Abstract: Recently, Stoimenow and Kidwell asked: 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 maxdeg_z P_{W(K)} - 1 = 2 maxdeg_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 a Möbius transform of the skein polynomial of the boundary link $\partial A$. This relation is shown to hold for algebraic alternating links, simultaneously solving the conjecture by Kidwell and Stoimenow and a related conjecture by Tripp for this class of knots and links. We also prove a simple formula for the Rudolph polynomial of a composite link using linear skein theory. This result reduces the conjecture in question to the case of prime links.