This is just a short post to draw attention to a new preprint by Friedl and Powell The presentation of the Blanchfield pairing of a knot via a Seifert matrix.
The Blanchfield pairing on the Alexander module occurs in various places in knot theory, including in quantum topology. Levine’s 1977 argument for its expression in terms of the Seifert matrix doesn’t make easy reading (the authors suggest it’s incomplete- I can’t judge), and it is notoriously difficult to prove that the Blanchfield pairing is Hermitian. The authors deal deftly with both problems using a more modern but clearly sensible toolbox. Time to rewrite the textbooks.
I wish there were more papers like this. Some aspects of low dimensional topology could use a careful, sensible, modern reboot such as that of this paper.
A few days ago, two people named Joshua (one Howie and one Greene) independently posted to arXiv a similar solution to an old question of Ralph Fox:
Question: What is an alternating knot?
The preprints are:
This post will briefly introduce the problem; I look forward to reading the solutions themselves! (more…)
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!
I’m sad to announce the untimely passing of Dongseok Kim, a specialist on Kuperberg spiders and their generalizations. He was also a really nice guy whose conference talks were always well-worth listening to.
Although he’s better known for his quantum and stuff, the part of Kim’s work which was most intriguing for me personally was his work on spiders for Lie superalgebras. His paper on the topic doesn’t seem to be cited, despite superalgebra-related quantum invariants being a hot topic (work of Geer, Patureau-Mirand, Costantino, Turaev…)- has anyone noticed it? (more…)
One of the main ways in which I keep my finger on the pulse of what is hot now in low dimensional topology is to write lots and lots of reviews, both for Zentralblatt MATH and also for MathSciNet. In the last year or so, what has been increasingly coming through the pipe is papers about knot homology and mirror symmetry. There seems to be a lot happening in this field right now. (more…)
Check out this exciting new preprint by Vaughan Jones!
V.F.R. Jones, Some Unitary Representations of Thompson’s Groups and , arXiv:1412.7740.
Yesterday I received the shocking news of the passing of Tim Cochran (1955-2014), a leader in the field of knot and link concordance. The Rice University obituary is here.
A groundbreaking paper which made a deep impression on a lot of people, including me, was Cochran-Orr-Teichner’s Knot concordance, Whitney towers and signatures. This paper revealed an unexpected geometric filtration of the topological knot concordance group, which formed the basis for much of Tim Cochran’s subsequent work with collaborators, and the work of many other people.
In this post, in memory of Tim, I will say a few words about roughly what all of this is about. (more…)
I would like to draw attention to a fascinating MO question by Dylan Thurston, originally asked, it seems, by John Conway:
Can a knot be monotonically simplified using under moves?
The question asks whether, rather than searching for Reidemeister moves to simplify a knot diagram, we could instead search for “big Reidemeister moves” in which we view a section which passes underneath the whole knot (only undercrossing) or over the whole knot (only overcrossing) as a single unit, and we replace it by another undersection (or oversection) which has the same endpoints.
This question (or more generally, the question of how to efficiently simplify knot diagrams in practice) loosely relates to a fantasy about being able to photograph a knot with a smartphone, and for the phone to be able to identify it and to tag it with the correct knot type. Incidentally, I’d like to also draw attention to a question by Ryan Budney on the topic of computer vision identification of knots, which is topic I speculated about here:
Algorithm to go from a picture (or pictures) of a string in space, to a piecewise-linear representation of the curve.
A core question to which all of this relates is:
Are there any very hard unknots?
And perhaps more generally, are there any very hard ambient isotopies of knots?
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.
I don’t know about you, but when I tell non-mathematicians what knot theory is, I often find myself telling a story about identifying a knotted protein by its knottedness- something about different proteins tending to be bendy to differing degrees, so that certain types of protein tend to form knots with higher writhe than others, and that this helps biologists and chemists to distinguish proteins which they would otherwise need a lot of time and money and an electron microscope to tell apart.
One major problem with this story, and with similar stories, is that the knot diagrams have to be photographed (and thus identified) by hand. The pictures are not always easy to interpret (e.g. distinguishing overcrossings from undercrossings):
Also resolution might be low, objects might be in the way…
This is a computer vision problem as opposed to a math problem- but wouldn’t it be nice if a computer could recognise a knot type from a suboptimal picture? If you could snap a picture of yourself standing in front of an knot making bunny ears behind it, and your computer would automatically tag it with the correct knot type? Furthermore, wouldn’t it be nice if a computer could recognise your knot on the basis of many noisy pictures, perhaps taken from different angles? (more…)