Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-29T18:19:31.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Some links with non-adequate minimal-crossing diagrams

Published online by Cambridge University Press:  24 October 2008

Masao Hara  and
Makoto Yamamoto
Masao Hara
Affiliation:
Department of Mathematics, School of Education, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku, Tokyo 16950, Japan
Makoto Yamamoto
Affiliation:
Department of Applied Mathematics, Osaka Women's University, 2-1, Daisen-cho, Sakai, Osaka 590, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

To investigate invariants of links derived from their diagrams, the recent new polynomial invariants of links play important roles. Murasugi6, 7, Kauffman3 and Thistlethwaite 9 independently showed that the number of crossings in a proper connected alternating diagram of a link is the minimal-crossing number of the link and that the writhe of the diagram is invariant. Murasugi 8 also determined the minimal-crossing number of torus links. In 5, Lickorish and Thistlethwaite introduced the concept of an adequate link diagram and showed that the number of crossings in an adequate diagram of a semi-alternating link is the minimal-crossing number of the link. They also determined the minimal-crossing number of almost all Montesinos links. In this paper we show that for some links represented by plats and braids which are not adequate, the numbers of crossings in the diagrams are the minimal-crossing numbers of the links.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 2 , March 1992 , pp. 283 - 289
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

1Brandt, R. D., Lickorish, W. B. R.Millett, K. C.. A polynomial invariant for unoriented knots and links. Invent. Math. 84 (1986), 563573.CrossRefGoogle Scholar
2Ho, C. F.. On the braid index of alternating links. Abstracts Amer. Math. Soc. 64 (1985) 300, 82157.Google Scholar
3Kauffman, L. H.. State models and the Jones polynomial. Topology 26 (1986), 395407.CrossRefGoogle Scholar
4Kidwell, M. E.. On the degree of the BrandtLickorishMillettHo polynomial of a link. Proc. Amer. Math. Soc. 100 (1987), 755762.CrossRefGoogle Scholar
5Lickokish, W. B. R. and Thistlethwaite, M. B.. Some links with non-trivial polynomials and their crossing-numbers. Comment. Math. Helv. 63 (1988), 527539.CrossRefGoogle Scholar
6Murasugi, K.. Jones polynomials and classical conjectures in knot theory. Topology 26 (1986), 187194.CrossRefGoogle Scholar
7Murasugi, K.. Jones polynomials and classical conjectures in knot theory. Math. Proc. Cambridge Philos. Soc. 102 (1987), 317318.CrossRefGoogle Scholar
8Murasugi, K.. On the braid index of alternating links. Preprint.Google Scholar
9Thistlethwaite, M. B.. A spanning tree expansion of the Jones polynomial. Topology 26 (1987), 297309.CrossRefGoogle Scholar