/ Vijay Higgins: Central elements in the SL(d) skein algebra

Vijay Higgins: Central elements in the SL(d) skein algebra

February 27, 2024
4:30 pm - 5:30 pm

Abstract: The skein algebra of a surface is spanned by links in the thickened surface, subject to skein relations which diagrammatically encode the data of a quantum group. The multiplication in the algebra is induced by stacking links in the thickened surface. This product is generally noncommutative. When the quantum parameter q is generic, the center of the skein algebra is essentially trivial. However, when q is a root of unity, interesting central elements arise. When the quantum group is quantum SL(2), the work of Bonahon-Wong shows that these central elements can be obtained by a topological operation of threading Chebyshev polynomials along knots. In this talk, I will discuss joint work with F. Bonahon in which we use analogous multi-variable ‘threading’ polynomials to obtain central elements in higher rank SL(d) skein algebras. Time permitting, I will discuss how a finer version of the skein algebra, called the stated skein algebra, can be used to show that the threading operation yields a well-defined algebra embedding of the coordinate ring of the character variety of the surface into the root-of-unity skein algebra in the case of SL(3).