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 be rings with unity, let be a
ring homomorphism, and let be the chain complex
such that is a finitely-generated free module for all . The Cohn localization of is the initial ring with the property
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 is -torsion, where denotes the complement of a knot , and and is the augmentation map, sending to .
Let be a meridian of and let be the infinite cyclic cover of in which (a loop) lifts to (an infinite line). Consider the chain complex of free modules. Then
by Alexander duality or Mayer-Vietoris (the point here is that
is a homology circle generated by ). The following purely algebraic fact is given without proof.
The Cohn localization of is . It is a flat module.
Now by the low-dimensional topologist’s definition of Cohn localization
Since is contractible, we get that is torsion, which is QED.
Simple, elegant, and general!
There is a point I’m uncomfortable with. You see, 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.