Geometrical relations and plethysms in the Homfly skein of the annulus

Morton, HR ORCID: 0000-0002-8524-2695 and Manchon, PMG (2008) Geometrical relations and plethysms in the Homfly skein of the annulus. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 78 (2). pp. 305-328.

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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.

Item Type:	Article
Additional Information:	28 pages, 15 figures
Uncontrolled Keywords:	math.GT, math.GT, 57M25; 57M27
Depositing User:	Symplectic Admin
Date Deposited:	21 Apr 2016 09:15
Last Modified:	16 Dec 2022 12:56
DOI:	10.1112/jlms/jdn026
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3000521