A few days ago, my co-blogger Nathan Dunfield posted a counterexample to the Strong Neuwirth Conjecture.
N. Dunfield, A knot without a nonorientable essential spanning surface, arXiv:1509.06653
The Neuwirth Conjecture, posed by Neuwirth in 1963, asks roughly whether all knots can be embedded in surfaces in a way analogous to how a torus knot can be embedded in an unknotted torus. A weaker version, the “Weak Neuwirth Conjecture”, asks whether the knot group of any non-trivial knot in the 3-sphere can be presented as a product of free groups amalgameted along some subgroup. This was proven by Culler and Shalen in 1984. But nothing is proven about the ranks of these groups. The Neuwirth Conjecture would give the ranks as the genus of the surface. Thus, the Neuwirth Conjecture is an important conjecture for the structure theory of knot groups.
The Neuwirth Conjecture has been proven for many classes of knots, all via basically the same construction using a nonorientable essential spanning surface. The “Strong Neuwirth Conjecture” of Ozawa and Rubinstein asserts that this construction is always applicable because such a surface always exists.
Dunfield’s counterexample, verified by Snappea, indicates that we will need a different technique to prove the Neuwirth conjecture. Neuwirth’s Conjecture has just become even more alluring and interesting!
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:
- 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.
Norwegian duo Ylvis have just released a music video about… well, essentially it’s about physical knot theory. It’s about tying “the greatest knot of all”, the Trucker’s hitch.
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.
Marc Lackenby has several state-of-the-art theorems which shows a disconnect between studying knots via planar diagrams and studying their complement.
One such theorem states if you take n knots, K_1, K_2, …, K_n, and their connect-sum K = K_1 # K_2 # … # K_n, and if we let c(K) be the minimial number of crossings in a planar diagram for the knot K, we have the relationship:
(c(K_1)+c(K_2)+…+c(K_n))/152 <= c(K) <= c(K_1)+c(K_2)+…+c(K_n)
The 152 on the left-hand side is what gets my attention. Why isn’t it simply 1? (more…)