### Witten Looking Anew at the Jones Polynomial

#### Posted by David Corfield

*Guest post by Bruce Bartlett*

Today at the Clay Research Conference, Edward Witten gave a talk on *A new look at the Jones polynomial of a knot*. It was an opportune moment, 25 years after his original original paper.

Let me give a quick report-back. Hopefully the video and slides will be available on the Clay website at some point, but that may take some years!

The main result seems to be that, using the gradient flow lines of the analytically-continued Chern-Simons action, the coefficients $b_n$ of the coloured Jones polynomial of a knot $K \subset \mathbb{R}^3$,

can be interpreted as the “number of solutions” to a certain 4-dimensional super Yang-Mills theory with given second Chern class $n$. He is giving a second talk tomorrow morning on Khovanov homology and probably this formula forms part of a bigger picture.

This work of Witten can be seen as `part II’ of his 125 page paper on Analytic Continuation of Chern-Simons Theory from 2010 (see here or my own report here, or the video here).

More precisely, I believe Witten’s talk today is the follow-up to the closing paragraph of that paper, entitled “Four or Five-Dimensional Interpretation”. Those ideas seem to have crystallized more clearly now. The upshot is that the Jones polynomial for a knot in $\mathbb{R}^3$ (interpreted as a path integral over all connections on $\mathbb{R}^3$ of the holonomy around the knot) can be computed from a path integral of $N = 4$ super Yang-Mills theory on $X = \mathbb{R}^3 \times \mathbb{R}^+$. Of course, to a mathematician this is still problematic since the right hand side is still a nonrigorous path integral! However, this statement can be made more cogent in two ways.

Firstly, the *perturbative* expansion of both sides can be computed and compared, and apparently they are the same. On the left hand side, this will be the Vassilliev invariants of Chern-Simons theory, while on the right hand side, it will presumably be some other perturbative expansion. I’m not sure how “deep” this particular statement is (that the two perturbative expansions are the same). No doubt though it is pretty cool.

Secondly, one can use “electric-magnetic duality” to *compute* the right hand side (the path integral of the 4-dimensional supersymmetric theory). This results in the statement that I gave at the top, that the coefficients $b_n$ of the Jones polynomial

can be interpreted as the “number of solutions of the 4d super Yang-Mills equations with given second Chern class $n$”. In particular, this gives a long-awaited quantum field theory explanation to the question, “Why is the Jones polynomial a polynomial”?

Tomorrow we hear part II of this talk, relating this to Khovanov homology.

## Re: Witten Looking Anew at the Jones Polynomial

Witten gave his second talk on

Khovanov homology and gauge theory. In fact, I understand now that his two talks were really an exposition of his recent past work on a gauge theory explanation for Khovanov homology and the Jones polynomial as contained in Analytic continuation of Chern-Simons theory and Khovanov homology and gauge theory. I just haven’t kept up (I’m still trying to understand his original paper :-)). It also contains elements of this paper on complexifying the path integral and this more technical paper on the details of the gauge theory realization of Khovanov homology.So, it’s not new to the experts, but it was new and beautiful to me.