#### 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.

Notes on file formats
Hugh R. Morton

Department of Mathematical Sciences, University of Liverpool

Peach St, Liverpool, L69 7ZL, UK

Email: morton@liv.ac.uk

URL: http://www.liv.ac.uk/~su14

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