nLab Jones polynomial

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).

Related concepts

• colored Jones polynomial

References

General

See also:

• Wikipedia, Jones polynomial

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

• Samuel J. Lomonaco, Louis Kauffman, Topological Quantum Computing and the Jones Polynomial, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (12 May 2006) (arXiv:quant-ph/0605004)

Jones polynomial as Wilson loop observables

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

• Edward Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121,351-399 (1989) (euclid.cmp/1104178138)

see also

• Jürg Fröhlich, C. King, The Chern-Simons theory and knot polynomials, Comm. Math. Phys.

  Volume 126, Number 1 (1989), 167-199 (euclid.cmp/1104179728)

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:

• Edward Witten, Khovanov homology and gauge theory, arxiv/1108.3103

• Edward Witten, Fivebranes and Knots (arXiv:1101.3216)

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