The Homfly polynomial of the decorated Hopf link

TL;DR

This paper computes the Homfly polynomial for links formed by decorating the Hopf link with elements from the annulus skein, expressing it via Schur functions and relating it to quantum invariants of colored sl(N)_q modules.

Contribution

It provides a method to compute the Homfly polynomial of decorated Hopf links using Schur functions and relates it to quantum invariants of colored sl(N)_q modules.

Findings

Homfly polynomial expressed via Schur functions and explicit power series.

Quantum invariants related to minors of the Vandermonde matrix.

Explicit formulas for decorated Hopf links in skein theory.

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_\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}).