Around Chebyshev's polynomial and the skein algebra of the torus

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• Hoel Queffelec (CNRS (Centre national de la recherche scientifique); Université de Montpellier)
• Monday 26 June 2017, 16:00-17:00
• Seminar Room 1, Newton Institute.

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The diagrammatic version of the Jones polynomial, based on the Kauffman bracket skein module, extends to knots in any 3-manifold. In the case of thickened surfaces, it can be endowed with the structure of an algebra by stacking. The case of the torus is of particular interest, and C. Frohman and R. Gelca exhibited in 1998 a basis of the skein module for which the multiplication is governed by the particularly simple “product-to-sum” formula.
I'll present a diagrammatic proof of this formula that highlights the role of the Chebyshev's polynomials, before turning to categorification perspectives and their interactions with representation theory.

Joint work with H. Russell, D. Rose and P. Wedrich.

This talk is part of the Isaac Newton Institute Seminar Series series.

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