Skein theory and deformations

Abstract: We use skein theoretic `recognition' theorems to classify deformations of certain pivotal or ribbon monoidal categories. We first show that a slight generalization of Kuperberg's characterization of quantum $G_2$ shows that any infinitesimal deformation of quantum $G_2$ as a pivotal category arises by varying $q$. This result applies both at generic $q$, and for the category of tilting modules at $q$ a root of unity (provided we exclude a few small roots of unity). Second, we prove a new Kuperberg-like characterization of Deligne's $S_t$ and use it to show that any infinitesimal deformation of $S_t$ (excluding $t=0$) as a ribbon category comes from varying $t$.