The quantum sl(3) invariants of cubic bipartite planar graphs

Temperley-Lieb algebras have been generalized to sl(3, C) web spaces. Since a cubic bipartite planar graph with suitable directions on edges is a web, the quantum sl( 3) invariants naturally extend to all cubic bipartite planar graphs. First we completely classify them as a connected sum of prime webs. We provide a method to find all prime webs. Using the quantum sl( 3) invariants, we provide a criterion which determines the symmetry of cubic bipartite planar graphs. We also provide an example that our criterions for graphs is different from that of links.