The Alexander Polynomial as a Quantum Invariant: Part 2

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.

3-manifold groups are known, right?

During the nice talk that Ian Biringer gave on the structure of hyperbolic 3-manifolds at Caltech on Friday, a 4-manifold theorist in the back was heard to ask ‘3-manifold groups are known, right?’

I know what he meant. Any finitely presented group can arise as the fundamental group of a 4-manifold (this is a nice exercise that you can do for yourself, involving surgery on a connect sum of copies of S³ x S¹). With a little care, you can deduce that classifying topological 4-manifolds is at least as impossible as classifying finitely presented groups (which is impossible).

In contrast, there are constraints on the fundamental groups of closed 3-manifolds. One of the first is an easy consequence of the existence of Heegaard splittings: any closed 3-manifold group admits a balanced presentation, meaning that there are no more relations than generators.

What does it mean to ‘know’ a class of finitely presentable groups? Do we really ‘know’ the class of (orientable) 3-manifold groups? The point of view that I will adopt in addressing these questions is algorithmic, and comes from combinatorial group theory.

Postdocs in the UK

Saul Schleimer has asked me to advertise the the Leverhulme Early Career Fellowships, which are essentially postdocs that can be taken to any University in the UK in any discipline.  I think Saul is hoping that some promising young topologists apply to take one of these to Warwick.  The deadline is March 11 (4PM GMT).  However, Daniel pointed out in the comments that to be eligible one must have a degree from a UK University, or currently be a postdoc in the UK.  That probably narrows down the applicant pool quite a bit.  If any readers would like to advertise other job oportunities that aren’t on the usual job listing sites, feel free to mention them in the comments.

Dani Wise’s Lectures at Park City

Dani Wise gave a series of lectures at the Wasatch Topology Conference back in December about his very exciting work on quasi-convex hierarchies and residual properties of groups (like subgroup separability). Thomas Koberda took detailed notes which he polished into a useful 9 page summary of Dani’s work. (I’d been thinking I should blog about Dani’s talks, which I also attended, and I’m very glad that Thomas saved me the trouble…)

The Alexander Polynomial as a Quantum Invariant: Part 1

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.

Spring and Summer Conferences

I haven’t been updating the conference list on the blog lately, and in fact the link to it disappeared for a few months when we switched to the new look.  But I wanted to point out that the list is back now, I just updated it, and there are some good conferences coming up in the next few months.  If you know of any topology conferences that I missed (whether or not you’re an organizer), please let me know in the comments on that page and Ill add them as soon as I can.

Google wave and LaTeX

I’m not sure if any of the other math blogs have pointed this out yet (I haven’t been very good about writing OR reading blogs the last few months) but Google Wave is now up and running.  Terry Tao discussed the potential use of wave for math collaborations back when it was first announced.   While wave is only available by invitation at the moment, there are a number of mathematicians who are now using it and if you ask around you can probably find someone with an extra invite to send you.

Mystery lecture notes

Number theorist E. Kowalski found an old set of typewritten lecture notes on 3-manifold topology in a bin at Penn. There’s no author indicated, but hopefully someone here will recognize them. The first page is shown below.

Garoufalidis’s Jones Slope conjecture

Last week, Stavros Garoufalidis posted [1] on ArXiv, which got me daydreaming:
The coloured Jones function is an invariant of a knot consisting of a sequence of elements in indexed by . The boundary slope of an incompressible surface is where and are the number of times the surface winds around the meridian and longitude of correspondingly. The conjecture is that the limit points of are boundary slopes of incompressible surfaces in . One doesn’t need to know much quantum topology to find this intriguing… why should boundary slopes of incompressible surfaces be visible in such a straightforward way to the coloured Jones function?

Alexander polynomials and noncommutative localization

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