Geometrical relations and plethysms in the Homfly skein of the annulus

TL;DR

This paper explores the algebraic and geometric structures within the Homfly skein of the annulus, connecting symmetric functions, plethysms, and quantum invariants to advance understanding of knot invariants.

Contribution

It reformulates quantum sl(N) invariants of cables using plethysms and relates them to the Homfly skein algebra, providing new computational formulas.

Findings

Derived formulas relating symmetric functions to geometric bases in the skein

Reformulated quantum sl(N) invariants in terms of plethysms

Established relations between Homfly invariants and power sum decorations

Abstract

The oriented framed Homfly skein C of the annulus provides the natural parameter space for the Homfly satellite invariants of a knot. It contains a submodule C+ isomorphic to the algebra of the symmetric functions. We collect and expand formulae relating elements expressed in terms of symmetric functions to Turaev's geometrical basis of C+. We reformulate the formulae of Rosso and Jones for quantum sl(N) invariants of cables in terms of plethysms of symmetric functions, and use the connection between quantum sl(N) invariants and C+ to give a formula for the satellite of a cable as an element of C+. We then analyse the case where a cable is decorated by the pattern which corresponds to a power sum in the symmetric function interpretation of C+ to get direct relations between the Homfly invariants of some diagrams decorated by power sums.