Low Dimensional Topology

March 30, 2015

MOO is classical

Filed under: 3-manifolds,Dehn surgery,Quantum topology — dmoskovich @ 9:43 am

The simplest quantum 3-manifold invariant is the Murakami-Ohtsuki-Okada (MOO) invariant. It comes from \mathrm{U}(1) Chern-Simons theory in the way that the \mathrm{SU}(2) Reshetikhin-Turaev invariant comes from \mathrm{SU}(2) Chern-Simons Theory. It has a closed formula in terms of the order of the first cohomology class of the 3-manifold M and an eighth root of unity. Witten’s Chern-Simons theory for gauge group \mathrm{U}(1) shows that the MOO invariant can be reformulated in terms of classical Riemann theta functions with characteristic, but the relationship is by way of quantum field theory.

A recently published paper by Gelca and Uribe, which is also the topic of a book by Gelca and some nice slides, constructs the MOO invariant from theta functions completely classically essentially without using anything quantum at all (although the representation theory behind it was originally developed for quantum mechanical purposes). Thus, like the Alexander polynomial and the linking number, MOO is seen to be quantum but also classical.

There is also a more analytic, heat-equation-based way of seeing the same thing due to Andersen, but I haven’t read Andersen’s paper and therefore I can’t say anything about that. (more…)

November 26, 2013

What’s Next? A conference in question form

Mark your calendars now: in June 2014, Cornell University will host “What’s Next? The mathematical legacy of Bill Thurston”.  It looks like it will be a very exciting event, see the (lightly edited) announcement from the organizers below the fold.

Conference banner

September 30, 2013

SnapPy 2.0 released

Marc Culler and I pleased to announce version 2.0 of SnapPy, a program for studying the topology and geometry of 3-manifolds. Many of the new features are graphical in nature, so we made a new tutorial video to show them off. Highlights include

September 8, 2012

ICERM Fall 2013: Topology, geometry, and dynamics

I’ve mentioned before that the fall semester program at ICERM for 2013 will focus on computation in low-dimensional topology, geometry, and dynamics.   You can now apply to be a long-term visitor for this as a graduate student, postdoc, or other.   The deadline for the postdoctoral positions is January 14, 2013; the early deadline for everyone else is December 1, 2012 and the second deadline March 15, 2013.

There will also be three week-long workshops associated with this, so mark your calendars for these exciting events:

  1. Exotic Geometric Structures. September 15-20, 2013.
  2. Topology, Geometry, and Group Theory: Informed by Experiment. October 21-25, 2013.
  3. Geometric Structures in Low-Dimensional Dynamics. November 18-22, 2013.

August 22, 2012

Bill Thurston is dead at age 65.

Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years.   I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology.  Almost everything we blog about here has the imprint of his amazing mathematics.    Bill was always very generous with his ideas, and his presence in the community will be horribly missed.    Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:

April 13, 2012

The virtual Haken conjecture

Agol’s preprint, which includes a long appendix joint with Groves and Manning, is now on the arXiv.

October 5, 2011

Dehn filling and genus dropping

Filed under: 3-manifolds,Dehn surgery,Heegaard splittings,Knot theory — Jesse Johnson @ 11:00 am

A common problem in low-dimensional topology is to ask how the topology and geometry of a manifold changes if you glue a solid torus into one of its torus boundary components (also known as Dehn filling) or more generally, if you glue a handlebody into a higher genus boundary component.  One topological version of this problem is to ask how the isotopy classes of Heegaard surfaces change. Every Heegaard surface  for the unfilled manifold becomes a Heegaard surface for the filled manifold, but there may be other properly embedded non-Heegaard surfaces that also become Heegaard surfaces if you cap them off after the filling. In particular these new Heegaard surfaces may have lower genus, so the Heegaard genus of the manifold could drop after filling. The quintessential example of this is a knot complement in the 3-sphere: There are knot complements with arbitrarily high Heegaard genus, but if you Dehn fill to produce the 3-sphere, then the genus drops to zero.

Of course, for such a manifold there is exactly one filling that produces the 3-sphere and one can ask how much the genus can drop for the other fillings. There are examples where Heegaard genus drops by one for a line of slopes, and the resulting Heegaard surfaces are often called horizontal.  However, Moriah-Rubinstein [1] (and later Rieck-Sedgwick [2]) showed that there are only finitely many slopes for which the genus can drop by more than one (and only finitely many lines of slopes where it drops by one.) As far as I know there are no examples where there are two slopes for which the genus drops by more than one. So one can ask:

Question: Is there a 3-manifold M with Heegaard genus g, a torus boundary component T and two slopes on T such that Dehn filling along each slope produces a 3-manifold with Heegaard genus less than or equal to g - 2?


September 6, 2009

Some thoughts on the Kirby Theorem

Filed under: 3-manifolds,Dehn surgery,Mapping class groups,Quantum topology — dmoskovich @ 2:44 am

I don’t think it would be too controversial to assert that the Kirby Theorem is an important theorem in low dimensional topology. Given a 3-manifold M and an framed link L\subset M (by “framed” let’s mean “integer framed”), let M_L denote the 3-manifold obtained from M by surgery around L. The Kirby Theorem states that, given two framed links L_1,L_2\subset M, the 3-manifolds M_{L_1} and M_{L_2} are homeomorphic if and only if L_1 and L_2 are related by a finite sequence of the following local moves:

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