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## "Quantum field theory and the Jones polynomial", Edward Witten, 1989

*Reviewed August 31, 2023*

*Citation:* Witten, Edward. "Quantum field theory and the Jones polynomial." Communications in Mathematical Physics 121.3 (1989): 351-399.

*Web:* https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-121/issue-3/Quantum-field-theory-and-the-Jones-polynomial/cmp/1104178138.full

*Tags:* Foundational, Physical, Mathematical

This is the groundbreaking paper by Edward Witten which connected quantum field theory and the Jones polynomial. Namely, Witten shows that the knot invariants arising from certain (2+1) dimensional Chern-Simons are exactly the Jones invariant.

One of the reasons this is useful is applications to TQC. Another reason was that before all of the definitions of the Jones polynomial were two dimensional - you project your knot onto two dimensions, do something with every crossing, and show its invariant under Reidemeister movies. The beautiful thing about this new QFT definition is that it is inherently 3 dimensional, and does not require one to do any projections.