Blogs

The view from inside a mirrored tetrahedron.

What would it look like being inside a regular tetrahedron, but with all the walls made from mirrors?   Glad you asked.

Here is a little movie of the above scene:

Canada has reached a little threshold for topologist-density recently.  Every province except PEI has a topologist in either a permanent position or a tenure-track job. As far as I know this is unprecedented.

BC: Rolfsen, Adem, Pettet, Ben Williams, myself.

Alberta: Bauer, Zvengrowski, Peschke.

Manitoba: Adam Clay. (this is recent news to me!)

Some videos of me

Two videos of me. The first has plenty of editing to cut out my prolific usage of "uh". It's a general description of topology, intended for inquisitive non-experts.

Milnor's interchange symmetry argument

I'm teaching a differential geometry course this semester and having fun with it. This is kind of over-the-top use of computer graphics, but...

A small compilation of topology videos

Recently a bunch of nice classic topology videos have been appearing in various locations. I thought I'd put together a little list of some of them.

Most common small 3-manifolds in the 4-sphere

I've been enumerating vertex-normal 3-manifolds in triangulated homotopy 4-spheres recently. I thought I'd put a list of the most commonly found 3-manifolds somewhere. This seems like a good a place as any.

New 3-manifolds in the 4-sphere

I've been kicking around a project, trying to discover which 3-manifolds embed in the 4-sphere for five years now. This preprint represents the state of the art. V5 should appear on Thursday, Sept. 27th. There's two big changes in the preprint from V4.

A classical conjecture in knot theory says that when you have a connect-sum of knots, you obtain a minimal-crossing planar diagram for it by taking the minimal-crossing planar diagrams of the prime summands, and take the connect-sum of their diagrams. Said another way "crossing number of knots is additive". That got me to wondering, are there similar possibilities for other operations on knots and links, related to crossing number?

An analogue of the figure-8 knot, in four dimensions

The figure-8 knot is a lovely example of a knot in 3-dimensions. It's one of the simplest knots you can draw. The complement admits a hyperbolic structure of finite volume. And the hyperbolic structure is very nice -- it is the union of two regular ideal tetrahedra, moreover, the n-th face of tetrahedron 1 is glued to the n-th face of tetrahedron 2, with two of the gluing maps by right-handed 2π/3 twists, and two by left-handed 2π/3 twists. The complement of the figure-8 knot (thought of as a knot in the 3-sphere) fibers over the circle, and the fibre is a once-punctured torus.

The most commonly asked question after a presentation

The most frequently asked question I get after a presentation in seminars is "how did you draw those pictures?"

For example, the main image in this slide