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Surgery description of colored knots

R A Litherland and Steven D Wallace

Algebraic & Geometric Topology 8 (2008) 1295–1332

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 then used to establish the upper bound as well as to obtain a complete invariant of p–colored knot surgery equivalence.

p-colored knot, Fox coloring, surgery, bordism
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27, 55N22, 57M12
Received: 7 October 2007
Revised: 29 May 2008
Accepted: 1 June 2008
Published: 8 August 2008
R A Litherland
Department of Mathematics
Louisiana State University
Baton Rouge
Louisiana 70803
Steven D Wallace
Department of Mathematics
Louisiana State University
Baton Rouge
Louisiana 70803