On knot polynomials of annular surfaces and their boundary links

Abstract

AbstractStoimenow and Kidwell asked the following question: letKbe a non-trivial knot, and letW(K) be a Whitehead double ofK. LetF(a,z) be the Kauffman polynomial andP(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 maximumz-degrees of the Kauffman polynomial of an annular surfaceAon 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 linkK, 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.