My original intention for this post was to summarize Chapter 2.3 and Appendix C of Quantum Invariants by Tomotada Ohtsuki, with some commentary. But it looks like an opaque calculation to me at the moment- calculate the R-matrix of the Alexander polynomial from its Skein relation, and lo-and-behold it coincides with the R-matrix you get from a certain representation of the braid group on strands to , where is a 2-dimensional vector space over . It’s mathematics I can calculate as opposed to mathematics I understand (to some extent anyway), and I’d much rather blog about the latter than about the former. Ohtsuki writes it up better than I ever could anyway.
Luckily, thanks to a number of very good reader responses, I can post instead about two very beautiful and profound ideas, which, while falling short of conceptually explaining the quantum nature of the Alexander polynomial (to my eyes anyway), certainly point very much in the right direction.
Let’s begin with the reference Ryan sent me:
A Lagrangian Representation of Tangles by D. Cimasoni and V. Turaev.
This paper extends the Burau and Gassner representations from braids and string links to tangles. It gives rise to the Alexander polynomial of the closure if the tangle is straight, meaning that it has no closed components. In this way, it explains the local nature of the Alexander polynomial (restricted to straight tangles at least) in a conceptual way. But, because the tangles must be straight, and you have to take the closure “by hand”, it doesn’t really answer the question. The “spinning” operation of taking the closure isn’t built into the construction. In any event, this is rather a large step in the right direction, I think, and any conceptual explanation worth its salt of the quantum nature of the Alexander polynomial should surely recover Cimasoni-Turaev.
The tools used are completely classical, although the setup is one which quantum topologists use all the time. Consider your tangle to live inside a cylinder , and let denote (the complement of the tangle in the cylinder). Perturb a height function so that at each height slice you have a crossing, a cup, or a cap. Think of this as a cobordism between two oriented discs with holes and . In physics, the cobordism might be a time evolution from one state of the world to another, your cup might be the birth of a particle-antiparticle pair, and the cap might be their annihilation. As far as the Alexander polynomial and suchlike invariants are concerned, we only care about homology (by the way, in my opinion this implies that a good explanation of anything to do with the Alexander polynomial of a knot should work for a homology embedded in a homology ).
In think I’d like to think about what Cimasoni and Turaev do next in terms of homology wiith local coefficients (thanks one more time, Ryan!). Let be the infinite cyclic covering space of the knot complement, and consider where is the bundle of groups where the fibre over a point is . Starting with the case that is a braid, as a warmup to the key idea, induces a homomorphism from to , acting on homology classes sort of like an egg-beater. This homomorphism turns out to be a unitary isomorphism, and mapping to the matrix representing gives rise to the Burau representation of the braid group. Next consider to be a string link (strands go from bottom to top, but not necessarily monotonically). This time you have to localize local coefficients, but after you do that, the same construction again gives a unitary homomorphism to , and mapping to the matrix representing gives Le Dimet’s extension of the Gassner respresentation.
In both these cases, a cobordism induced a homomorphism from the homology of the bottom disc to the homology of the top disc, with appropriate local coefficients. But in general a cobordism won’t do that. But a standard result about concordance is that what a cobordism will do is that the kernel of the inclusion homomorphism will induce a morphism between and (coefficients in the bundle of groups ). This kernel is Lagrangian with respect to the intersection form on homology (their definition of Lagrangian is a bit algebraic, but it means just what you think it means- maximal isotropic).
This idea reminds me of chess players like Karpov and Capablanca- the moves look simple and obvious, but only after they have been played. That the idea looks so obvious in retrospect seems to me to be a sign that it is the right one, and a testimony to the genius of its authors.
Then, of course, you have the details- needs to be free, or the algebra doesn’t seem to work. Therefore needs to be a straight, and the “spinning operation” needs to be worked in by hand. So it fails to be an explanation of why the Alexander polynomial of a knot or of a link is a quantum invariant, or at least has the right locality properties (it doesn’t bother calculating the R-matrix, although it’s obvious how to do so).
The second reference I was sent, by one of it’s authors (thanks!) is
The Alexander polynomial via topological quantum field theory. by C. Frohman and A. Nicas, in Differential geometry, global analysis, and topology (Halifax, NS, 1990), 27–40, CMS Conf. Proc., 12, Amer. Math. Soc., Providence, RI, 1991.
It’s a bit terrifying to try to summarize the main ideas here when the author might well be reading what I write… well, blogging is really all about advertising one’s ignorance to the world anyway, so here goes…
Consider a knot . Frohman and Nicas consider the Alexander polynomial to as an invariant of the homology of a Seifert surface (independent of the choice of Seifert surface), so they choose a Seifert surface with bicollar whose complement is . Consider as a cobordism between and . You now have the inclusion which is represented by the Seifert matrix (after you choose bases for homology). This is pretty much the classical setup for the Alexander polynomial.
What Frohman and Nicas do now, which is very interesting, is to reproduce this setup on the Jacobian of the Seifert surfaces, that is the -representation variety . If is not connected, is the cartesian product of these over its connected components. It turns out that and are isomorphic, as are and . This is either miraculous or obvious, depending how you think about it- obvious because is just the circle group, and this is kind-of the universal coefficient theorem, and miraculous because it lets you construct the Alexander polynomial by using, instead of homology of Seifert surfaces, cohomology of their Jacobians. A graded Lefshetz trace takes the place of taking the determinant, their equivalence proved by linear algebra. It bothers me that the linear algebra has to take place over (or at least ) for this linear algebra to work, because we need to take square roots, although Kerler makes it work over , using a bit of representation theory (I think).
By the way, I first read Kerler’s paper in 2002 for a reading course on Reidemeister torsion at Tokyo University. While reading it, I got tired of the use of acronyms like TQFT in mathematics and physics, and resolved to call TQFTs topoquafts. Therefore, because a blog is a good place to showcase one’s silliness, I think I shall continue to call them topoquafts in this post.
So why is representing the Alexander polynomial in terms of Jacobian varieties a good thing? Because you can obtain a topoquaft from this construction in the following way. First, perform -surgery on the knot in , making the two copies of the Seifert surface into a pair of closed surfaces and in a –manifold . So now is a cobordism between closed surfaces. The construction is easier to describe (see Kerler) when is an epimorphism, so let’s restrict to that case. Then is a half-dimension immersion (is it Lagrangian with respect to some natural non-degenerate skew-Hermitian form?). So the top form defines a middle-dimensional homology class in . Using Poincare duality, this space is isomorphic to the space of linear maps between homology rings of the Jacobian varieties. The functor which associates to the linear map which comes from now satisfies the axioms of a topoquaft. In general, if fails to be onto, Frohman and Nicas consider the (graded) intersection number of the Jacobian of a thinkened Heegaard surface for with the Jacobian for handles separated by that surface. This makes the Alexander polynomial look very much like the Casson invariant, which is good!
This construction puts the Alexander polynomial firmly inside the world of quantum topology, but doesn’t explain why it is a quantum invariant in the sense discussed in the last post- namely, it can’t be calculated from decomposing your knot into crossings, cups, and caps in any obvious conceptual way, and it doesn’t even give rise to an -matrix. I don’t even see how it extends to tangles, string links, and braids. Actually, Kerler shows that you can recover an -matrix from the Frohman-Nicas topoquaft by proving its isomorphism with the Hennings topoquaft, which does give all these good things, for the non-semisimple Hopf algebra . This is a horrible Hopf algebra to have to work with. Worse still, Kerler’s proof is a calculation and imparts no conceptual understanding (to me anyway) as to why it should hold. In this sense, Frohman-Nicas both completely answers our question about how to see the Alexander polynomial as a quantum invariant, and completely fails to answer it. But it’s a beautiful and profound idea in any case.
What strikes me most about this paper is that nobody besides Kerler seems to have followed up on it, despite the fact that it’s almost 20 years old and that the ideas in it are so natural. There are so many really silly, naive questions which I have, such as:
- Does this work in any setting where one constructs the Alexander polynomial using a Seifert surface? How about homology boundary -links in a homology -sphere?
- Can you get rid of the Seifert surface? I have some silly ideas which don’t seen to work about this…
- Categorify. Surely. This setting is pure Knot Floer Homology Land.
- As the authors mention- different gauge groups! Maybe stuff like .
So, in conclusion, I’m surprised to say that Ryan’s request for a conceptual explanation of why the Alexander polynomial is a quantum invariant (in the sense I mean, at least) seems to be an open problem. And a very interesting one at that, if you ask me!