The Homfly polynomial of the decorated Hopf link

Morton, HR ORCID: 0000-0002-8524-2695 and Lukac, SG (2003) The Homfly polynomial of the decorated Hopf link Journal of Knot Theory and Its Ramifications, 12 (3). pp. 395-416. ISSN 0218-2165, 1793-6527

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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<inf>λ</inf>, depending on partitions λ. We show how the 2-variable Homfly invariant 〈λ, μ〉 of the Hopf link arising from decorations Q<inf>λ</inf> and Q<inf>μ</inf> can be found from the Schur symmetric function s<inf>μ</inf> of an explicit power series depending on λ. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)<inf>q</inf> modules V<inf>λ</inf> and V<inf>μ</inf>, which is a 1-variable specialisation of 〈λ, μ〉, can be expressed in terms of an N × N minor of the Vandermonde matrix (q^ij).

Item Type:	Article
Additional Information:	25 pages
Uncontrolled Keywords:	skein theory, Hopf link, Homfly polynomial, quantum sl(N) invariants, symmetric functions, Schur functions, annulus, Hecke algebras
Depositing User:	Symplectic Admin
Date Deposited:	23 Jan 2017 10:03
Last Modified:	01 Mar 2026 10:17
DOI:	10.1142/S0218216503002536
Related Websites:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3005316
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