One of the main ways in which I keep my finger on the pulse of what is hot now in low dimensional topology is to write lots and lots of reviews, both for Zentralblatt MATH and also for MathSciNet. In the last year or so, what has been increasingly coming through the pipe is papers about knot homology and mirror symmetry. There seems to be a lot happening in this field right now.
There are a number of different strands coming together, and the one that is pulling the others right now is physics, and string theory in particular. In the physics approach, knot polynomials are seen as Wilson-loop averages in some version of –dimensional Chern-Simons theory.The knot polynomials which play a particular role are the coloured Jones polynomials and their generalizations, coloured HOMFLYPT polynomials.
Ooguri and Vafa, building on work of Witten, showed how knot and link polynomials are associated to string theories. For an polynomial, certain topological strings are defined, of which wrap around and one corresponds each component of the knot, and the polynomial invariant is thought of as a amplitude of the resulting string in some sense. It turns out that the relevant thing is how the branes corresponding to the knot complement behave at infinity. Mirror symmetry relates this behaviour at infinity with geometric quantities at infinity. We can view several big conjectures in quantum topology at the moment, such as the A-J conjecture and various flavours of the Volume Conjecture. One of the recent strands of work has been on extending these conjectures to links from the physical perspective, which turns out to involve a generalization of the very notion of mirror symmetry itself! Thus, in some sense, low dimensional topology is providing an impetus for deeper physical research into mirror symmetry. This is of course one major aspect of why knot theory has proven historically important inside topology and elsewhere- it provides very nice toy models for other theories. This is the paper I’ve been looking at.
Another active direction has to do with understanding knot homology (Khovanov homology). A lot of sophisticated mathematics and physics seems to be going into this. Everyone seems to be focussing on torus knots and links right now, because the circle action simplifies things- but the concensus seems to be that the results should all generalize somehow. I’ll just mention some things I’ve looked at:
- From the physics perspective, there’s work of Aganagic and Shakirov reframing knot homology (of torus knots at least) in terms of refinements of Chern-Simons theory, rather than more exotic objects. This is again a string theory approach, and follows a conjectural picture of Gukov, Vafa, and Schwarz.
- There’s a lot of work right now on finding lots and lots of gradings (and coloured differentials) on coloured HOMFLYPT homology, in order to make it universal in an interesting way and in order to facilitate its computation. http://arxiv.org/pdf/1304.3481.pdf
- Topological recursion, which is a way of accessing the asymptotic expansion of colored Jones or HOMFLYPT polynomials by looking at the curve associated to the A-polynomial, is a hot topic right now. Various new ideas are popping up, but my general impression looking at this is that what’s going on right now in this field is mainly a lot of groping around in the dark, and that there’s a lot more work to be done in this direction.
- From a mathematical perspective, what I find most exciting are the connections to known deep mathematical structures. Perhaps most intriguing of all is the connection to rational doubly-affine Hecke algebras, AKA rational Cherednik algebras, AKA rational DAHA, which you can read about HERE. Knots and links are arising here via their diagrams, as diagrammatic formalisms for aspects of representation theory, and all kinds of number theory and theory of hypergeometric functions is entering the picture. This direction seems white-hot right now.
I can’t really say that I understand the endgame of any of this, but, on the one hand, the invariants coming out of these lines of work are among the most powerful that we know, and on the other hand, there’s hope that they will become easy (or at least easier) to compute. Beyond all of this, all of this categorified coloured HOMFLYPT theory represents a second entry of quantum knot theory into really deep mainstream mathematics.
Maybe knots and links are nothing more than nice toy objects for all sorts of serious mathematical tools. But I like to dream that maybe the algebraic structure of a knot diagram genuinely does encode some central aspect of the mathematical universe.