Skein theory and the Murphy operators

Details Skein theory and the Murphy operators Skein theory and the Murphy operators

2001-02-13

arxiv.org/abs/math/0102098v1

J. Knot Theory Ramif. 11 (2002), 475-492

Mathematics

Geometric Topology

22 pages, 34 interspersed figures. Submitted for the proceedings of Knot 2000 conference, Korea

Scientific paper

The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of H_n. In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulus, C. Symmetric functions of the Murphy operators are also central in H_n. I define geometrically a homomorphism from C to the centre of each algebra H_n, and find an element in C, independent of n, whose image is the m-th power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of C in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.

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