Low Dimensional Topology

December 19, 2014

Concordance Champion Tim Cochran 1955-2014.

Filed under: 4-manifolds,knot concordance,Knot theory,Misc. — dmoskovich @ 8:08 am

Yesterday I received the shocking news of the passing of Tim Cochran (1955-2014), a leader in the field of knot and link concordance. The Rice University obituary is here.

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 L^2 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. (more…)

May 16, 2013

Organizing knot concordance

Filed under: 3-manifolds,4-manifolds,knot concordance — Ryan Budney @ 10:10 am
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I have a rather naive question for the participants here.  I’m at the Max Planck 4-manifolds semester, currently sitting through many talks about knot concordance and various filtrations of the knot concordance group.   

Do any of you have a feeling for how knot concordance should be organized, say if one was looking for some global structure?    In the purely 3-dimensional world there are many very “tidy” ways to organize knots and links.  There’s the associated 3-manifold, geometrization.  There’s double branched covers and equivariant geometrization, arborescent knots and tangle decompositions.  I find these perspectives to be rather rich in insights and frequently they’re computable for reasonable-sized objects.  

But knot concordance as a field feels much more like the Vassiliev invariant perspective on knots: graded vector spaces of invariants.  Typically these vector spaces are very large and it’s difficult to compute anything beyond the simplest objects. 

My initial inclination is that if one is looking for elegant structure in knot concordance, perhaps it would be at the level of concordance categories.  But what kind of structure would you be looking for on these objects?   I don’t think I’ve seen much in the way of natural operations on slice discs or concordances in general, beyond Morse-theoretic cutting and pasting.   Have you? 

June 22, 2012

Knots as symmetric Poincaré triads

Filed under: knot concordance,Knot theory,noncommutative localization — dmoskovich @ 4:08 am

Life is easy. If you find it to be hard, it means you’re looking at it in the wrong way.– Tomer, Kathmandu (Israeli TV series).

Can the same be said of mathematics? In mathematics, it’s a common occurance to run up against a problem which one finds to be difficult. But then it’s often the case that a logically equivalent problem in a different model is much easier.

In Low-Dimensional Topology, particularly in Knot Theory, we have many competing models for our objects, and we can ask logically equivalent questions in each one. I don’t know if any one of these categories is inherently complexity-theoretically easier than any other- as far as I know it may all depend on the specific context. But I’ve recently been reading Mark Powell’s thesis A second order algebraic knot concordance group, in which he puts forward a new model for knots which seems well-suited to a certain sort of problem which I care about, and which I am excited about. (more…)

May 3, 2010

What is L-theory and why should I care: Part I

Filed under: Algebraic topology,knot concordance — dmoskovich @ 10:49 am

In the middle of February, Mark Powell was in Kyoto, where he taught me about algebraic surgery and other interesting topics. These are tools which were developed by high-dimensional algebraic topologists to tackle the kind of problems which they are interested in, and moreover they have a rather fierce reputation. So why should the low-dimensional topologist care about L-theory and suchlike? In this short series of posts, I’ll summarize the basics of what Mark taught me in February, and I’ll tell you why I care about L-theory and why I think you should care about it too (for those who can’t wait: because of Cochran-Orr-Teichner and because it provides a natural language to think about Blanchfield pairings).

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