Computation in geometric topology

Complete lecture videos for last week’s workshop Computation in Geometric Topology at Warwick are now posted on YouTube. The complete list of talks with abstracts and video links is here.

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A gorgeous but incomplete proof of “The Smale Conjecture”

In 1959 Stephen Smale gave a proof that the group of diffeomorphisms of the 2-sphere has the homotopy-type of the subgroup of linear diffeomorphisms, i.e. the Lie Group O_3.  His proof went in two steps:

SnapPy 2.4 released

A new version of SnapPy, a program for studying the topology and geometry of 3-manifolds, is available.  Added features include a census of Platonic manifolds, rigorous computation of cusp translations, and substantial improvements to its link diagram component.

Simple loop conjecture for Sol manifolds

Drew Zemke, who is a grad student of Jason Manning, posted a proof of the Simple Loop Conjecture for 3-manifolds modeled on Sol last week.

The Simple Loop Conjecture fits into that family of statements such as Dehn’s Lemma and the Sphere Theorem which translate statements about fundamental groups into statements about 3-manifolds. Such theorems allow us to trade 3-manifolds for their fundamental groups (which are much simpler mathematical objects).

MOO is classical

The simplest quantum 3-manifold invariant is the Murakami-Ohtsuki-Okada (MOO) invariant. It comes from Chern-Simons theory in the way that the Reshetikhin-Turaev invariant comes from Chern-Simons Theory. It has a closed formula in terms of the order of the first cohomology class of the -manifold and an eighth root of unity. Witten’s Chern-Simons theory for gauge group 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.

SnapPy 2.3 released

Marc Culler and I are pleased to announce version 2.3 of SnapPy. New features include:

• Major improvements to the link and planar diagram component, including link simplification, random links, and better documentation.
• Basic support for spun normal surfaces.
• New extra features when used inside of Sage:
  • HIKMOT-style rigorous verification of hyperbolic structures,
    contributed by Matthias Goerner.
  • Many basic knot/link invariants, contributed by Robert
    Lipschitz and Jennet Dickinson.
  • Sage-specific functions are now more easily accessible as
    methods of Manifold and better documented.
  • Improved number field recognition, thanks to Matthias.
• Better compatibility with OS X Yosemite and Windows 8.1.
• Development changes:
  • Major source code reorganization/cleanup.
  • Source code repository moved to Bitbucket.
  • Python modules now hosted on PyPI, simplifying installation.

All available at the usual place.

Complex hyperbolic geometry of knot complements

This morning there was a paper which caught my eye:

Deraux, M. & Falbel, E. 2015 Complex hyperbolic geometry of the figure-eight knot.
Geometry & Topology 19, 237–293.

In it, the authors study a very different geometric structure for the figure-eight knot complement, as the manifold at infinity of a complex hyperbolic orbifold.

Understanding the anomaly

I’ve recently been looking at the following paper in which -TQFT anomalies are treated carefully and various old constructions of Turaev and Walker are elucidated:

Gilmer, P.M. and Masbaum, G., Maslov Index, Mapping Class Groups, and TQFT, Forum Math. 25 (2013), 1067-1106.

It makes me think a lot about just what the anomaly `actually means’…

A celebration of diagrammatic algebra

Relaxing from my forays into information and computation, I’ve recently been glancing through my mathematical sibling Kenta Okazaki’s thesis, published as:

K. Okazaki, The state sum invariant of 3–manifolds constructed from the linear skein.
Algebraic & Geometric Topology 13 (2013) 3469–3536.

It’s a wonderful piece of diagrammatic algebra, and I’d like to tell you a bit about it!

SnapPy 2.1: Now with extra precision!

Marc Culler and I released SnapPy 2.1 today. The main new feature is the ManifoldHP variant of Manifold which does all floating-point calculations in quad-double precision, which has four times as many significant digits as the ordinary double precision numbers used by Manifold. More precisely, numbers used in ManifoldHP have 212 bits for the mantissa/significand (roughly 63 decimal digits) versus 53 bits with Manifold.