Surgery description of colored knots

Litherland, Richard A¹ ; Wallace, Steven D¹

¹Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Algebraic and Geometric Topology, Tome 8 (2008) no. 3, pp. 1295-1332

Résumé

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.

Détail

DOI : 10.2140/agt.2008.8.1295

Keywords: p-colored knot, Fox coloring, surgery, bordism

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