Low Dimensional Topology

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? 

July 12, 2012

Symmetric decompositions of the 4-sphere

Filed under: 3-manifolds,4-manifolds,Knot theory — Ryan Budney @ 4:43 pm

Rob Kusner recently pointed out to me that the 4-sphere has a very natural differential-geometric decomposition as a double mapping cylinder S^3/Q_8 \to \mathbb RP^2. Here Q_8 is the group \{\pm 1, \pm i, \pm j, \pm k\} in the unit quaternions and \mathbb RP^2 is the real projective plane. Another way to say this is take the Voronese projective plane in S^4, a regular neighbourhood of it is a mapping cylinder S^3/Q_8 \to \mathbb RP^2. Moreover, the *complement* of that regular neighbourhood is another such mapping cylinder.


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