TITLE: A categorical approach to knot theory?
QUESTION [5 upvotes]: I want to learn some knot theory. I already know some algebraic topology and have teeny-tiny bits of understanding of model categories, homotopy theory, that jazz. (To estimate the level of abstraction I'm comfortable with.)
Specifically, I want to read something that takes on knot theory from a categorical/sufficiently abstract POV. Does such a thing exist?
Naively, I thought there would be some neat 'category of knots,' but after pushing around my own candidate definitions for it, I realized such a category would be much trickier to define than I thought. And a quick Google search did not recover anything.
This is a reference-request.

REPLY [5 votes]: One part of knot theory where category theory is useful is in the study of knot invariants.  An early method was to use Markov's theorem, that every knot is the closure of a braid.  By finding representations of the braid group that have a "trace" satisfying particular properties, you can get knot invariants such as the Jones and HOMFLY polynomials.  The braid group ends up being the group of invertible endomorphisms of a larger category, the tangle category, and the representations each generalize to being functors from the tangle category to the category of representations of a quasitriangular Hopf algebra (for example, a so-called quantum groups, certain deformations of universal enveloping algebras of Lie algebras).  This all falls under the heading of "topological quantum field theory," or TQFT.
Knots and representation theory are deeply intertwined.  Here's one example.  In classical representation theory, one aspect of Schur-Weyl duality is that every endomorphism of $V^{\otimes n}$ that commutes with the $SL(V)$ action can be written as a linear combination of permutations.  For $U_q(\mathfrak{sl}(2))$, the quantum deformation of $\mathfrak{sl}(2)$, permutations are replaced with linear combinations of braids.  (The linear maps from permutations and braids are not injective, though.)
Khovanov homology is a "categorification" of the Jones polynomial.  Polynomials are replaced with modules (chain complexes in particular), and certain kinds of maps between knots (knot concordance) are sent to homomorphisms of chain complexes.  There is a paper by Lauda and Pfeiffer that seems to give a construction of Khovanov homology for tangles, but I'm not sure if this gives a functorial construction in the above sense (I haven't really read it).
I know people who work on TQFTs with little geometric intuition.  They have a hard time understanding my diagrams, and I have a hard time following their manipulations with Sweedler notation. Maybe this suggests that these things aren't "really" knot theory, but knot theory is a big subject.  I myself would find it all to be very difficult if I didn't have a firm grounding in geometric topology and 3-manifolds.
Another book(s) that might be interesting to you are ones by Kauffman.  He likes to work with knots as formal combinatorial objects, and that lends itself well to categorical treatment.