nLab Jones polynomial




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


Relation to 3d Chern-Simons theory

In (Witten 89) it was shown that the Jones polynomial as a polynomial in qq is equivalently the partition function of SU(2)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).



See also:

An algorithm for computing the Jones polynomial on a topological quantum computer based on anyon statistics:

Jones polynomial as Wilson loop observables

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

see also

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)

See also

  • Vaughan Jones, Index for subfactors, Invent. Math. 72, I (I983); A polynomial invariant for links via yon Neumann algebras, Bull. AMS 12, 103 (1985); Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126, 335 (1987)
category: knot theory

Last revised on May 21, 2021 at 06:56:32. See the history of this page for a list of all contributions to it.