rybu's blog

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

Is there a rings operad?

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.

Homology of spaces of knots, known unknowns part 2: the coming-together of loose ends

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.

Group completion of string links - or a cute idea of Salvatore's

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.

Why can you turn a shirt inside-out?

My take, written up here.

The wild frontier, in Brisbane

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.

the new IPMU

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.

Some news from Stoimenow

Back when I was a grad student, the Lawrence-Krammer representations of the braid groups were a relatively new thing.

Some roads are longer than others

The creative process for papers has always been murky to me. I rarely set out to write a particular type of paper. Quite often results appear as by-products of computations I'm working on for some other reason.


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