The planar algebra of a bipartite graph. (English) Zbl 1021.46047

Jones, Vaughan F. R.

Gordon, Cameron McA. (ed.) et al., Knots in Hellas ’98. Proceedings of the international conference on knot theory and its ramifications, European Cultural Centre of Delphi, Greece, August 7–15, 1998. Singapore: World Scientific. Ser. Knots Everything 24, 94-117 (2000).

Summary: We review the definition of a general planar algebra \(V=\cup V_k\). 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.
For the entire collection see [Zbl 0959.00034].

MSC:

46L37	Subfactors and their classification
57-XX	Manifolds and cell complexes

Keywords:

planar algebra; bipartite graph