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

Kidwell, Mark E.

doi:10.1090/S0002-9939-1987-0894450-0

On the degree of the Brandt-Lickorish-Millett-Ho polynomial of a link
HTML articles powered by AMS MathViewer

by Mark E. Kidwell PDF

Proc. Amer. Math. Soc. 100 (1987), 755-762 Request permission

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

Robert D. Brandt, W. B. R. Lickorish, and Kenneth C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986), no. 3, 563–573. MR 837528, DOI 10.1007/BF01388747
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014

A new polynomial invariant for knots and links

6

Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/0040-9383(87)90009-7
Kunio Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187–194. MR 895570, DOI 10.1016/0040-9383(87)90058-9
Morwen B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297–309. MR 899051, DOI 10.1016/0040-9383(87)90003-6