# Categories generated by a trivalent vertex

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## 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 Open image in new window 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 \(\dim {\text {Hom}}(1 \rightarrow X^{\otimes n})\) bounded by 1, 0, 1, 1, 4, 11, 40 for \(0 \le n \le 6\) 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 *H*3 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 the sequence 1, 0, 1, 1, 4. Our main techniques are a new approach to finding skein relations which can be easily automated using Gröbner bases, and evaluation algorithms which use the discharging method developed in the proof of the 4-color theorem.

## Mathematics Subject Classification

18D10 (Monoidal Categories) 05C10 (Planar graphs; geometric and topological aspects of graph theory) 57M27 (Invariants of knots and 3-manifolds)## Notes

### Acknowledgments

Scott Morrison was supported by an Australian Research Council Discovery Early Career Researcher Award DE120100232, and Discovery Projects DP140100732 and DP160103479. Emily Peters was supported by the NSF Grant DMS-1501116. Noah Snyder was supported by the NSF Grant DMS-1454767. All three authors were supported by DOD-DARPA Grant HR0011-12-1-0009. Scott Morrison would like to thank the Erwin Schrödinger Institute and its 2014 programme on “Modern Trends in Topological Quantum Field Theory” for their hospitality. We would like to thank Greg Kuperberg for a blog comment [37] suggesting applying the discharging method to skein theory, Victor Ostrik for explaining his construction of the twisted Haagerup categories, and David Roe and Dylan Thurston for helpful suggestions.

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