SnapPy 2.3 released

Marc Culler and I are pleased to announce version 2.3 of SnapPy. New features include:

• Major improvements to the link and planar diagram component, including link simplification, random links, and better documentation.
• Basic support for spun normal surfaces.
• New extra features when used inside of Sage:
  • HIKMOT-style rigorous verification of hyperbolic structures,
    contributed by Matthias Goerner.
  • Many basic knot/link invariants, contributed by Robert
    Lipschitz and Jennet Dickinson.
  • Sage-specific functions are now more easily accessible as
    methods of Manifold and better documented.
  • Improved number field recognition, thanks to Matthias.
• Better compatibility with OS X Yosemite and Windows 8.1.
• Development changes:
  • Major source code reorganization/cleanup.
  • Source code repository moved to Bitbucket.
  • Python modules now hosted on PyPI, simplifying installation.

All available at the usual place.

Spaces of knots and low-dimensional topology

Over the past 10-12 years, geometric topology has entered a new era.  Most of the foundational problems are solved, and there’s a fairly isolated collection of foundational problems remaining.  In my mind, the two most representative ones would be the smooth 4-dimensional Poincare hypothesis, and getting a better understanding of the homotopy-type of the group of diffeomorphisms of the n-sphere (especially for n=4, but for n large as well).   I want to talk about what I’d call second-order problems in low-dimensional topology, less foundational in nature and more oriented towards other goals, like relating low-dimensional topology to other areas of science.  Specifically, this is an attempt to describe the “spaces of knots” subject in a way that might entice low-dimensional topologists to think about the subject.

The algorithm to recognise the 3-sphere

The purpose of this post is to convince you the 3-sphere recognition algorithm is simple.  Not the proof!  Just the statement of the algorithm itself.  I find in conversations with topologists, it’s fairly rare that people know the broad outline of the algorithm.  That’s a shame, because anything this simple should be understood by everyone.   

Organizing knot concordance

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?