  In this paper, we use skein-theoretic techniques to classify all virtual knot
polynomials and trivalent graph invariants with certain smallness conditions.
The first half of the paper classifies all virtual knot polynomials giving
non-trivial invariants strictly smaller than the one given by the Higman-Sims
spin model. In particular, we exhibit a family of skein theories coming from
$\text{Rep}(O(2))$ with an interesting braiding. In addition, all skein
theories of oriented virtual tangles with some smallness conditions are
classified. In the second half of the paper, we classify all non-trivial
invariants of (perhaps non-planar) trivalent graphs coming from symmetric
trivalent categories. For each of these categories, we also classify when the
sub-category generated by only the trivalent vertex is braided. An interesting
example of this arise from the tensor category Deligne's $S_t$.
