A categorical approach to knot theory?

Question

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.

Answer 1

To "study X from the perspective of Y" you need to have a good reason to believe the language of Y is effective to study X. (Imagine reading a question titled "a knot-theoretic approach to category theory". Why should knots help us understand categories?) Without an argument that something is fruitful, we are chasing ghosts.

Ultimately you need to define what an embedding of knots is, you need to define what an isotopy of knots is, to talk about knot groups you need to know what a knot complement is, to talk about Reidemeister's theorem you need to know what knot projections are (and Reidemeister's theorem itself is hardly pure formality, it requires actually thinking about what PL isotopies look like when projected to the plane). To talk about the fancy stuff you might be excited about, like the Jones polynomial or Khovanov homology, you need to know what a crossing diagram is and what the Reidemeister moves are and the relation of knot isotopy to crossing diagrams modulo Reidemeister moves. And so on, and so on.

I see no reason that model categories or homotopy theory can provide a single piece of insight into the above.

Sure, there is, say, a simplicial set whose vertices are embeddings $S^1 \to S^3$ and whose edges are isotopies (and k-cells are embedded sections of the projection $S^3 \times \Delta^k \to \Delta^k$). That gives you a space of knots, which is interesting, but provides no help with anything I mentioned above.

There is also, say, a category ('the concordance category') whose objects are embeddings $j: S^1 \to S^3$ and whose morphisms $\text{Hom}(i,j)$ are embeddings $H: [0,1] \times S^1 \to [0,1] \times S^3$ so that $H(t,z) = (t, i(z))$ for $t \in [0, \epsilon)$ while $H(t, z) = (t, j(z))$ for $t \in (1-\epsilon, 1]$. But this is an additional thing you can study / investigate, connecting 3-dimensional knot theory and 4-dimensional (surface) knot theory. It doesn't somehow help you with the basics.

You need to get your hands dirty. There is no magic categorical ideas that will avoid that.

Answer 2

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.