His work on knot polynomials, with the discovery of what is now called the Jones polynomial, was from an unexpected direction with origins in the theory of von Neumann algebras. But how is his work related to statistical mechanics?
A brief version: certain algebras arising in Jones' work also occur in the study of exactly solvable models in statistical mechanics. See here for details:
J.S. Birman, The Work of Vaughan F. R. Jones, in ICM'1990 proceedings:
http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0009.0018.ocr.pdf
another connection is through the Jones polynomials (among other knot invariants) - in Polymer Physics the appearance of natural knotted polymers is very relevant and an important tool for diagnosing is the Jones polynomials. Now, the relation between Polymer Physics and statistical mechanics is quite straightforward - see for example the part on polymer physics in
http://www.amazon.com/Integrals-Quantum-Mechanics-Statistics-Financial/dp/9814273562
This is also relevant in fluid dynamics which is intimately related to stat mech.
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