Low Dimensional Topology

April 6, 2013

New connection between geometric and quantum realms

Filed under: Hyperbolic geometry,Knot theory,Quantum topology — dmoskovich @ 9:41 am

A paper by Thomas Fiedler has just appeared on arXiv, describing a new link between geometric and quantum topology of knots. http://arxiv.org/abs/1304.0970

This is big news!! (more…)

June 10, 2012

Colin Day, in memoriam

Filed under: Knot theory,Quantum topology — dmoskovich @ 7:40 am

On 16 February 2012, Colin Day passed away after a year long bout with stomach cancer. In his memory, I would like to discuss his thesis

Day, Colin
A topological construction of Vassiliev style invariants of links.
Thesis (Ph.D.)–The University of North Carolina at Chapel Hill. 1993. 99 pp.
Available from Jim Stasheff’s homepage (new and improved scan; scroll down to Colin).

In his thesis, Colin Day extends Vassiliev’s construction of finite-type invariants from knots to links, and elucidates Vassiliev’s construction in the process. I don’t think that Vassiliev’s construction ever really got enough attention, and Colin’s thesis is surely the best entry-point to it, and deserves to be widely read.

March 5, 2012

Rethinking the basics

Filed under: Misc.,Quantum topology,Triangulations — dmoskovich @ 3:36 am

Some nights, one gazes up at the stars, and thinks about philosophy. Who are we? What is the meaning of life? What is reality? What are manifolds really?

This morning, I looked at Poincaré’s original definition in Papers on Topology: Analysis Situs and Its Five Supplements, translated by John Stillwell. His original definition was pretty-much that a manifold is a quotient of \mathbb{R}^n by a properly discontinuous group action, that group being his original fundamental group. Implicitly, his smooth, PL, and topological categories were all the same thing (indeed true for 1-manifolds, and for dimensions 2 and 3 PL and smooth categories still “coincide” in a sense that can be made fully precise); nowadays we understand that the situation is more subtle. But I’m still not sure that I understand what a manifold is- what it really is.

In some non-mathematical, philosophical (theological?) sense, I believe that both smooth and PL manifolds actually exist, in the sense that natural numbers exist, and tangles exist. Our clumsy formal definitions are attempts at describing something that is actually out there, as the Peano axioms describe the natural numbers. I also believe that Physics is a guide to Mathematics, because things that really exist might also be observed… so ideas from Physics (topological invariants defined by means of path integrals) ought to be taken very seriously, and it is my irrational belief that these will eventually turn out to be the most fundamental invariants in some precise mathematical sense.

It is fascinating to me, then, that input from physics seems to be leading towards a fundamental rethink of the basic definitions of smooth and PL manifolds. I feel like we had some sub-optimal definitions, which we worked with for sociological reasons (definitions are made by people, and people are not perfect), and maybe in the not too distant future there will be a chance to put more convenient definitions in place. Maybe the real world (physics) will force it on us. Let me tell you, then, about some of the papers I’ve been (casually) flicking through recently (the one I’m most excited about is Kirillov’s On piecewise linear cell decompositions). (more…)

January 15, 2012

Beyond the trivial connection

Filed under: 3-manifolds,4-manifolds,Quantum topology — dmoskovich @ 10:21 pm

One of the foundational papers in Quantum Topology, and one of the main reasons that the subject is called Quantum Topology, is Edward Witten’s landmark paper Quantum field theory and the Jones polynomial. One of the things Witten did in that paper was to define a 3–manifold invariant as a partition function with action functional proportional to the Chern-Simons 3–form. A partition function is a path integral, so Witten’s invariant is a physical construction rather than a mathematical one. Quantum topology of 3–manifolds is, to a large extent, the field whose goal is to mathematically reconstruct, and to understand, Witten’s invariant. Meanwhile, for 4–manifolds with a metric, Witten defined a 4–manifold invariant as a partition function in another landmark paper Topological quantum field theory.

I should warn you that I don’t know any physics so some (all?) of what I say below might be rubbish. Still, pressing boldly ahead…

Up until recently, mathematicians only understood tiny corners of Witten’s invariants, or, more broadly, of invariants (topological or otherwise) of manifolds (with or without extra structure) which come from quantum field theory partition functions. But I’ve recently glanced through two papers which seem to finally be going further, seeing more. The tiny corners we have seen already give rise to mathematical invariants of preternatural power (surely that’s the best word to describe it!), such as Ohtsuki series of rational homology 3–spheres (\mathbb{Q}HS), Donaldson invariants, and Seiberg–Witten invariants.

November 11, 2011

Who cares about the Volume Conjecture?

Filed under: Hyperbolic geometry,Knot theory,Quantum topology — dmoskovich @ 9:49 am

Yesterday, I attended a very interesting informal talk by Roland van der Veen in which, among other things, he told me a little bit about why he cares about the Volume Conjecture. The Volume Conjecture is considered somehow to be the `big open problem’ in quantum topology. I had never understood why though (I had even asked an MO question but hadn’t really been convinced by any of the very good answers). Why, after all, should people care about any mathematical conjecture? (more…)

June 26, 2011

Hopf algebra structure on 3-manifolds!?

Filed under: 3-manifolds,Quantum topology — dmoskovich @ 10:29 am

The main question in quantum topology has always been a curious one- “What do quantum invariants mean topologically?”
For 3-manifolds, part of the mystery has been that quantum invariants, such as the Reshetikhin-Turaev and Turaev-Viro invariants, are constructed using the representation theory of Hopf algebras. But wait a minute- 3-manifolds don’t admit a natural Hopf algebra structure, do they? What’s going on? Why should Hopf algebras have anything at all to say about 3-manifold topology? Why should quantum invariants exist?

January 20, 2011

Newsflash: Witten’s new preprint

Filed under: Knot theory,Quantum topology — dmoskovich @ 12:03 pm

A couple of days ago, Edward Witten uploaded a preprint titled Fivebranes and Knots. Based on Witten’s record on such topics, and on a preliminary visual scan of the introduction, it would not be unreasonable to surmise that this preprint could change history. Khovanov homology will never look the same again.

October 14, 2010

A problem with LMO?

Filed under: 3-manifolds,Quantum topology — dmoskovich @ 9:35 pm

Renaud Gauthier from KSU posted this preprint on ArXiv a few days ago, in which he claims to have found a serious problem with the construction of the LMO invariant, the universal finite-type invariant for rational homology spheres (it’s defined for all 3-manifolds, but I think of it as an invariant of homology spheres). What a headline that makes! A possible hole in the definition of the LMO invariant, with the potential to wash large swaths of quantum topology of 3-manifolds down the drain! Indeed, this is the topic of his second preprint.
Tomotada Ohtsuki was my PhD advisor, and he’s a careful mathematician with tremendous technical ability, who checks his answers against computational data to make absolutely sure no errors creep into his work. Le and Murakami are similar. Gauthier’s claim is that they made a fatal error calculating the effect of the second Kirby move on the framed Kontsevich invariant, which is used to construct the LMO.
Without having read Gauthier’s preprint (which is 81 pages long), my bias is to be skeptical of his claim. Maybe he found a typo, but surely not more. But what a headline it would make if a substantial error was there! This is math drama in the making.
I think it might be fun and educational to crowdsource peer review Gauthier’s claim. It’s important if it’s right, and if it’s wrong, at least it’s an opportunity to get into the kishkes of the LMO.

UPDATE: A MathOverflow question about this.

UPDATE: A third paper by Gauthier was uploaded on Thursday.

UPDATE: Gwenael Massuyeau shows a major flaw in Gauthier’s arguments in the comments. Another two large flaws are noted by Dylan Thurston. Due to these problems, Gauthier’s claim of an error in LMO does not appear to hold up.

September 26, 2010

What is the right context for the Alexander polynomial?

Filed under: Knot theory,Quantum topology — dmoskovich @ 1:45 pm

I’m now a postdoc at the University of Toronto, and I have a new baby daughter (named Chana or Ann), hence I haven’t posted for a while.
A common theme in topology, and in mathematics in general I suppose, is that it is important to work out the natural setting of a problem. This would be the setting in which the problem is stated most naturally, and in which it’s solution makes most sense conceptually.

For instance, in knot theory, does your problem make more sense for knots or for links? (Example: skein theoretic invariants) Is there a reason that you are restricting to knotted S^1‘s in S^3, or would the problem make more sense for knotted S^n‘s in S^{n+2} (Example: Seifert surfaces), or perhaps for more general knotted objects in more general manifolds (Example: fundamental group of the complement)? Do you need the manifold at all? Maybe your problem is one which involves group presentations, and is most naturally stated as a combinatorial group theory problem regarding groups with presentations of deficiency 1? Maybe your problem is homological, and makes sense for any homology manifold? (Example: homological invariants) Maybe it’s a problem which makes most conceptual sense in the context of symmetric chain complexes and L-theory? (Example: linking forms)
Sometimes the search for the right setting for a problem might also involve restricting the class of objects you might want to consider. And sometimes it might involve changing it completely.
In this post, I’d like to discuss the natural setting for the Alexander polynomial. (more…)

September 2, 2010

Combinatorial Heegaard Floer Homology, Part II

Filed under: 3-manifolds,Heegaard splittings,Quantum topology — Jesse Johnson @ 12:39 pm

In my previous post about combinatorial Heegaard floer homology, I described the basic structure for how we will define the Homology groups \widehat{HF}.  We need an Abelian group C with a relative grading and a boundary map \partial that sends each graded subgroup into the subgroup one index lower, and whose square is zero.  The homology group is then defined as the kernel of \partial quotiented by the image of \partial. Last time I described the group C: Given a Heegaard diagram for a given 3-manifold, consisting of loops \{\alpha_i\}, \{\beta_j\}, then C is a direct product of copies of \mathbf{Z}_2, with one copy for each pairing of the curves.  (A pairing is a collection of points such that each \alpha_i contains exactly one of the points and each \beta_j contains exactly one point.)  In this post, I will describe the grading on this group and the boundary map \partial. (more…)

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