Algebraic & Geometric Topology

Surgery description of colored knots

Richard A Litherland and Steven D Wallace

Full-text: Open access


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

p-colored knot Fox coloring surgery bordism


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.

Export citation


  • A Adem, R J Milgram, Cohomology of finite groups, volume 309 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Berlin (1994)
  • G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter & Co., Berlin (1985)
  • T D Cochran, A Gerges, K Orr, Dehn surgery equivalence relations on 3–manifolds, Math. Proc. Cambridge Philos. Soc. 131 (2001) 97–127
  • P E Conner, E E Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 33, Academic Press, Publishers, New York (1964)
  • R H Crowell, R H Fox, Introduction to knot theory, Springer, New York (1977) \qua Reprint of the 1963 original, Graduate Texts in Mathematics, No. 57
  • J F Davis, P Kirk, Lecture notes in algebraic topology, Graduate Studies in Mathematics 35, American Mathematical Society, Providence, RI (2001)
  • P Gilmer, Classical knot and link concordance, Comment. Math. Helv. 68 (1993) 1–19
  • R E Gompf, A Stipsicz, $4$–manifolds and Kirby Calculus, Graduate Studies in Mathematics 20, AMS Providence (1991)
  • C M Gordon, R A Litherland, On the signature of a link, Invent. Math. 47 (1978) 53–69
  • R C Kirby, The topology of $4$–manifolds, Lecture Notes in Mathematics 1374, Springer, Berlin (1989)
  • KnotInfo Table of Knots\qua\char'176knotinfo/
  • W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer, New York (1997)
  • C Livingston, Knot theory, Carus Mathematical Monographs 24, Mathematical Association of America, Washington, DC (1993)
  • D Moskovich, Surgery untying of coloured knots, Algebr. Geom. Topol. 6 (2006) 673–697
  • J R Munkres, Elements of algebraic topology, Addison–Wesley Publishing Company, Menlo Park, CA (1984)
  • D Rolfsen, Knots and links, Publish or Perish, Berkeley, CA (1976) \qua Mathematics Lecture Series, No. 7
  • G W Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics 61, Springer, New York (1978)