# Contents

## Idea

The Jones polynomial is a knot invariant. It is a special case of the HOMFLY-PT polynomial. See there for more details.

## Properties

### Relation to 3d Chern-Simons theory

In (Witten 89) it was shown that the Jones polynomial as a polynomial in $q$ is equivalently the partition function of $SU(2)$-Chern-Simons theory with a Wilson loop specified by the given knot as a function of the exponentiated level of the Chern-Simons theory. Extensive lecture notes on this are in (Witten 13a).

### Relation to 4d super Yang-Mills theory

Later in (Witten 11) this identification was further refined to a correspondence between Khovanov homology and observables in 4-dimensional super Yang-Mills theory. Extensive lectures notes on this are in (Witten 13b).

## References

The identification of the Jones polynomial with the partition function of Chern-Simons theory with Wilson loops is due to

• Edward Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121,351-399 (1989) pdf

The further identification of this with via the Khovanov homology induced by a 4-dimensional super Yang-Mills theory is due to

Lecture notes on this are in

• Edward Witten, A New Look At The Jones Polynomial of a Knot, Clay Conference, Oxford, October 1, 2013 (pdf)

• Edward Witten, Khovanov Homology And Gauge Theory, Clay Conference, Oxford, October 2, 2013 (pdf)