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Quantum field theory and the Jones polynomial. (English) Zbl 0726.57010
Braid group, knot theory and statistical mechanics, Adv. Ser. Math. Phys. 9, 239-329 (1989).
[For the entire collection see Zbl 0716.00010.]
This paper describes an intrinsically 3-dimensional definition of the Jones polynomial of a knot in \(S^ 3\) in terms of quantum field theory. The theory in question is \(2+1\) dimensional quantum Yang-Mills theory with an action given by the integral over a 3-manifold M (such as \(S^ 3)\) of the Chern-Simons 3-form. This leads to a generalization of the Jones polynomial to give invariants of arbitrary 3-manifolds (by replacing \(S^ 3\) by the 3-manifold M and taking the empty knot). These invariants can be computed in terms of a surgery presentation of M.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
81T13 Yang-Mills and other gauge theories in quantum field theory
57R65 Surgery and handlebodies
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics