Link Invariants
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The Jones polynomial is a knot invariant. It is a special case of the HOMFLY-PT polynomial. See there for more details.
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).
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).
The identification of the Jones polynomial with the partition function of Chern-Simons theory with Wilson loops is due to
The further identification of this with via the Khovanov homology induced by a 4-dimensional super Yang-Mills theory is due to
Edward Witten, Khovanov homology and gauge theory, arxiv/1108.3103
Edward Witten, Fivebranes and Knots (arXiv:1101.3216)
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