The Homfly polynomial of the decorated Hopf link

Details The Homfly polynomial of the decorated Hopf link The Homfly polynomial of the decorated Hopf link

2001-08-02

arxiv.org/abs/math/0108011v1

J. Knot Theory Ramif. 12 (2003), 395-416

Mathematics

Geometric Topology

25 pages

Scientific paper

The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_\lambda, depending on partitions \lambda. We show how to find the 2-variable Homfly invariant <\lambda,\mu> of the Hopf link arising from decorations Q_\lambda and Q_\mu in terms of the Schur symmetric function s_\mu of an explicit power series depending on \lambda. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)_q modules V_\lambda and V_\mu, which is a 1-variable specialisation of <\lambda,\mu>, can be expressed in terms of an N x N minor of the Vandermonde matrix (q^{ij}).

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