An old corker on the unknotting of knots

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. 

New connection between geometric and quantum realms

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!!

Update on subadditivity of tunnel number

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 [1], gives new examples of knots in which the tunnel number degenerates by a large amount. The second paper, by Trent Schirmer [2] (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.

The Bridge Spectrum

A knot in a three-manifold is said to be in bridge position with respect to a Heegaard surface if the intersection of with each of the two handlebody components of the complement of is a collection of boundary parallel arcs, or if is contained in . The bridge number of a knot in bridge position is the number of arcs in each intersection (or zero if if is contained in ) and the genus  bridge number of  is the minimum bridge number of over all bridge positions relative to genus Heegaard surfaces for . The classical notion of bridge number is the genus-zero bridge number, i.e. bridge number with respect to a sphere in , but a number of very interesting results in the last few years have examined the higher genus bridge numbers. Yo’av Rieck defined the bridge spectrum of a knot as the sequence where is the genus bridge number of and asked the question: What sequences can appear as the bridge spectrum of a knot? (At least, I first heard this term from Yo’av at the AMS section meeting in Iowa City in 2011 – as far as I know, he was the first to formulate the question like this.)

Morse-Novikov number and tunnel number

Someone recently pointed out to me a paper by A. J. Pajitnov [1] 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.

Untangling a knot

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!”.

Subadditivity of complexities under gluing

A recent talk in our topology seminar by Trent Schirmer (who just joined OSU as a postdoc this year) got me thinking about three closely related (almost equivalent) problems in three-dimensional topology. Trent spoke about the following problem in knot theory: Given two knots in with tunnel numbers and , what would you expect the tunnel number of their connect sum to be? Recall that the tunnel number of a knot is the Heegaard genus of the knot complement minus one. With a little work, one can show that the tunnel number of the connect sum is at most . However, there are also examples where it is much lower and Trent has constructed links where the connect sum has tunnel number around  [1]. This is fairly interesting on its own, but it turns out there are (at least) two other situations with similar phenomenon that appear to have the same underlying reasons.

ICERM Fall 2013: Topology, geometry, and dynamics

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:

1. Exotic Geometric Structures. September 15-20, 2013.
2. Topology, Geometry, and Group Theory: Informed by Experiment. October 21-25, 2013.
3. Geometric Structures in Low-Dimensional Dynamics. November 18-22, 2013.

Bill Thurston is dead at age 65.

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:

• Wikipedia.
• 2010 lecture on The mystery of 3-manifolds.
• On proof and progress in mathematics.

SnapPy 1.6: Now with more links and precision!

Marc Culler and I have released version 1.6 of SnapPy.  There are two sets of new features:

1. Creating links formulaically, e.g. via combining tangles algebraically.  See our page of examples.
2. 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.