nLab Jones polynomial

Contents

Context

Knot theory

knot theory

Examples/classes:

knot invariants

Related concepts:

category: knot theory

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

Jones polynomial as Wilson loop observables

The identification of the Jones polynomial with Wilson loop observables in Chern-Simons theory is due to

Lecture notes:

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

The categorification of this relation to an identification of Khovanov homology with observables in D=4 super Yang-Mills theory:

Lecture notes:

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