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An upper bound for the breadth of the Jones polynomial

  • Morwen B. Thistlethwaite (a1)

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 〈D〉 ∈ Z[A, A−1 of a link diagram D.

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[3] W. Menasco . Closed incompressible surfaces in alternating knot and link complements. Topology 23 (1984), 3744.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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