I imagine many readers of this blog are familiar with the fact that you can knot a circle in 3-space, but not in 4-space. If you enjoy thinking about why that is true, please read on!
Think of euclidean 3-space, as a linear subspace of euclidean 4-space, . So if you have a knotted circle in 3-space, you can consider it as an embedded circle in 4-space. And you can unknot it! I think one of the simplest explanations of of this would be the idea to push the knot up into the 4-th dimension every time a strand is close to being an overcrossing (in a planar diagram). At this stage you could in effect change the crossing to be anything you want, after you’re done modifying the crossings, you could push the knot back into 3-space to get a different knot.
A paper by Thomas Fiedler has just appeared on arXiv, describing a new link between geometric and quantum topology of knots. http://arxiv.org/abs/1304.0970
This is big news!! (more…)
A few months ago, I wrote a blog post about the interesting phenomenon that the tunnel number of a connect sum of two knots may be anywhere from one more than the sum of the tunnel numbers to a relatively small fraction of the sum of the tunnel numbers. Since then, a couple of related papers have been posted to the arXiv, so I thought that justifies another post on the subject. The first preprint I’ll discuss, by João Miguel Nogueira , gives new examples of knots in which the tunnel number degenerates by a large amount. The second paper, by Trent Schirmer  (who is currently a postdoc here at OSU), gives a new bound on the amount tunnel number and Heegaard genus can degenerate by under connect sum/torus gluing, respectively, in certain situations.
Someone recently pointed out to me a paper by A. J. Pajitnov  proving a very interesting connection between circular Morse functions and (linear) Morse functions on knot complements. (A similar result is probably true in general three-manifolds as well.) Recall that a (linear) Morse function is a smooth function from a manifold to the line in which there are a finite number of critical points (where the gradient of the function is zero), and each critical point has one of a number of possible forms. For a two-dimensional manifold the possible forms are the familiar local minimum, saddle or local maximum. This post is about three-dimensional Morse functions, in which case the possible forms are slight generalizations of local minima, maxima and saddles. A circular Morse function is a function with the same conditions on critical points, but whose range is the circle rather than the line. For a three-dimensional manifold, the minimal number of critical points in a linear Morse function is twice the Heegaard genus plus two, and for knot complements it’s twice the tunnel number plus two. (In particular, one can construct a Heegaard splitting or unknotting tunnel system directly from a Morse function, but that’s for another post.) The minimal number of critical points in a circular Morse function is called the Morse-Novikov number, and is equal to the minimal number of handles in a circular thin position for the manifold (usually a knot complement). Pajitnov has a very clever argument to show that the (circular) Morse-Novikov number of a knot complement is bounded above by twice its (linear) tunnel number. Below, I want to outline a slightly different formulation of this proof in terms of double sweep-outs, though I should stress that the underlying idea is the same.
Chad Musick made a video in which he untangles a complicated trivial knot due to Ochiai. The procedure is described in his paper Recognising Trivial Links in Polynomial Time. My reaction was “Sweet!”.
I’ve mentioned before that the fall semester program at ICERM for 2013 will focus on computation in low-dimensional topology, geometry, and dynamics. You can now apply to be a long-term visitor for this as a graduate student, postdoc, or other. The deadline for the postdoctoral positions is January 14, 2013; the early deadline for everyone else is December 1, 2012 and the second deadline March 15, 2013.
There will also be three week-long workshops associated with this, so mark your calendars for these exciting events:
- Exotic Geometric Structures. September 15-20, 2013.
- Topology, Geometry, and Group Theory: Informed by Experiment. October 21-25, 2013.
- Geometric Structures in Low-Dimensional Dynamics. November 18-22, 2013.
Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years. I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology. Almost everything we blog about here has the imprint of his amazing mathematics. Bill was always very generous with his ideas, and his presence in the community will be horribly missed. Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:
Marc Culler and I have released version 1.6 of SnapPy. There are two sets of new features:
- Creating links formulaically, e.g. via combining tangles algebraically. See our page of examples.
- Arbitrary precision calculation of certain things (e.g. tetrahedra shapes) and finding associated number fields, a la Snap. Very basic at this point compared to what Snap can to, but here are examples of what we have so far. To use this, you need to install SnapPy in Sage, which should be easy.