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Are there any good short expositions of planar algebras out there? I am interested primarily in seeing the main definition and some explicit examples.

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up vote 9 down vote accepted

My paper with Emily and Noah about the D_{2n} planar algebra includes our attempt at a friendly explanation. It's all about arguably the simplest non-trivial example of a subfactor planar algebra.

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This looks great. Thanks! – Elisha Peterson Oct 17 '09 at 16:01

Emily and Noah wrote some really nice expository posts on planar algebras at the Secret Blogging Seminar. I also like Chris's posts on TQFTs and Planar Algebras to explain why I might care.

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Vaughan Jones' paper "The planar algebra of a bipartite graph" is short and example-focused. It's got the "right" definition (phrased with operads) and the mechanics of the bipartite graph planar algebra. While the bipartite graph planar algebra isn't a subfactor planar algebra, it's pretty darn close.

A subfactor planar algebra, by the way, is one which meets restrictions on the sizes of the spaces involved and has an inner product. Unsurprisingly, they give rise to subfactors (through machinery of Popa).

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Vijay Kodiyalam gave a series of polished lectures introducing to the planar algebras theory, the videos (HD) are available in the following link: Vijay Kodiyalam - Planar algebras - IMSc 2015
By watching the first lectures, you will quickly get a good understanding of the definition and examples.

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