Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T07:47:32.000Z Has data issue: false hasContentIssue false
  • English
  • Français

PL Link Isotopy, Essential Knotting and Quotients of Polynomials

Published online by Cambridge University Press:  20 November 2018

Dale Rolfsen
Dale Rolfsen*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Y4
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.

Piecewise-linear (nonambient) isotopy of classical links may be regarded as link theory modulo knot theory. This note considers an adaptation of new (and old) polynomial link invariants to this theory, obtained simply by dividing a link's polynomial by the polynomials of the individual components. The resulting rational functions are effective in distinguishing isotopy classes of links, and in demonstrating that certain links are essentially knotted in the sense that every link in its isotopy class has a knotted component. We also establish geometric criteria for essential knotting of links.

Keywords

57M2557R52
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 4 , 01 December 1991 , pp. 536 - 541
Copyright
Copyright © Canadian Mathematical Society 1991

References

[BLM] Brandt, R. D., Lickorish, W. B. R., Millett, K. C., A polynomial invariant for unoriented links, Invent. Math. 84 (1986), 563–573.Google Scholar
[C] Conway, J. H., An enumeration of knots and links. In Computational problems in abstract algebra (ed. J. Leech), Pergamon Press, 1969.Google Scholar
[FYHLMO] P. Freyd, Yetter, D., Hoste, J., Lickorish, W. B. R., Millett, K. C., Ocneanu, A., A new polynomial invariant of knots and links, Bull. A.M.S. 12 (1985), 239–246.Google Scholar
[H] Ho, C. F., A new polynomial invariant of knots and links, A.M.S. Abstracts 6(1985), 300.Google Scholar
[J] Jones, V., A polynomial invariant for knots via Von Neumann algebras, Bull. A.M.S. 12(1985), 103–111.Google Scholar
[JR] Jin, G. T., Rolfsen, D., Some remarks on rotors in link theory, this volume, 480-484.Google Scholar
[K] Kauffman, L, On knots, Annals of Math. Studies 115(1987).Google Scholar
[R] Rolfsen, D., Isotopy of links in codimension two, J. Indian Math. Soc. 36 (1972), 263–278.Google Scholar
[S] Schubert, H., Knoten und Vollringe, Acta Math. 90 (1953), 131–286.Google Scholar