I have read according list of below papers a basic connection between [Jones polynomial](https://en.wikipedia.org/wiki/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 :



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[1]: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>


[2]:"Statistical Mechanics and the Jones Polynomial" by Louis Kauffman [http://www.maths.ed.ac.uk/~aar/papers/kauffmanjones.pdf](http://www.maths.ed.ac.uk/%7Eaar/papers/kauffmanjones.pdf)


[3]: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>


[4]: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>



What are applications of Jones polynomial on von Neumann algebras?
I don't think it's quite right to think of knot polynomials as having applications to von Neumann algebras. Instead I think it's more accurate to say that the Temperley-Lieb-Jones algebras (and more generally "towers of algebras with Markov traces" or equivalently quantum groups or tensor categories) have applications *both* to von Neumann algebras (via the theory of standard invariants) and to low-dimensional topology (via their connection to the braid group).


A great place to start on reading about applications from TLJ to subfactor theory is Vaughan's paper "Index for Subfactors" where he rediscovered the Temperley-Lieb algebras in the context of subfactors. (I believe it was David Evans who pointed out that they'd appeared previously in the context of statistical mechanics in Temperley-Lieb's work.)


