Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

An upper bound for the breadth of the Jones polynomial

Morwen B. Thistlethwaitea1

a1 Department of Computing and Mathematics, Polytechnic of the South Bank, London SE1 0AA

In the recent article [2], a kind of connected link diagram called adequate was investigated, and it was shown that the Jones polynomial is never trivial for such a diagram. Here, on the other hand, upper bounds are considered for the breadth of the Jones polynomial of an arbitrary connected diagram, thus extending some of the results of [1,4,5]. Also, in Theorem 2 below, a characterization is given of those connected, prime diagrams for which the breadth of the Jones polynomial is one less than the number of crossings; recall from [1,4,5] that the breadth equals the number of crossings if and only if that diagram is reduced alternating. The article is concluded with a simple proof, using the Jones polynomial, of W. Menasco's theorem [3] that a connected, alternating diagram cannot represent a split link. We shall work with the Kauffman bracket polynomial xs3008Dxs3009 xs2208 Z[A, A−1 of a link diagram D.

(Received June 03 1987)