  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.
