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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
MathSciNet review: 894450
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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.

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Keywords: Brandt-Lickorish-Millett-Ho polynomial, alternating link projection, bridge length, arborescent link
Article copyright: © Copyright 1987 American Mathematical Society