Categories generated by a trivalent vertex

• Scott Morrison ^[1] ; Emily Peters ^[2] ; Noah Snyder ^[3]
  1. [1] Australian National University

     Australian National University

     Australia

     
  2. [2] Loyola University Chicago
  3. [3] Indiana University Bloomington

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 23, Nº. 2, 2017, págs. 817-868
• Idioma: inglés
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• Resumen

  • 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 dim 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 TemperleyLieb 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 the sequence 1, 0, 1, 1, 4. Our main techniques are a new approach to finding skein relations which can be easily automated using Gröbner bases, and evaluation algorithms which use the discharging method developed in the proof of the 4-color theorem