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A Very Non-Planar Very Planar Algebra

Workshop on Free Probability, Random Matrices, and Planar Algebras

The Fields Institute, Toronto, September 2007

Abstract. A long time ago I read somewhere that when you write a scientific article you may use the word "very" very freely. Then at proof-reading you just have to go over everything a second time and scratch out every, really every, occurrence of that word from your article. Anyway, my talk will be about a class of knotted trivalent graphs which is obviously very 3D and nothing planar, yet it forms a planar algebra. Indeed, a very planar algebra - so planar that unlike, say, tangles, it cannot be generalized to higher genus ambient surfaces. The resulting planar algebra is quite useful - it provides a Reidemeister-move planar context for a universal finite type invariant. I'll mention that and explain a bit, but my main reason for talking about this very non-planar very planar algebra is simply that it is amusing.

Handout. JavaScript browser, PNG.

Audio. At Fields

"God created the knots, all else in topology is the work of mortals."

Leopold Kronecker (modified)