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Jones polynomial

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 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).

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)

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

Revised on October 4, 2013 12:42:47 by Urs Schreiber (82.169.114.243)