Low Dimensional Topology

January 13, 2010

The Alexander Polynomial as a Quantum Invariant: Part 1

Filed under: Knot theory,Quantum topology — dmoskovich @ 4:07 am

In this series of posts, I would like to lay open my confusion regarding how the Alexander polynomial is a quantum invariant (I’ll explain what that means in the next installment). The Alexander polynomial is the archetypal knot invariant in many ways. It’s also the archetypal quantum invariant. Let’s define it to be the order of the Alexander module (the first homology of the infinite cyclic cover of a knot complement as a module over deck transformations). This is what one would call a “big construction”- it works in great generality, for any homology circle and with any coefficients, and it gives a clear conceptual picture of what the Alexander polynomial is measuring.
Quantum invariants were discovered through physicists, via a procedure which looks completely ad-hoc and seems to have nothing to do with topology. Without expecting anyone to understand at this point, a quantum invariant assigns an R-matrix to crossings of braid diagrams (you can do it for tangles, knots, and links but it’s native to braids), and composes them “vertically”. These R-matrices arise from representations of ribbon Hopf algebras such as quantum groups. One of these R-matrices, associated to a certain representation of the braid group B_n on n strands onto V^{\otimes n}, where V is a 2-dimensional vector space over \mathbb{C}, happens to recover the Alexander polynomial. The conceptual reason why this happens is a total mystery to me, and the topic of a question I asked on Math Overflow.
A quantum knot invariant is typically defined through a small construction- via an R-matrix or a skein relation. Such a construction may be good for calculation, but it depends on a lot of choices. In this case, the Alexander polynomial as a quantum invariant seems to only be defined over \mathbb{C}, and seems to be an invariant of a braid which just happens by lucky coincidence to be invariant under Markov moves. This is deeply unsatisfying.
In the third installment of this series of posts, I plan to outline the construction of the Alexander polynomial as a quantum invariant, and the proof (from Ohtsuki’s book “Quantum Invariants”) that what we are actually getting is the Alexander polynomial. I will warn you that this proof (which factors through the Burau representation) gives me no insight into the conceptual reason why the statement should be true… any insights and ideas would be appreciated!
So what would I want a conceptual proof to look like in a dream world? Here’s one fantasy: Consider a cobordism between a knot complement and a solid torus (the complement of the unknot). Consider an appropriately chosen 3+1 TQFT, a functor from such objects to vector spaces satisfying Atiyah’s axioms. In one direction, if we present our TQFT as T\times I (solid torus times unit interval) with 2-handles attached (a Dehn surgery presentation of our knot), we should recover the Alexander polynomial as in Chapter 7 of Rolfsen, through surgery. Somehow, the TQFT should be recording the lifts of the handles in the infinite cyclic cover, together with the deck transformations, and the question is how to construct that. On the other hand, there’s a standard way to derive an R-matrix from a TQFT. I have a feeling physicists know something like this, and that it’s probably somewhere in some paper of Rozansky or Saleur (do you, the reader, know?), although perhaps only on the physical level of rigour.
I have the feeling that the lack of a natural, intuitively obvious proof that the Alexander polynomial is a quantum invariant, should really strike right to the core of the central problem in quantum topology, which is how to interpret quantum invariants topologically. At least, it is a critical gap in my own personal understanding.



  1. Hi Daniel, I’m looking forward to seeing where you go with this.

    One question I’d like answered (thought I’m not asking you or anyone in particular) is this: if the Alexander polynomial is a quantum invariant, what kind of quantum information (excuse the double-meaning there) is it giving us?

    The topological information provided by the Alexander polynomial is well-trodden territory. For example, I know that if the Alexander polynomial has symmetric quadratic factors, there’s potentially a non-trivial signature invariant in the Alexander module so I can hope to find a slice obstruction, etc.

    But when someone tells me something is a quantum invariant, other than giving me some idea on how the invariant is constructed all I really know is it’s an invariant. So as far as I can tell the scope for what can be done with it is highly constrained. For example, compare the proofs of how the Alexander polynomial and Jones polynomials give obstructions to knots having various types of symmetries. The Alexander polynomial proofs basically just lift the automorphism to the abelian cover and it follows. But for the Jones polynomial, you have to put the knot into a symmetric position and then find the corresponding symmetric knot diagram. So the technical details are far more fierce.

    Comment by Ryan Budney — January 16, 2010 @ 6:55 pm | Reply

  2. A refinement: are there simpler proofs of the properties of the Jones polynomial of symmetric knots? Ones that avoid the use of special knot diagrams — arguments that stick to “native” Chern-Simons theory?

    Comment by Ryan Budney — January 16, 2010 @ 7:17 pm | Reply

    • I know a partial answer to your question- the Jones polynomial is the sl_2 reduction of the Kontsevich invariant, which has a surgery formula. Symmetric knot complements (any orientable DIFF manifold admitting a faithful properly discontinuous periodic orientation-preserving action) have symmetric surgery presentations (Sakuma proves this in general, but it’s probably even simpler for knot complements), and the properties of the Jones polynomial follow. I have no idea whether any of this is written-up…

      Comment by Daniel Moskovich — January 17, 2010 @ 12:35 am | Reply

  3. I guess what I mean to ask is, “is there any conceivable way to prove these theorems (Jones polynomial properties for symmetric knots) without using symmetric knot/link/surgery diagrams?”

    Comment by Ryan Budney — January 17, 2010 @ 7:06 pm | Reply

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