Some Planar Algebras Related to Graphs

Brian Curtin Vaughan F. R. Jones

To appear at Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26, 2001


A planar algebra consists of a collection of vector spaces which is closed under certain operators defined by diagrams known as planar tangles. Although planar algebras were introduced to study problems in subfactors, they have applications in combinatorics. We shall discuss planar algebras as a framework for graph rewriting systems similar to the skein theory of knots. We focus on the prototypical example: the Reidemeister moves and spin models for link invariants. We discuss the planar algebras related to Higman-Sims graph and the Petersen graph. Despite many similarities, the former admits Reidemeister-like reductions while the latter does not. We discuss some tools used to study these planar algebras and their growth functions.

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