Generalized n-colorings of links

Daniel Silver; Susan Williams

Banach Center Publications (1998)

  • Volume: 42, Issue: 1, page 381-394
  • ISSN: 0137-6934

Abstract

top
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.

How to cite

top

Silver, Daniel, and Williams, Susan. "Generalized n-colorings of links." Banach Center Publications 42.1 (1998): 381-394. <http://eudml.org/doc/208818>.

@article{Silver1998,
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.},
author = {Silver, Daniel, Williams, Susan},
journal = {Banach Center Publications},
keywords = {-coloring; satellite knot},
language = {eng},
number = {1},
pages = {381-394},
title = {Generalized n-colorings of links},
url = {http://eudml.org/doc/208818},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Silver, Daniel
AU - Williams, Susan
TI - Generalized n-colorings of links
JO - Banach Center Publications
PY - 1998
VL - 42
IS - 1
SP - 381
EP - 394
AB - 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.
LA - eng
KW - -coloring; satellite knot
UR - http://eudml.org/doc/208818
ER -

References

top
  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. Zbl0126.39105
  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. Zbl0195.54002
  5. [Ha] R. Hartley, Metabelian representations of knot groups, Pacific J. Math. 82 (1979), 93-104. Zbl0404.20032
  6. [HaKe] J. C. Hausmann and M. Kervaire, Sous-groupes dérivés des groupes de noeuds, L'Enseign. Math. 24 (1978), 111-123. Zbl0414.57012
  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. Zbl0339.55004
  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. Zbl0035.11103
  15. [SiWi1] D. S. Silver and S. G. Williams, Augmented group systems and shifts of finite type, Israel J. Math. 95 (1996), 231-251. Zbl0899.20011
  16. [SiWi2] D. S. Silver and S. G. Williams, Knot invariants from symbolic dynamical systems, Trans. Amer. Math. Soc., to appear. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.