TITLE: Laymans explanation of the relation between QFT and knot theory
QUESTION [16 upvotes]: Could someone give an laymans explanation of the relation between QFT and knot theory? What are the central ideas in Wittens work on the Jones polynomial?

REPLY [8 votes]: This will not be by any means a "layman's" explanation, but here's something short that might be helpful. There also won't be much physics here, if that's what you're looking for. 
Let's take for granted, following Witten, that if $G$ is a simply connected compact Lie group (for simplicity) and $k$ is an integer, then there is a 3d topological field theory called "Chern-Simons theory with gauge group $G$ and level $k$." Chern-Simons theory is so named because its heuristic physical description involves path integrals where the action has something to do with the Chern-Simons 3-form. I won't get into all of the structure that this TFT has; it's enough for now that it assigns (up to some subtleties I don't want to get into)

a finite-dimensional complex vector space $Z(\Sigma)$ to every closed surface $\Sigma$, and
a linear map $Z(\Sigma_1) \to Z(\Sigma_2)$ to every 3d cobordism $M$ between $\Sigma_1$ and $\Sigma_2$

and these assignments satisfy various compatibilities. Also, you'll need to know one more thing: the vector space $Z(T^2)$ assigned to the torus has a natural basis that can be identified with either the irreducible representations of the corresponding loop group ("at level $k$") or with the irreducible representations of a version of the corresponding quantum group (for some root of unity $q$ depending on $k$). In physics language these are "Wilson loop operators." 
How do you get knot invariants out of this? Given a knot $K$ sitting inside $S^3$, consider a tubular neighborhood around it. The complement of this tubular neighborhood describes a cobordism from the torus $T^2$ to the empty manifold, and hence hitting it with Chern-Simons gives a linear map
$$Z(T^2) \to \mathbb{C}.$$
Now evaluating this linear map at an irreducible representation of, according to taste, either the loop group or the quantum group produces a knot invariant. When $G = SU(2)$ and the representation is the standard one, you get the Jones polynomial evaluated at some root of unity $q$ depending on $k$.