I have read according list of below papers a basic connection between Jones polynomial and statistical mechanics is that the Kauffman bracket or Kauffman polynomial a polynomial invariant of knots is in different special cases the Jones polynomial for knots and the partition function for the Potts model in statistical mechanics. The Jones polynomial and its relations to the Yang-Baxter equations in Statistical mechanics, has been generalized to other invariants of knot theory by Kauffman via the Kauffman bracket .Witten has shown that one can use knot theory in the context of quantum field theory to produce invariants of 3- dimensional manifolds. Michael Atiyah also is using the Jones-Witten theory to explore functional integration in gauge theories and quantization. Now my question here is :
Question What are applications of Jones polynomial on von von Neumann algebras ? or what the Jones polynomials has to do with von Neumann algebras?
:The book "Exactly Solved Models in Statistical Mechanics" by Baxter is a really good source if you are interested in the connection between statistical physics and the work of Jones http://physics.anu.edu.au/theophys/_files/Exactly.pdf
:"Statistical Mechanics and the Jones Polynomial" by Louis Kauffman http://www.maths.ed.ac.uk/~aar/papers/kauffmanjones.pdf
:A good source of information on the connection between QFT and the Jones polynominal is Witten's paper "Quantum field theory and the Jones polynomial" http://projecteuclid.org/download/pdf_1/euclid.cmp/1104178138
: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