Résumé : Since the original construction of the Jones polynomial, the Temperley-Lieb algebra has become a cornerstone of a fruitful interaction between Knot theory and Representation theory. The Temperley-Lieb algebra was introduced by N. Temperley and E. Lieb in a statistical mechanical context in 1971 and was rediscovered by V.F.R. Jones as a knot algebra in 1983. A knot algebra comprises an algebra A, an appropriate representation of the braid group in A and a Markov trace function defined on A. The Temperley-Lieb algebra, the Iwahori-Hecke algebra and the BMW algebra are the most known examples of knot algebras.
In this talk we will present a new 2-variable generalization θ of the Jones polynomial that is derived from the framization of the Temperley-Lieb algebra. Τhe framization of a knot algebra is a relatively new technique that was proposed by J. Juyumaya and S. Lambropoulou and it consists in an extension of a knot algebra via the addition of framing generators which are intrinsically involved in the algebra relations. In this way one obtains a new algebra which is related to framed braids and framed knots. The basic example of framization is the Yokonuma-Hecke algebra which can be regarded as a framization of the Iwahori-Hecke algebra. We will start from the basic definitions of the framization of the Temperley-Lieb algebra and we will our ways towards proving the well-definedness of the new invariant θ both algebraically and skein theoretically. The 2-variable invariant θ coincides with the Jones polynomial on knots but is stronger than the Jones polynomial on links, as it can detect more pairs of non-isotopic links.
Lieu : Fermat - Salle 2205