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An upper bound for the breadth of the Jones polynomial

Published online by Cambridge University Press:  24 October 2008

Morwen B. Thistlethwaite
Morwen B. Thistlethwaite
Affiliation:
Department of Computing and Mathematics, Polytechnic of the South Bank, London SE1 0AA
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 3 , May 1988 , pp. 451 - 456
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Kauffman, L. H.. State models for knot polynomials. Topology. (In the Press.)Google Scholar
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[6]Thistlethwaite, M. B.. On the Kauffman polynomial of an adequate link. Invent. Math. (In the Press.)Google Scholar