Tensor categories generated by a trivalent vertex

In an attempt to explore part of the landscape of tensor categories, with Emily Peters and Noah Snyder I've been investigating "small" tensor categories generated by a "small" morphism.

A prototypical example is studying a tensor category whose objects are tensor powers of a fixed object X, and the vector spaces $\operatorname{hom}(X^{\otimes n} \to X^{\otimes m})$ are spanned by braids (or perhaps tangles). If we ask that $\dim \operatorname{hom}(X^{\otimes n} \to X^{\otimes m})$ is small when $n$ and $m$ are small, we find that the tensor category must essentially be one of the tensor categories describing the famous knot polynomials (the Jones polynomial, the HOMFLY polynomial, or the Kauffman polynomial).

As a first step beyond these examples, we decided to study tensor categories generated by an object $X$, and a "trivalent" morphism $m:X \otimes X \to X$. Again, putting bounds on $\dim \operatorname{hom}(1 \to X^{\otimes n})$ for small $n$, we find that we can classify the possible examples. Our methods involve a combination of category theory, skein theory, graph combinatorics, and homeopathic dose of algebraic geometry. The first few examples that fall out of the classification are strangely satisfying, and strangely various: one is explained by the chromatic polynomial, another by the group $G_2$, and another by the exotic subfactor discovered by Haagerup in the 90s.

Although our motivation was towards tensor categories, everything can be explained in terms of studying planar graphs modulo local (linear) relations, and I'll take advantage of this to keep the talk as elementary as possible.