On the Degree of the Brandt-Lickorish-Millett-Ho Polynomial of a Link

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.