Comments for Low Dimensional Topology

Comment on Alexander polynomials and noncommutative localization by Ryan Budney

Ryan Budney — Fri, 13 Nov 2009 03:53:54 +0000

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 on Alexander polynomials and noncommutative localization by Daniel Moskovich

Daniel Moskovich — Fri, 13 Nov 2009 03:30:38 +0000

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 on Alexander polynomials and noncommutative localization by Ryan Budney

Ryan Budney — Fri, 13 Nov 2009 02:52:52 +0000

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 on Alexander polynomials and noncommutative localization by Ryan Budney

Ryan Budney — Fri, 13 Nov 2009 02:39:23 +0000

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 on Alexander polynomials and noncommutative localization by Ryan Budney

Ryan Budney — Fri, 13 Nov 2009 02:03:27 +0000

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 on Alexander polynomials and noncommutative localization by Ryan Budney

Ryan Budney — Fri, 13 Nov 2009 02:02:17 +0000

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 on Alexander polynomials and noncommutative localization by Daniel Moskovich

Daniel Moskovich — Thu, 12 Nov 2009 23:57:29 +0000

It’s saying a bit more than that… it’s saying it’s a module over Laurent polynomials which augment to . 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 on Alexander polynomials and noncommutative localization by Ryan Budney

Ryan Budney — Thu, 12 Nov 2009 23:34:25 +0000

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?

-ryan

Comment on Which knotted objects are worthy of study? by Daniel Moskovich

Daniel Moskovich — Tue, 10 Nov 2009 04:28:49 +0000

That’s correct… more than that, the stable homeomorphism class only depends on the germ of the ambient surface near the curve. Therefore, what happens outside a regular neighbourhood does not matter, and the surface itself is not really a part of the structure.

Comment on Which knotted objects are worthy of study? by JungHoon Lee

JungHoon Lee — Mon, 02 Nov 2009 07:12:36 +0000

I also think knotted graphs are interesting. (… because I have been thinking about it recently.) Someone nearby me talked about tunnel number one, theta curve. At first, it was an unfamiliar terminology.
It can be regarded as part of Heegaard splitting theory with just genus increased by one. But, there was an exmaple called Kinoshita’s theta curve that attracted my attention.

As I think, whether a knotted object is natural or not depends on individual taste to some extent.