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

Triangle Topology Seminar

Hoel Queffelec (CNRS and Institut Montpelliérain Alexander Grothendieck)

Thursday, April 13, 2017 -
4:30pm to 5:30pm
at UNC (Phillips 332)

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. (This is joint work with H. Russell, D. Rose, and P. Wedrich)