Download PDF - Generalized n-colorings of linksAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Generalized n-colorings of links

Authors 1, 1

Affiliations

  1. Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688, U.S.A.

Abstract

The notion of an (n,r)-coloring for a link diagram generalizes the idea of an n-coloring introduced by R. H. Fox. For any positive integer n the various (n,r)-colorings of a diagram for an oriented link l correspond in a natural way to the periodic points of the representation shift Φ/n(l) of the link. The number of (n,r)-colorings of a diagram for a satellite knot is determined by the colorings of its pattern and companion knots together with the winding number.

Bibliography

  1. [BuZi] G. Burde and H. Zieschang, Knots, de Gruyter Stud. in Math. 5, de Gruyter, Berlin, 1985.
  2. [CrFo] R. H. Crowell and R. H. Fox, An Introduction to Knot Theory, Ginn and Co., 1963.
  3. [Fo1] R. H. Fox, A quick trip through knot theory, in: Topology of 3-Manifolds and Related Topics, M. K. Fort (ed.), Prentice-Hall, N.J. (1961), 120-167.
  4. [Fo2] R. H. Fox, Metacyclic invariants of knots and links, Canad. J. Math. 22 (1970), 193-201.
  5. [Ha] R. Hartley, Metabelian representations of knot groups, Pacific J. Math. 82 (1979), 93-104.
  6. [HaKe] J. C. Hausmann and M. Kervaire, Sous-groupes dérivés des groupes de noeuds, L'Enseign. Math. 24 (1978), 111-123.
  7. [Lt] R. A. Litherland, Cobordism of satellite knots, Contemp. Math. 35 (1984), 327-362.
  8. [LvMe] C. Livingston and P. Melvin, Abelian invariants of satellite knots, in: Geometry and Topology, C. McA. Gordon (ed.), Lecture Notes in Math. 1167, Springer, Berlin, 1985, 217-227.
  9. [LySc] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer, Berlin, 1977.
  10. [Pr] J. H. Przytycki, 3-coloring and other elementary invariants of knots, these proceedings.
  11. [Re] K. Reidemeister, Knotentheorie, Ergeb. Math. Grenzgeb. 1, Springer, Berlin, 1932; English translation: Knot Theory, BCS Associates, Moscow, Idaho, 1983.
  12. [Ro] D. Rolfsen, Knots and Links, Math. Lecture Ser. 7, Publish or Perish Inc., Berkeley, 1976.
  13. [Sc] H. Schubert, Knoten und Vollringe, Acta Math. 90 (1953), 131-286.
  14. [Se] H. Seifert, On the homology invariants of knots, Quart. J. Math. Oxford 2 (1950), 23-32.
  15. [SiWi1] D. S. Silver and S. G. Williams, Augmented group systems and shifts of finite type, Israel J. Math. 95 (1996), 231-251.
  16. [SiWi2] D. S. Silver and S. G. Williams, Knot invariants from symbolic dynamical systems, Trans. Amer. Math. Soc., to appear.
Banach Center Publications
1998/42/1
Export to PBN, Opens in new tab
Pages:
381-394
Main language of publication
English
Published
1998
Exact and natural sciences