Annular webs and central elements in skein algebras

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 how these threading operations extend to other kinds of skein theories, including SL(n), Sp(2n), and G_2 webs. I will discuss a method for checking that the elements produced are central in the skein algebra by studying webs in the annulus. Some of these works are joint with Francis Bonahon, Haihan Wu, and an REU group.