Surgery description of colored knots

Litherland, Richard A; Wallace, Steven D

doi:10.2140/agt.2008.8.1295

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Surgery description of colored knots

R A Litherland and Steven D Wallace

Algebraic & Geometric Topology 8 (2008) 1295–1332

DOI: 10.2140/agt.2008.8.1295

Abstract

The pair $(K, ρ)$ consisting of a knot $K \subset S^{3}$ and a surjective map $ρ$ from the knot group onto a dihedral group of order $2 p$ 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 $2 p$ 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.

Keywords

p-colored knot, Fox coloring, surgery, bordism

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 57M27, 55N22, 57M12

References

Publication

Received: 7 October 2007

Revised: 29 May 2008

Accepted: 1 June 2008

Published: 8 August 2008

Authors

R A Litherland
Department of Mathematics Louisiana State University Baton Rouge Louisiana 70803
http://www.math.lsu.edu/~lither
Steven D Wallace
Department of Mathematics Louisiana State University Baton Rouge Louisiana 70803
http://www.math.lsu.edu/~wallace

Volume 8, issue 3 (2008)