# What are applications of Jones polynomial on von Neumann algebras?

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?

Reference list

[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

[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

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.)

In this paper, Vaughan attributes the observation of the similarity of the Temperley-Lieb relations and the braid group relations to D. Hatt, P. de la Harpe and N. Stoltzfus:

Jones, Vaughan, Groupes de tresses, algèbres de Hecke et facteurs de type (II_ 1). (Braid groups, Hecke algebras and type (II_ 1) factors), C. R. Acad. Sci., Paris, Sér. I 298, 505-508 (1984). ZBL0597.20034.

(Hatt and de la Harpe are also mentioned in the paper "A POLYNOMIAL INVARIANT FOR KNOTS VIA VON NEUMANN ALGEBRAS"). He attributes the observation of the similarity between the Temperley-Lieb algebra presentation and the Hecke algebra to R. Steinberg. Since the Hecke algebra is a quotient of the group ring of the braid group, one obtains finite dimensional representations of the braid group into Hecke algebras, and similarly into the Temperley-Lieb algebra, satisfying the skein relation. These braid group representations were discovered by Jones in 1983 (see "Braid groups, Hecke algebras and type II1 factors").

He announced the knot polynomial in 1985 in the paper cited above. In this paper, he acknowledges Joan Birman's help for identifying the trace that is invariant under the Markov move (this is essentially fixing the dependence of the trace on the writhe).

Clearly Vaughan discovered the Temperley-Lieb algebra from his study of subfactors. But I speculate that his derivation of representations of the braid group and the knot polynomials was more fortuitous stemming from the suggestions of people that he acknowledges.

• I think one of Vaughan's strengths as a mathematician was that he was very persistent in wringing out all that he could from a technique. There's obviously something fortuitous in that the Temperley-Lieb algebra leads to interesting knot invariants (when it could just have well have given boring ones), but I think there's also a highly non-fortuitous part which is that once he had the Temperley-Lieb algebras in hand he really stuck with them to see everything you could do with them including knot polynomials. – Noah Snyder Sep 28 at 21:09
• That is, both Jones and Temperley-Lieb independently discovered the Temperley-Lieb algebras in contexts completely unrelated to knot theory. In principal Temperley-Lieb's work could have lead to the discovery of the Jones polynomial if someone had kept at thinking about what else you could with those algebras the same way Vaughan did a few years later. (That said, Vaughan did have a critical advantage, the subfactor approach gives not only the algebras themselves but also a natural trace on them, which plays a critical role.) – Noah Snyder Sep 28 at 21:19
• Yes, I was merely postulating that he did not, as far as I know, intend to derive braid representations and knot invariants from his investigations of von Neumann algebras. But as you indicate, luck favors the prepared mind (and he certainly mastered these and related subjects quickly after discovering these connections). As far as I know, a direct connection between the Jones polynomial and von Neumann algebras / subfactors is still mysterious. Henriques and collaborators have derived conformal field theories from von Neumann algebras, but I don't know if this gives any relation. – Ian Agol Sep 28 at 21:55