The Homfly polynomial of the decorated Hopf link

Hugh R. Morton; Sascha G. Lukac

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arXiv · math/0108011

The Homfly polynomial of the decorated Hopf link

Hugh R. Morton·Sascha G. Lukac

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Abstract

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_λ, depending on partitions λ. We show how to find the 2-variable Homfly invariant <λ,μ> of the Hopf link arising from decorations Q_λand Q_μin terms of the Schur symmetric function s_μof an explicit power series depending on λ. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)_q modules V_λand V_μ, which is a 1-variable specialisation of <λ,μ>, can be expressed in terms of an N x N minor of the Vandermonde matrix (q^{ij}).

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Hugh R. Morton, Sascha G. Lukac. 2001-08-02. The Homfly polynomial of the decorated Hopf link. https://arxiv.org/abs/math/0108011

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