Thompson links are links arising from elements of the Thompson group. They were introduced by Vaughan Jones as part of his effort to construct a conformal field theory for every finite index subfactor. It was shown that every link is isotopic to some Thompson link, so the Thompson group is as good as the braid groups at producing links. In this talk I will discuss a method to study Thompson links from the viewpoint of contact geometry. I will first talk about grid diagrams and their relationship with Legendrian links. We will also review Jones' construction of Thompson links based on a unitary representation of the Thompson group. Next I will introduce a notion of half grid diagram to give an equivalent construction of Thompson links and further associate with each Thompson link a canonical Legendrian type. Lastly, I will talk about some applications, including a lower bound of the maximal Thurston-Bennequin number in terms of the Thompson index. This is joint work with Shunyu Wan.

For a Lie group G, the G-skein module of a 3-dimensional manifold M is a fundamental object in Witten’s interpretation of quantum knot invariants in the framework of a topological quantum field theory. It depends on a parameter q and, when this parameter q is a root of unity, the G-skein module contains elements with a surprising “invisibility” property, in the sense that they can be traversed by any other skein without changing the resulting total skein. I will describe some of these invisible elements in the case of the special linear group SL_n. The construction is based on the very classical theory of symmetric polynomials in n variables.