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On knot polynomials of annular surfaces and their boundary links

Published online by Cambridge University Press:  01 July 2009

HERMANN GRUBER
HERMANN GRUBER*
Affiliation:
Institut für Informatik, Justus-Liebig-Universität Giessen Arndtstr. 2, 35392 Giessen, Germany. e-mail: hermann.k.gruber@informatik.uni-giessen.de
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Abstract

Stoimenow and Kidwell asked the following question: let K be a non-trivial knot, and let W(K) be a Whitehead double of K. Let F(a, z) be the Kauffman polynomial and P(v, z) the skein polynomial. Is then always max degzPW(K) − 1 = 2 max degzFK? Here this question is rephrased in more general terms as a conjectured relation between the maximum z-degrees of the Kauffman polynomial of an annular surface A on the one hand, and the Rudolph polynomial on the other hand, the latter being defined as a certain Möbius transform of the skein polynomial of the boundary link ∂ A. That relation is shown to hold for algebraic alternating links, thus simultaneously solving the conjecture by Kidwell and Stoimenow and a related conjecture by Tripp for this class of links. Also, in spite of the heavyweight definition of the Rudolph polynomial {K} of a link K, the remarkably simple formula {◯}{L#M} = {L}{M} for link composition is established. This last result can be used to reduce the conjecture in question to the case of prime links.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 1 , July 2009 , pp. 173 - 183
Copyright
Copyright © Cambridge Philosophical Society 2009

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