Four-dimensional manifold theory is remarkable for a variety of reasons. It has the only outstanding generalized smooth Poincare conjecture. It is the only dimension where vector spaces have more than one smooth structure. The only dimension with an unresolved generalized Shoenflies problem. The list goes on. One issue that is perhaps not discussed enough is the paucity of theorems about smooth isotopy. In dimensions 2 and 3, the Schoenflies and Alexander theorems are the backbone of all theorems about isotopy, allowing one to work from the ground-up.

## January 1, 2020

## May 24, 2016

### SnapPy 2.4 released

## April 9, 2016

### Arithmetic Chern-Simons

The most compelling aspect of Quantum Topology for me is its connection to analytic number theory. Today I’d like to draw your attention to recent work of Minhyong Kim on Arithmetic Chern-Simons Theory (see his paper for more details). I was fortunate to hear him give a talk about this last Wednesday and a colloquium talk on related subjects the day before. People have been talking about such “quantum topological number theory” for a long time- *e.g.* this 2010 MO question – but we haven’t seen much of an uptake so far. This isn’t an easy direction to pursue because one needs to know both quantum topology and analytic number theory, but I was left with the strong feeling that “There’s gold in them thar hills”, both for topologists and for number theorists. (more…)

## December 20, 2015

### The Blanchfield pairing done right

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.

## November 22, 2015

### What is an alternating knot? A tale of two Joshuas.

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:

- Joshua Evan Greene, Alternating links and definite surfaces, arXiv:1511.06329
- Joshua Howie, A characterisation of alternating knot exteriors, arXiv:1511.04945

This post will briefly introduce the problem; I look forward to reading the solutions themselves! (more…)

## September 27, 2015

### A counterexample to the Strong Neuwirth Conjecture

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!

## August 19, 2015

### Dongseok Kim 1968-2015

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…)

## May 22, 2015

### Recent coloured HOMFLYPT-related stuff

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…)

## January 14, 2015

### Jones’s new polynomial

Check out this exciting new preprint by Vaughan Jones!

V.F.R. Jones,

Some Unitary Representations of Thompson’s Groups and, arXiv:1412.7740.

## December 19, 2014

### Concordance Champion Tim Cochran 1955-2014.

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…)