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Tutte's golden identity from a fusion category

Presented by: 
Paul Fendley University of Oxford
Tuesday 14th March 2017 - 14:00 to 15:00
INI Seminar Room 2
The chromatic polynomial \chi(Q) can be defined on any graph, such that for Q integer it counts the number of colourings. In statistical mechanics, it is known as the partition function of the antiferromagnetic Potts model on that graph. It has many remarkable properties, and Tutte's golden identity is one of the more unusual ones. For any planar triangulation, it relates \chi(\phi+2) to the square of \chi(\phi+1), where \phi is the golden mean. Tutte's original proof is purely combinatorial. I will give here an elementary proof using fusion categories, which are familiar for example from topological quantum field theory, anyonic quantum mechanics, and integrable statistical mechanics. In this setup, the golden identity follows by simple manipulations of a topological invariant related to the Jones polynomial. I will also mention recent work by Agol and Krushkal on understanding what happens to the identity for graphs on more general surfaces.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons