On Knot Polynomials of Annular Surfaces and their Boundary Links

Hermann Gruber

Published 2004-06-06, updated 2008-10-23Version 4

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 $\max\deg_z P_{W(K)} - 1 = 2 \max\deg_z F_K$? 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 M\"obius 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.