Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-16T17:12:57.717Z Has data issue: false hasContentIssue false
  • English
  • Français

On two-bridged knot polynomials

Published online by Cambridge University Press:  09 April 2009

R. I. Hartley
R. I. Hartley
Affiliation:
Department of Mathematics University of MelbourneParkville 3052 Victoria, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The extended diagram of a two-bridged knot is introduced, and it is shown how the coefficients of the Alexander polynomial of the knot may be read straight from this diagram. Using this result, it is shown by diagram manipulation that a conjecture of Fox about the coefficients of the Alexander polynomial of an alternating knot is true at least for two-bridged knots (which are all alternating).

1980 Mathematics subject classification (Amer. Math. Soc.): 57 M 25.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 2 , September 1979 , pp. 241 - 249
Copyright
Copyright © Australian Mathematical Society 1979

References

Crowell, Richard H. (1959), ‘Genus of alternating link types’, Ann. of Math. (2) 69, 258–275.CrossRefGoogle Scholar
Fox, R. H. (1957), ‘Review of Schubert (1956)’, Math. Reviews 18, no. 6, p. 498.Google Scholar
Fox, R. H. (1962), ‘Problems in knot theory’, in Topology of three-manifolds and related topics, Proceedings of the University of Georgia Institute, 1961, edited by Fort, M. K. Jr (Prentice Hall Inc., Englewood Cliffs, N.J.).Google Scholar
Funke, Klaus (1978), ‘Geschlecht von Knoten und die Faserbarkeit ihrer Aussenräume’, Math. Z. 159, 3–24.CrossRefGoogle Scholar
Murasugi, Kunio (1958), ‘On the Alexander polynomial of the alternating knot’, Osaka Math. J. 10, 181–189.Google Scholar
Schubert, Horst (1956), ‘Knoten mit 2 Brücken’, Math. Z. 65, 133–170.CrossRefGoogle Scholar