Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the degree of the Brandt-Lickorish-Millett-Ho polynomial of a link


Author: Mark E. Kidwell
Journal: Proc. Amer. Math. Soc. 100 (1987), 755-762
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9939-1987-0894450-0
MathSciNet review: 894450
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {Q_L}$ be the link polynomial defined by Brandt, Lickorish, Millett, and Ho. Let $ \deg {Q_L}$ be the maximum degree of a nonzero term. If $ p(L)$ is any regular link projection and $ B$ is any bridge (maximal connected component after undercrossing points are deleted), define the length of $ B$ as the number of crossings in which the overcrossing segment is a part of $ B$.

Theorem 1. Let $ p(L)$ be a connected, regular link projection with $ N$ crossing points. Let $ K$ be the maximal length of any bridge in $ p(L)$. Then $ \deg {Q_L} \leq N - K$.

Theorem 2. If $ p(L)$ is a prime, connected alternating projection with $ N > 0$ crossing points, then the coefficient of $ {x^{N - 1}}$ is a positive number.


References [Enhancements On Off] (What's this?)

  • [BLM] R. D. Brandt, W. B. R Lickorish and K. C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986), 563-573. MR 837528 (87m:57003)
  • [C] J. H. Conway, An enumeration of knots and links and some of their algebraic properties, Computational Problems in Abstract Algebra (J. Leech, ed.), Pergamon Press, Oxford and Elmsford, N.Y., 1969, pp. 329-358. MR 0258014 (41:2661)
  • [H] C. F. Ho, A new polynomial invariant for knots and links, Abstracts Amer. Math. Soc. 6 (1985), 300.
  • [K] L. Kauffman, State models and the Jones polynomial (to appear). MR 899057 (88f:57006)
  • [M] K. Murasugi, Jones polynomials and classical conjectures in knot theory, Topology (to appear). MR 895570 (88m:57010)
  • [T] M. B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology (to appear). MR 899051 (88h:57007)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57M25

Retrieve articles in all journals with MSC: 57M25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0894450-0
Keywords: Brandt-Lickorish-Millett-Ho polynomial, alternating link projection, bridge length, arborescent link
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society