rybu's blog

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?

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 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

One of the things I find pretty neat about the splicing operad is how much goes into it. It's a relatively complicated structure and I'm a little surprised there aren't many similar, related structures in the literature.

Shortly after arriving as a postdoc at the University of Rochester I started getting results on the global topology of spaces of knots. This was, in a sense, the first topic I had wanted to work on for my dissertation. But I got stuck, and my Ph.D evolved to a different topic. At Rochester, sparked by Fred Cohen's curiosity, the project regained momentum.

One afternoon I was sitting around with Paolo Salvatore, trying to prove that an operad of little cubes had no hope of acting on spaces of string links. I was certain the operad couldn't act and kept on insisting on it. Eventually Paolo came up with a proof. This post is about Paolo's idea.

My take, written up here.

Towards the end of June the University of Brisbane is hosting a conference on a subject which as the quote above suggests, is something of a barren landscape scattered with unsavory characters.

IPMU (the Institute for the Physics and Mathematics of the Universe) officially opened its new building this week. Ben Burton and I were in Tokyo visiting and I took some photos.