A groundbreaking paper which made a deep impression on a lot of people, including me, was Cochran-Orr-Teichner’s Knot concordance, Whitney towers and signatures. This paper revealed an unexpected geometric filtration of the topological knot concordance group, which formed the basis for much of Tim Cochran’s subsequent work with collaborators, and the work of many other people.
In this post, in memory of Tim, I will say a few words about roughly what all of this is about.
It would be very nice to be able to equip the space of knots with a good algebraic structure. Somehow, the natural binary operations on knots seem to be the satelite operations, of which the connect sum may be considered a special (degenerate) case.
Unfortunately, unless is the trivial knot, there is no knot which can be `satelited’ to to obtain a trivial knot. Thus, the set of knots under connect sum, or indeed under any satelite operations, does not form a group.
But the set of knots can be quotiented by an equivalence relation called concordance, and concordance classes of knots do form a group under the connect sum. Concordance takes place in an ambient 4-dimensional space, and so it provides an avenue for knot theory to be used to study 4-dimensional topology. For most of the world, this is the ultimate motivation to study link concordance. This point of view is beautifully laid out in Freedman-Quinn’s Topology of 4-Manifolds.
The way the field has gone, every conjecture about how `good’ the structure of the link concordance group is has turned out to be wrong. Almost every paper which has come out about knot concordance in the last 20 years, as far as I know, has been a negative result. Cochran’s work has been instrumental in showing `how bad things are’. The group isn’t trivial, and its non-triviality is detected by the Casson-Gordon invariants. The next step was taken in Cochran-Orr-Teichner; Casson-Gordon invariants do not detect knots up to concordance. They’re just the first step in an infinite geometric filtration of invariants, which is non-trivial at every step.
Stavros Garoufalidis suggested a long time ago that the Cochran-Orr-Teichner filtration should be investigated through the lens of quantum topology. This was a major research interest of mine at one point, and to the best of my knowledge, nobody has yet achieved this aim. I remain convinced that this is an interesting avenue of research worthy of future investigation.
A recent paper of Tim Cochran which captured by imagination was his joint work with his mathematical daughter Shelly Harvey on The Geometry of the Knot Concordance Space. In it, Cochran and Harvey suggest viewing the topological knot concordance space in a metric space in various different ways, and suggest investigating its coarse geometry. Again, the structure isn’t neat- it isn’t quasi-isometric to a finite product of hyperbolic spaces- but it is possible to address the question of whether it is what the authors call a `fractal space’, that is roughly a space which admits a natural system of self-similarities. The conjecture that the knot concordance space is a fractal space looks intuitively highly plausible to me; and the investigation of the coarse geometry of the knot concordance space looks to me like a marvelous research project which will surely lead to many fruitful results in the future, both positive and negative.
And apart from all of his fantastic and groundbreaking ideas, Tim was an inspiring teacher, lecturer, and colleague: a true powerhouse of good mathematics.
RIP, Tim Cochran.