Since Witten's seminal work in the 80's showing how to obtain an invariant of knots in a 3-manifold using the Chern-Simons topological quantum field theory (TQFT,) the close relationship between TQFTs and low-dimensional topology has been much studied by mathematicians and physicists leading to many new insights in both fields. I will begin by explaining one approach to constructing this invariant which can then, despite the lack of an adequate mathematical theory of the path-integrals used to come up with the invariant, be proven by other methods to be well-defined and invariant under ambient isotopy.
Along the way, we will begin to see what appear at first to just be ad hoc tricks which were invented to deal with the gauge symmetries appearing in TQFTs, known to physicists as the BRST formalism. While these tricks have a nice intuitive interpretation in the case of Chern-Simons theory, I will give the example of BF-theory, an $n$-dimensional TQFT which can be used to define invariants of knotted $(n-2)$-dimensional submanifolds of an n-dimensional manifold, for which the BRST formalism fails and an even more powerful one, the Batalin-Vilkovisky (BV) formalism must be used. I will conclude by discussing recent interpretations by mathematicians of the BV-formalism, originally invented by physicists in the 80s, as describing possibly non-free quotients and possibly non-transverse intersections of manifolds in the recently developed world of derived differential geometry.