Papers › Categories generated by a trivalent vertex
Categories generated by a trivalent vertex
Scott Morrison, Emily Peters, Noah Snyder
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
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 ℂ generated by a symmetric self-dual simple object X and a rotationally invariant morphism 1 →X ⊗X ⊗X. Our main result is that the only trivalent categories with Hom(1, X^(⊗n)) bounded by $1,0,1,1,4,11,40$ for 0 ≤n ≤6 are quantum SO(3), quantum G₂, 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 →X^(⊗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.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections