Categories generated by a trivalent vertex

TL;DR

This paper classifies certain pivotal tensor categories generated by a symmetric self-dual object, identifying specific quantum groups, fusion categories, and free product categories through skein theoretic invariants and automated algebraic techniques.

Contribution

It introduces a novel automated approach to classify trivalent categories using skein relations and Gr"obner bases, expanding understanding of quantum and fusion categories.

Findings

Classified all trivalent categories with specific invariant space dimensions.

Identified quantum $SO(3)$, quantum $G_2$, and Haagerup fusion categories among solutions.

Developed automated methods for finding skein relations using Gr"obner bases.

Abstract

This is the first paper in a general program to automate skein theoretic arguments. In this paper, we study skein theoretic invariants of planar trivalent graphs. Equivalently, we classify trivalent categories, which are nondegenerate pivotal tensor categories over $\mathbb C$ generated by a symmetric self-dual simple object $X$ and a rotationally invariant morphism $1 \rightarrow X \otimes X \otimes X$. Our main result is that the only trivalent categories with $\dim \operatorname{Hom}(1, X^{\otimes n})$ bounded by $1,0,1,1,4,11,40$ for $0 \leq n \leq 6$ are quantum $SO(3)$, quantum $G_2$, a one-parameter family of free products of certain Temperley-Lieb categories (which we call ABA categories), and the $H3$ Haagerup fusion category. We also prove similar results where the map $1 \rightarrow X^{\otimes 3}$ is not rotationally invariant, and we give a complete classification of nondegenerate braided trivalent categories with dimensions of invariant spaces bounded by $1,0,1,1,4$. Our main techniques are a new approach to finding skein relations which can be easily automated using Gr\"obner bases, and evaluation algorithms which use the discharging method developed in the proof of the $4$-color theorem.