On Knot Polynomials of Annular Surfaces and their Boundary Links

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 $\max {°}_z{P}_{W(K)}-1=2\max {°}_z{F}_K$? 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 $\partial A$. 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 $\{\bigcirc\}\{L#M\}=\{L\}\{M\}$ for link composition is established. This last result can be used to reduce the conjecture in question to the case of prime links.

Version 4: revision as of October 10, 2008. Fixed several errors and inaccuracies. 11 pages, 1 figure. To appear in Mathematical Proceedings of the Cambridge Philosophical Society