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.

Incidentally, if you haven’t already (and if you’re not planning on answered them yourself in your next post), you should ask those questions on MathOverflow! I’d certainly love to know what Alexander(point) is. My memory is that people say that Alexander comes from U_q(gl(1|1)), so Alexander(point) is the q-deformed (1|1)-dimensional “defining” representation. But I haven’t studied it myself, and I certainly don’t know how to put it into a full extended TQFT.

Comment by Theo — February 2, 2010 @ 3:47 pm |

Thanks! I asked on Math Overflow following your suggestion.

Comment by dmoskovich — February 2, 2010 @ 8:20 pm |

Hi, these posts are really interesting, keep them up (I want to hear about this hypothetical TQFT measuring things about 2-handles, etc!) My initial reaction is that there is no good understanding of why the Alexander polynomial is a quantum invariant; as you say, this would seem to be a big step towards understanding the topology of the quantum invariants. Once we understood the arrow “topology of Alex. Polynomial —-> quantumness of Alex. polynomial” very well, we could presumably apply the inverse arrow to the Jones polynomial…

A few random thoughts. Heegaard Floer homology decategorifies to the Alexander polynomial, and the new Bordered Floer homology of Ozsvath-Thurston-Lipschitz “extends” the corresponding 3+1 TQFT by assigning a category C_S to closed 2-manifolds S. So the object that the Alexander polynomial QFT assigns to a surface should be the Grothendieck group of V_S. It would be very interesting for someone to go through the construction and see what this gives you… I can’t remember, but general heuristics might suggest it should be HF(S X S^1). As for Alex(point), this should be a 3-category or something, no? I don’t see any good reason that it should have been written down already, but maybe it has been.

As far as understanding intrinsically why the Alex. polynomial can be broken up as a tangle invariant, I presume that this could be deduced explicitly from the topological definition, which would be a worthy exercise, I think, maybe half the battle (one of those things I always think I should take a weekend and do…) One could start with the baby case where the Hopf Algebra is just the group ring of some finite group; then the invariant one obtains is just the set of representations from the knot group into G. Why is this local? If we upgrade G to a Lie Group (SU(2)?), what’s the relationship with the Jones polynomial? I would like to see the answers to these (possibly naive) questions.

Last thing: you say the V from A —> End(V^\otimes n) has a homological interpretation — in what sense? This is the same V which carries the defining representation for U_q(sl_2) in the case of the Jones polynomial, right?

Comment by Sam Lewallen — February 3, 2010 @ 12:51 pm |

MR1158467 (93a:57007) Frohman, Charles; Nicas, Andrew The Alexander polynomial via topological quantum field theory. Differential geometry, global analysis, and topology (Halifax, NS, 1990), 27–40, CMS Conf. Proc., 12, Amer. Math. Soc., Providence, RI, 1991

Comment by Charlie Frohman — February 19, 2010 @ 2:10 pm |

Thank you very much for this reference! I forgot all about that idea, and in one sense it completely answers my question- the Alexander polynomial is shown to have a natural, conceptual interpretation in terms of TQFT. But it’s still not shown to be a quantum knot invariant in the sense above- where is the R-matrix? Theorem 1 of Thomas Kerler’s

“Homology TQFT’s and the Alexander-Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory”. Canad. J. Math. 55 (2003) 766-821.

shows that it is equivalent to the Hennings TQFT for a certain non-semisimple Hopf algebra, but again, the proof is not at all conceptual as far as I can see- it’s a calculation, and I have no intuition as to why it should be true.

Do you know whether there has been any progress since Kerler? The idea in your paper with Nicas is beautiful and simple, and makes the Alexander polynomial look just like a graded U(1) Casson invariant, which looks completely natural! To ask the question provocatively, why has the paper been “forgotten” for 20 years?

Comment by dmoskovich — February 23, 2010 @ 7:12 pm |

Not all quantum invariants come from R-matrices, but an awful lot do. The point of spherical categories is to define quantum invariants of three manifolds in the absence of R-matrices, though I guess you can work backwards and find them anyways.

I am not sure there is an inclusive answer to the question: What is a quantum invariant? The problem is that quantum has so many meanings. Maybe it means coming from a noncommutative deformation of a commutative invariant. Maybe it means that it comes from a state sum. Maybe it just means it’s inspired by a construction having its roots in quantum mechanics.

I think people didn’t read our paper because it wasn’t widely available.

Our (meaning Andy’s and my) other paper in that time period on knot invariants from the intersection homology of moduli spaces of semistable bundles of fixed determinant over a nonsingular algebraic curve, was something that even fewer people read. However, the base idea, that the homotopy theory of the gauge group factors through the homology of the underlying surface, is why Seiberg-Witten invariants in dimension three report about the Alexander polynomial.

At the time, our work was sort of a bummer, because people though they were going to get the Jones polynomial out of the action of the mapping class group on the moduli space of U(n)-bundles. It was something that needed to be done though,

and it could be done because of Atiyah and Bott’s work on Yang-Mills on Riemann surfaces and Kirwan’s computation of the intersection homology of the moduli spaces.

It was later that Lawrence and Krammer found the right place to look for an action on homology that recovered the Jones polynomial. That work was fantastic, as it led to Bigelow’s proof that the braid group is linear.

One of my students Adam McDougall recently put a paper on the Arxiv titled “A diagramless link homology” that explains the Jones polynomial as a weighted sum of equivalence classes of unknotted surfaces having the knot as boundary. I think its a cool theorem. I like Stavros’ conjecture about the colored Jones polynomial and boundary slopes of incompressible surfaces,

which is moving even farther away from R-matrices and towards topology.

I also like the work of Ryzard Rubinstein and Magnus Jacobssen where they find a categorification of the Jones polynomial from the SU(2) representations of the fundamental group of the knot that send the meridian to a matrix of trace zero.

I guess the point is that all of this work is about a quantum invariant, but its significance is that it is moving away from the R matrix definition and towards things that can be analyzed from geometric, topological and analytic viewpoints.

Comment by Charlie Frohman — February 23, 2010 @ 8:41 pm |