Algebraic & Geometric Topology

Surgery description of colored knots

Richard A Litherland and Steven D Wallace

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Abstract

The pair (K,ρ) consisting of a knot KS3 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.

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 3 (2008), 1295-1332.

Dates
Received: 7 October 2007
Revised: 29 May 2008
Accepted: 1 June 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796888

Digital Object Identifier
doi:10.2140/agt.2008.8.1295

Mathematical Reviews number (MathSciNet)
MR2443245

Zentralblatt MATH identifier
1156.57009

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 57M12: Special coverings, e.g. branched

Keywords
p-colored knot Fox coloring surgery bordism

Citation

Litherland, Richard A; Wallace, Steven D. Surgery description of colored knots. Algebr. Geom. Topol. 8 (2008), no. 3, 1295--1332. doi:10.2140/agt.2008.8.1295. https://projecteuclid.org/euclid.agt/1513796888


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