Surgery description of colored knots

Abstract

The pair $\left(K,\rho \right)$ consisting of a knot $K\subset S^3$ and a surjective map $\rho$ 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 $S^3$ 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 as well as to obtain a complete invariant of $p$–colored knot surgery equivalence.