I’m now a postdoc at the University of Toronto, and I have a new baby daughter (named Chana or Ann), hence I haven’t posted for a while.
A common theme in topology, and in mathematics in general I suppose, is that it is important to work out the natural setting of a problem. This would be the setting in which the problem is stated most naturally, and in which it’s solution makes most sense conceptually.
For instance, in knot theory, does your problem make more sense for knots or for links? (Example: skein theoretic invariants) Is there a reason that you are restricting to knotted ‘s in , or would the problem make more sense for knotted ‘s in (Example: Seifert surfaces), or perhaps for more general knotted objects in more general manifolds (Example: fundamental group of the complement)? Do you need the manifold at all? Maybe your problem is one which involves group presentations, and is most naturally stated as a combinatorial group theory problem regarding groups with presentations of deficiency 1? Maybe your problem is homological, and makes sense for any homology manifold? (Example: homological invariants) Maybe it’s a problem which makes most conceptual sense in the context of symmetric chain complexes and L-theory? (Example: linking forms)
Sometimes the search for the right setting for a problem might also involve restricting the class of objects you might want to consider. And sometimes it might involve changing it completely.
In this post, I’d like to discuss the natural setting for the Alexander polynomial.
Setting 1: Modules over unique factorization domains
A knot complement has first homology , so the homology of its infinite cyclic cover carries the natural structure of a finitely presented -module. This module, which we call , has a presentation matrix
which is called the Alexander matrix. The th elementary ideal is the ideal generated by the minors of . The ring is a unique factorization domain. The maximal common divisor of the elements of (which are polynomials) is an element in , i.e. a polynomial. This is the Alexander polynomial.
In this approach, the Alexander polynomial is an invariant of a finitely presented module over a unique factorization domain ( could have been any such module). This was Alexander’s original approach. It was not conducive to proving theorems about the form of Alexander polynomials, and therefore I don’t think that this is the natural setting in which Alexander polynomials should be considered.
Setting 2: Modules over principal ideal domains
The next approach, historically, was to localize and to consider as a module over its ring of fractions Λ. The ring Λ is a principal ideal domain. The module is torsion when considered as a module over Λ (Lemma 4 in Milnor’s A duality theorem for Reidemeister torsion), and the generator of its annihilator is (roughly) the Alexander polynomial.
Localizing (the process of considering as a module over Λ instead of over ) was a process which took me a while to conceptually understand… but now I think I can even understand it well enough to explain it! Let be the set of polynomials which we are inverting to get from to Λ, which are those polynomials which have constant term 1. The conceptual idea behind localizing is that the Alexander polynomial should really be thought of as an invariant of the non--torsion part of . In other words, the Alexander polynomial is really an invariant of finitely presented modules over principal ideal domains, not an invariant of modules over unique factorization domains at all… or is it?
Setting 3: (Non)planar algebras
The third approach which I’d like to discuss is the quantum topology approach. I don’t know the conceptual reason that the Alexander polynomial is a quantum invariant (I blogged about it, though here, here, and here), but I have a better grip of the question now. A quantum invariant is not really an invariant of a knot or a link, as far as I can make out- conceptually it’s an invariant of a planar algebra, or of a non-planar generalization of such. You can project a knot in to a plane to obtain a knot diagram. Knot diagrams are made up of over- and under- crossings (the generators for our planar algebra), which are joined together in a plane. Two knot diagrams represent the same knot if and only if they are related by a finite sequence of Reidemeister moves. The planar algebra generated by crossings modulo Reidemeister moves is what the Alexander polynomial (of a knot), in its quantum aspect, is an invariant of.
Via Melvin-Morton-Rozansky, the Alexander polynomial occurs as a certain part (the “wheels part”) of the universal quantum invariant, a very complicated invariant called the Kontsevich invariant. So to understand the Alexander polynomial, one has to drag it out of the mud- the Kontsevich invariant mud. But now, because we’re thinking of the Alexander polynomial as an invariant of planar algebras and suchlike, maybe we could find a nicer planar algebra for which the Alexander polynomial itself was the universal finite-type invariant? This would allow us to directly attempt to understand the conceptual reason why the Alexander polynomial is a quantum invariant.
Dror Bar-Natan has done just this (click on any of his newer talks), showing that the Alexander polynomial is the universal finite type invariant of -knots. A -knot topologically corresponds to a knotted torus in . But from the point of view of quantum topology, we think of it as a diagram in a (non-planar) algebra generated by over- and under- crossings, whose endpoints are connected by strands. These strands are now no longer required to live in the plane- they are just associations between endpoints of crossings, and are free to intersect. Two such diagrams are considered equivalent if they are related by a certain extended family of Reidemeister moves.
The whole setup is designed to make the quantum picture as simple as possible. And indeed, the universal quantum invariant for such objects turns out to be the Alexander polynomial. Knots map into -knots by inclusion (remember that a knot is now a certain sort of diagram).
So, from a quantum perspective, are –knots in fact the natural setting for thinking about the Alexander polynomial?
Something completely different?
There is great scope for irresponsible fantasies. It isn’t hard to come up with a silly dream for a unified perspective which makes it clear that the Alexander polynomial is quantum. Perhaps the day will come when one of these silly dreams will come true.