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The Murphy operators in the Hecke algebra Hn 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 Hn. 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, [Formula: see text]. Symmetric functions of the Murphy operators are also central in Hn. I define geometrically a homomorphism from [Formula: see text] to the centre of each algebra Hn, and find an element in [Formula: see text], independent of n, whose image is the mth power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of [Formula: see text] in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.
57M25, 20C08, Geometric Topology (math.GT), Mathematics - Rings and Algebras, Invariants of knots and \(3\)-manifolds, Hecke algebras and their representations, Murphy operators, Mathematics - Geometric Topology, Rings and Algebras (math.RA), Mathematics - Quantum Algebra, FOS: Mathematics, Knots and links in the \(3\)-sphere, Quantum Algebra (math.QA), skein theory, Hecke algebras
57M25, 20C08, Geometric Topology (math.GT), Mathematics - Rings and Algebras, Invariants of knots and \(3\)-manifolds, Hecke algebras and their representations, Murphy operators, Mathematics - Geometric Topology, Rings and Algebras (math.RA), Mathematics - Quantum Algebra, FOS: Mathematics, Knots and links in the \(3\)-sphere, Quantum Algebra (math.QA), skein theory, Hecke algebras
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