Morrison, ScottPeters, EmilySnyder, Noah2021-05-201022-1824http://hdl.handle.net/1885/233371This 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.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.application/pdfen-AU© Springer International Publishing 201618D10 (Monoidal Categories)05C10 (Planar graphs; geometric and topological aspects of graph theory)57M27 (Invariants of knots and 3-manifolds)201710.1007/s00029-016-0240-32020-11-23