Hostname: page-component-5c6d5d7d68-wbk2r Total loading time: 0 Render date: 2024-08-08T14:15:14.747Z Has data issue: false hasContentIssue false
  • English
  • Français

On the braid index of links with nested diagrams

Published online by Cambridge University Press:  24 October 2008

D. A. Chalcraft
D. A. Chalcraft
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Cambridge CB2 1SB
Get access Rights & Permissions [Opens in a new window]

Abstract

The number of Seifert circuits in a diagram of a link is well known 9 to be an upper bound for the braid index of the link. The -breadth of the so-called P-polynomial 3 of the link is known 5, 2 to give a lower bound. In this paper we consider a large class of links diagrams, including all diagrams where the interior of every Seifert circuit is empty. We show that either these bounds coincide, or else the upper bound is not sharp, and we obtain a very simple criterion for distinguishing these cases.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 2 , March 1992 , pp. 273 - 281
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Birman, J. S.. Braids, Links and Mapping Class Groups. Ann. of Math. Studies no. 82 (Princeton University Press, 1974).Google Scholar
2Franks, J. and Williams, R.. Braids and the JonesConway polynomial. Preprint (1985).Google Scholar
3Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millett, K. C. and Ocneanu, A.. A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. 12 (1985), 239246.Google Scholar
4Jones, V. F. R.. Hecke algebra representation of braid groups and link polynomials. Ann. of Math. (2) 126 (1987), 335388.Google Scholar
5Morton, H. R.. Closed braid representations for a link, and its JonesConway polynomial. Preprint (1985).Google Scholar
6Morton, H. R.. Polynomials from braids. Contemp. Math. 78 (1988), 575585.Google Scholar
7Murasugi, K.. On the braid index of alternating links. Proc. Amer. Math. Soc., to appear.Google Scholar
8Reidemeister, K.. Knotentheorie. Ergebn. Math. Grenzgeb. no. 1 (Springer-Verlag, 1932).Google Scholar
9Yamada, S.. The minimum number of Seifert circles equals the braid index of a link. Invent. Math. 89 (1987), 347356.Google Scholar