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"Invariants of 3-manifolds via link polynomials and quantum groups", Nicolai Reshetikhin, Vladimir Turaev, 1991

Reviewed August 31, 2023

Citation: Reshetikhin, Nicolai, and Vladimir G. Turaev. "Invariants of 3-manifolds via link polynomials and quantum groups." Inventiones mathematicae 103.1 (1991): 547-597.

Web: https://link.springer.com/article/10.1007/BF01239527

Tags: Foundational, Mathematical

Witten's knot invariant paper was groundbreaking, but (to quote the authors) it was on a "physical level of rigor". In this paper, Reshetikhin-Turaev define a new family of knot invariants from manifolds (i.e. TQFTs). To quote the authors again: "our invariants may be viewed as a mathematical realization of the Witten's program".

This procedure takes in a modular tensor category, and spits out a topological quantum field theory. It is exactly this construction which allows us to say that there is a mathematical connection between MTCs and TQFTs. Of course, it is not proved in this paper that the correspondence induces a bijection. To make the bijection work you need to add extra data on both sides of the equation and work out a lot of details, which was finally done in

> Bartlett, Bruce, et al. "Modular categories as representations of the 3-dimensional bordism 2-category." arXiv preprint arXiv:1509.06811 (2015).

This paper mostly uses the language of Hopf algebras and quantum groups.