Mathematics – Geometric Topology
Scientific paper
2004-06-06
Mathematical Proceedings of the Cambridge Philosophical Society 147 (2009) 173-183
Mathematics
Geometric Topology
Version 4: revision as of October 10, 2008. Fixed several errors and inaccuracies. 11 pages, 1 figure. To appear in Mathematic
Scientific paper
10.1017/S0305004109002370
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.
No associations
LandOfFree
On Knot Polynomials of Annular Surfaces and their Boundary Links does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On Knot Polynomials of Annular Surfaces and their Boundary Links, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Knot Polynomials of Annular Surfaces and their Boundary Links will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-594859