Low Dimensional Topology

November 12, 2009

Alexander polynomials and noncommutative localization

Filed under: Knot theory,noncommutative localization — dmoskovich @ 11:20 am

This post comes along with a more detailed PDF version.

Kent Orr at Indiana University taught me a gorgeous, intrinsic (but not well-known) homological proof due to Pierre Vogel that the Alexander module of a knot is annihilated by an Alexander polynomial. The only fact it uses about a knot is that its complement is a homology circle. The starting point is:

Low-dimensional topologist’s definition of Cohn localization:
Let R,S be rings with unity, let R\overset{\epsilon}{\longrightarrow} S be a
ring homomorphism, and let C_\ast be the chain complex

\cdots \overset{\partial_2}{\longrightarrow} C_{2}\overset{\partial_1}{\longrightarrow} C_1 \overset{\partial_0}{\longrightarrow} C_0

such that C_n is a finitely-generated free R module for all n. The Cohn localization \Lambda of R\overset{\epsilon}{\longrightarrow} S is the initial ring with the property

C_\ast\otimes S\text{\ is acyclic}\Longleftrightarrow C_\ast\otimes \Lambda\text{\ is acyclic.}

An explanation of relevant terminology, alongside a proof that this definition of the Cohn localization is equivalent to the algebraist’s definition, is to be found in the PDF version of this post.

The goal of this post is to show that H_1(E_K;\mathbb{Z}[t^{\pm 1}]) is \Lambda-torsion, where E_K denotes the complement of a knot K\subset S^3, and \Lambda=\left\{1+\ker \epsilon\right\} and \mathbb{Z}[t^{\pm 1}]\overset{\epsilon}{\longrightarrow}\mathbb{Z} is the augmentation map, sending t to 1.

Let m be a meridian of K and let \tilde{E}_K be the infinite cyclic cover of E_K in which m (a loop) lifts to \tilde{m} (an infinite line). Consider the chain complex C_\ast(\tilde{E}_K,\tilde{m}) of free \mathbb{Z}[t^{\pm 1}] modules. Then
H_1(C_\ast(\tilde{E}_K,\tilde{m})\underset{\mathbb{Z}[t^{\pm 1}]}{\otimes} \mathbb{Z}) \cong H_1(C_\ast(E_K,m)) \cong 0 by Alexander duality or Mayer-Vietoris (the point here is that E_K
is a homology circle generated by m). The following purely algebraic fact is given without proof.

The Cohn localization of \mathbb{Z}[t^{\pm 1}]\overset{\epsilon}{\longrightarrow}\mathbb{Z} is \Lambda= \left\{1+\ker\epsilon\right\}. It is a flat \mathbb{Z}[t^{\pm 1}] module.

Now by the low-dimensional topologist’s definition of Cohn localization

0\cong H_1(C_\ast(\tilde{E}_K,\tilde{m})\underset{\mathbb{Z}[t^{\pm 1}]}{\otimes} \Lambda)\cong H_1(C_\ast(\tilde{E}_K,\tilde{m})) \underset{\mathbb{Z}[t^{\pm 1}]}{\otimes} \Lambda.

Since \tilde{m} is contractible, we get that H_1(C_\ast(\tilde{E}_K,\tilde{m}))=H_1(\tilde{E}_K) is \Lambda torsion, which is QED.

Simple, elegant, and general!

There is a point I’m uncomfortable with. You see, \Lambda is strictly larger than the set of all Alexander polynomials. This is not surprising, because the proof is purely homological, and thus works for CW-complexes which may not be manifolds, and where Poincare duality may not hold. But it is still disturbing.



  1. Hi Daniel,

    So you’re just talking about the proof that the Alexander module is torsion (thought of as a module over the Laurent polynomial ring), right?

    IMO there’s a simpler proof. Look at Cameron Gordon’s proof in LNM 685 called “some aspects of classical knot theory”. That uses rational coefficients, so all you’re left worrying about is whether or not there’s Z-torsion. That doesn’t exist by Poincare duality (of the Blanchfield variety). I think the torsion pairing in this setting is called the Farber-Levine pairing.

    I guess it’s a taste issue. In spirit the proofs look pretty similar.

    Related question, Fox’s theorem that the Alexander ideal (integer coefficients) is a principal ideal, have you seen proofs of that, other than Fox’s?


    Comment by Ryan Budney — November 12, 2009 @ 6:34 pm | Reply

  2. It’s saying a bit more than that… it’s saying it’s a module over Laurent polynomials which augment to \pm 1. Also, Vogel’s proof doesn’t use Poincare duality, because Poincare duality implies that we are in a Poincare duality space (e.g. a manifold), while this proof is purely homological. I think that’s what makes it so attractive… the statement is purely homological, so the proof should be as well.
    I need to think about your second question…

    Comment by Daniel Moskovich — November 12, 2009 @ 6:57 pm | Reply

  3. Oh, okay. But that the Alexander polynomial evaluates to \pm 1 at 1, that’s essentially the statement that a knot complement is a homology circle — you see it in the Gordon article I refer to.

    Comment by Ryan Budney — November 12, 2009 @ 9:02 pm | Reply

  4. I don’t know if you noticed, but I put that comment very explicitly in the Wikipedia article on the Alexander polynomial — that p(1) = \pm 1 is the same thing as saying the underlying topological space is a homology circle.

    Comment by Ryan Budney — November 12, 2009 @ 9:03 pm | Reply

  5. Oh, right, and if your space is a rational homology circle, p(1) is \pm the order of the torsion subgroup of H_1.

    Comment by Ryan Budney — November 12, 2009 @ 9:39 pm | Reply

  6. IMO the literature is a bit of a mess when it comes to the Alexander polynomial. The best references I’ve come across (in terms of elegance of exposition) are Cameron Gordon’s article, Jerry Levine’s “Knot Modules I” (1977) and the recent book of Hillman’s “Algebraic Invariants of Links”.

    Most other articles either assume a very restricted context, or only prove a small epsilon issue, neglecting to re-do what others have done.

    Comment by Ryan Budney — November 12, 2009 @ 9:52 pm | Reply

    • I agree :)
      I would add Milnor’s treatment of Reidemeister torsion to that list.
      It is definitely on my fantasy “to do” list to write a decent survey of the Alexander polynomial- this stuff, Reidemeister torsion, Burau representation, Fox calculus approach, Seifert matrix formula, skein theory stuff, wheels part of the Aarhus integral (which should morally be equivalent to what was in this post, although I don’t see why right now)…

      Comment by Daniel Moskovich — November 12, 2009 @ 10:30 pm | Reply

  7. I’ve taught an intro algebraic topology / knot theory course where the syllabus more or less followed the Gordon notes. Roughly it went like this:

    1) fundamental group, covering spaces
    2) homology, cohomology,
    3) basics of smooth manifolds, transversality, manifolds admit triangulations
    4) poincare duality in smooth manifolds (Poincare’s proof)
    5) Serre’s theorem that cohomology of a space is homotopy-classes of maps into an Eilenberg-Maclane spaces
    6) The proof that knots have Seifert surfaces using the interpretation H^1(M) = [M,S^1] and transversality.
    7) The Alexander polynomial

    IMO something like that could be fattened up into a really nice textbook that would complement books like Hatcher’s Algebraic Topology and Guillemin and Pollack’s Differential Topology well.

    Comment by Ryan Budney — November 12, 2009 @ 10:53 pm | Reply

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