Categories generated by a trivalent vertex

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.

• The paper establishes a full classification of trivalent categories by identifying quantum SO(3)/G₂, ABA, and H3 Haagerup fusion categories under specific diagrammatic dimension constraints.
• It employs innovative methods, including automated Gröbner bases and discharging techniques, to systematically derive skein relations for evaluating diagram invariants.
• The findings enhance our understanding of tensor categories in quantum topology and pave the way for future automated techniques in mathematical reasoning.

Overview of the Paper on Trivalent Categories

This paper explores skein theoretic invariants associated with trivalent planar graphs, focusing on developing and automating arguments for these invariants within the context of quantum algebra and topology. The main thrust involves categorizing trivalent categories, specifically pivotal tensor categories over the complex field $\mathbb{C}$ generated by a symmetric self-dual simple object $X$ and a rotationally invariant morphism $1 \to X \otimes X \otimes X$.

Key Results and Contributions

The paper establishes an exhaustive classification of trivalent categories under constraints on the dimensions of the spaces of diagrams, specifically the $n$-boundary point spaces denoted as $\dim \text{Hom}(1 \to X^{\otimes n})$. The classification identifies the trivalent categories which satisfy the dimension bounds $1,0,1,1,4,11,40$ for $0 \leq n \leq 6$:

1. Quantum $SO(3)$ and $G_2$: These categories fall within the defined dimensional restrictions, with explicit realization provided through skein theoretic methods and quantum algebra.
2. ABA Categories: A one-parameter family of free products of certain Temperley-Lieb categories identified as ABA categories is introduced.
3. H3 Haagerup Fusion Category: Recognized as noteworthy within the scope of this classification, further elaborating on its place within the overarching framework.

Methodological Innovations

The paper employs an innovative approach that integrates Gr\"obner bases for automating the discovery of skein relations and introduces evaluation algorithms inspired by the discharging method from the proof of the four-color theorem. These novel techniques are pivotal in ensuring that skein theoretic arguments can be automated, particularly in evaluating closed diagrams concerning their reducibility to multiples of the empty diagram.

Implications and Future Prospects

The findings have profound implications for the development of skein theoretic methods beyond traditional constraints, potentially affecting broad areas of quantum topology and algebra. The automatic techniques demonstrated in this paper stand as a testament to the potential for increased efficacy in exploring skein theory. Moreover, the paper outlines possible generalizations of these methodologies, such as applying them in non-trivalent graph settings and other combinatorially dense constructs within mathematics.

Speculatively, the automation of finding evaluation algorithms could shape future inquiries in AI through the rapid assessment of complex examples, providing tangible advancements in the understanding and classification of skein theoretic settings.

In synthesizing these results, the paper situates itself as a significant milestone in the skein theoretic narrative, inviting further exploration into automated mathematical reasoning across diverse structural settings in contemporary mathematical physics and abstract algebra.