  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.
