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

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
Copyright
Copyright © Cambridge Philosophical Society 2009

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