In this second installment of the series, I’m going to give you one interpretation of what we mean when we call something a quantum invariant.
Note: Two better quantum topologists than myself suggested to me (implicitly or explicitly) that a quantum invariant is an invariant given by a skein relation. I maintain that this is wrong- it’s an invariant given by an matrix. A counterexample in one direction is the Brandt-Lickorish-Millett-Ho polynomial, which is given by a skein relation but is not quantum (and may contain no quantum invariant besides polynomials in the Casson invariant). In the other direction, what of a quantum knot invariant not contained in the coloured HOMFLYPT polynomial of a knot, if such an invariant exists?
The first thing we are saying when we say that invariant is a quantum invariant is that it is “determined locally”. Let’s say we are talking about quantum invariants of an -manifold , although the analogous statements will hold for knots, links, tangles, braids, whatever… for any finite decomposition for -manifolds with boundary and closed -manifolds the value is determined by . In fact, even more than this is true- (with definition suitably extended) is determined by its value on a single point by the Baez-Dolan cobordism hypothesis.
Question: Why is the Baez-Dolan cobordism hypothesis true for the Alexander polynomial? What is the value of an “extended Alexander invariant” on a single point? Surely this must be well-known. Maybe it comes from some sort of Goodwillie calculus construction? Why is it “local”? Can you naturally interpret it in terms of homology with local coefficients, for example?
The basic idea in defining and computing a quantum invariant is to fix its values on a short list of “simple” -manifolds with boundary , such that any -manifold can be constructed by gluing together elements of the list along components of their boundaries. For example, a quantum invariant of an oriented surface is specified by the values it takes on disks, cylinders, and pairs of pants.
To reiterate, to say something is a quantum invariant means that it is determined by its local values on “simple” objects. As the dimension grows, any such construction becomes increasingly inadequate, because it becomes harder to simplify much by cutting along closed submanifolds of codimension 1, and these submanifolds themselves might be complicated objects. This brings us to the following limitation of quantum topology.
Limitation: Loosely speaking, quantum topology is only mathematically rigourous in dimensions for . Thus, mathematically, we are looking (for example) at knot diagrams rather than at knots. This is perhaps why quantum topology has been most useful to answer combinatorial questions about knot diagrams, but hasn’t yet given much insight into the geometry and topology of knot complements.
Thus, we look at knots as cobordisms between disks. So a knot becomes , and we have no tools to consider anything fancier like bordisms between knot complements. This is the best we can do at the moment. The most popular list of “simple” submanifolds for a knot is , with all possible orientations. Clearly this list is light years from being unique, and for all I know it might not even be best… anyway, to specify a quantum knot invariant , it turns out that it is enough to specify its value on each submanifold on this list. And the one that “really” matters is the value which we choose for an positive crossing, which we call the -matrix.
The second thing we are saying about when we say it is a knot invariant is that it factors through a representation for some finite-dimensional vector space over a field , some positive integer , and some -algebra . So we’re saying that it is a “representation-theoretic” invariant. Typically will be given by some kind of homology or cohomology.
So maps a positive crossing (for instance) to an element of , which is in turn mapped to a matrix over . In the same way, negative crossings are mapped to , and images of caps, cups, and the trivial line segment are mapped to elements of depending on orientation. Reidemeister moves and ambient isotopy on the knot induce relations between these elements. The most important of these is the Yang-Baxter equation for the -matrix, induced by Reidermeister 3. Translating ambient isotopy and Reidemeister moves into relations in is great fun, and is carried out in books by Kassel, Turaev, or Ohtsuki, and probably somewhere in This Week’s Finds in Mathematical Physics… anyway, the upshot of it all is that, for to factor through a representation of and be a knot invariant, must satisfy the conditions of a so-called ribbon Hopf algebra. The typical example of a ribbon Hopf algebra, by the way, is the awfully mis-named quantum group.
Question: Why is the Alexander polynomial “representation theoretic” in the sense described above? How does it relate to the ribbon Hopf algebras from whose representations it comes?
So that’s it… there were equivalent analytic ways of saying the same thing, but in a nutshell, when we say that an invariant is quantum, we are saying that it has both a “local” and a “representation theoretic” nature. I know but don’t understand why the Alexander polynomial should have either.