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Surgery description of colored knots
Richard A Litherland and Steven D Wallace
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Algebraic and Geometric Topology 8 (2008)
1295–1332
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Abstract
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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.
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Keywords
p-colored knot, Fox coloring, surgery,
bordism
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Mathematical Subject Classification
Primary: 57M25
Secondary: 55N22, 57M12, 57M27
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Publication
Received: 7 October 2007
Revised: 29 May 2008
Accepted: 1 June 2008
Published: 8 August 2008
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