Geometry & Topology Monographs, Vol. 4 (2002),
Invariants of knots and 3-manifolds (Kyoto 2001),
Paper no. 15, pages 235--244.

Power sums and Homfly skein theory

Hugh R. Morton

Abstract. The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements, whose symmetric functions are central in H_n. In [Skein theory and the Murphy operators, J. Knot Theory Ramif. 11 (2002), 475-492] I defined geometrically a homomorphism from the Homfly skein C of the annulus to the centre of each algebra H_n, and found an element P_m in C, independent of n, whose image, up to an explicit linear combination with the identity of H_n, is the m-th power sum of the Murphy operators. The aim of this paper is to give simple geometric representatives for the elements P_m, and to discuss their role in a similar construction for central elements of an extended family of algebras H_{n,p}.

Keywords. Homfly skein theory, Murphy operators, power sums, supersymmetric polynomials, annulus, Hecke algebras

AMS subject classification. Primary: 57M25. Secondary: 20C08.

E-print: arXiv:math.GT/0111101

Submitted to GT on 31 October 2001. (Revised 15 May 2002.) Paper accepted 22 July 2002. Paper published 13 October 2002.

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Hugh R. Morton
Department of Mathematical Sciences, University of Liverpool
Peach St, Liverpool, L69 7ZL, UK
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