TY - JOUR

T1 - Categories generated by a trivalent vertex

AU - Morrison, Scott

AU - Peters, Emily

AU - Snyder, Noah

N1 - Publisher Copyright:
© 2016, Springer International Publishing.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - 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 [InlineEquation not available: see fulltext.] 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 Temperley-Lieb 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.

AB - 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 [InlineEquation not available: see fulltext.] 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 Temperley-Lieb 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.

KW - 05C10 (Planar graphs; geometric and topological aspects of graph theory)

KW - 18D10 (Monoidal Categories)

KW - 57M27 (Invariants of knots and 3-manifolds)

UR - http://www.scopus.com/inward/record.url?scp=84978035498&partnerID=8YFLogxK

U2 - 10.1007/s00029-016-0240-3

DO - 10.1007/s00029-016-0240-3

M3 - Article

SN - 1022-1824

VL - 23

SP - 817

EP - 868

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

IS - 2

ER -