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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 Rep(O(2)) with an interesting braiding. In addition, all skein theories of oriented virtual tangles with somearXiv:2008.04294v1 fatcat:em5yidd3tbhe3htb5sh75jmare