title)
Algebraic and Geometric Topology
Litherland, R., & Wallace, S. (2008). Surgery description of colored knots. Algebraic and Geometric Topology, 8 (3), 1295-1332. https://doi.org/10.2140/agt.2008.8.1295