The Planar Algebra of a bipartite graph

We review the definition of a general planar algebra V = ∪V_k. We show how to construct a general planar algebra from a bipartite graph by creating a specific model using statistical mechanical sums defined by labelled tangles. These planar algebras support a partition function for a closed tangle which is spherically invariant and defines a positive definite inner product on each V_k. We then describe how any planar algebra is naturally a cylic module in the sense of Connes and do some computations.