Some thoughts on mathematics

This sequence of posts was inspired by some conversations with Mike Miller Eismeier In this post, I want to talk about some contractible four-manifolds and connections with Dehn surgery .   Recall that lots of homology 3-spheres bound contractible four-manifolds.  For example, $S^3_{1/n}(K)$ bounds a contractible four-manifold if $K$ is a smoothly slice knot, as do the Brieskorn spheres $\Sigma(3,4,5)$ and $\Sigma(2,3,13)$.  By a quick Euler characteristic computation, it is easy to see that every contractible four-manifold other than the four-ball requires at least three handles.  In this sequence of posts I want to give a proof of the following goofy result (and give it some context).  Proposition:  There exists an irreducible, i.e. prime, homology 3-sphere which bounds a contractible 4-manifold and no bounding contractible four-manifold can be built with fewer than five handles.  Alternatively, there are prime homology 3-spheres that bound con...

It's winter break, so here's a short post.  I am trying to spend some time doing new non-math things.  The other day I was trying to help out with a fence repair.  If you have spent time with me, you'd likely think that I would not know how to help someone repair a fence - if you thought that, you are indeed correct.  I was not particularly helpful, but while I was embarrassing myself not knowing how to use a hammer, I did gain some insight into the mathematical process, which I wanted to mention here. In low-dimensional topology, as I hope to convey in this blog, it is very beneficial to have a wide range of tricks and tools.  This includes knowing topological tools (e.g. the Montesinos trick or Kirby calculus), invariants (e.g. gauge theory, Khovanov homology), but also perspectives and connections from other fields (e.g. viewing torus knots as links of singularities of algebraic curves).  However, knowing that hammers exist is not the same as being able ...

In Excision part 1, we studied how Floer homology changes when we cut and reglue along a surface.  In this post, I wanted to discuss an analogue for annular Khovanov homology.  (As a warning, I am not an expert in Khovanov homology, and definitely not knowledgeable about annular Khovanov homology.  Make sure to really fact check this!)   First, recall that given an oriented link $L$ in $S^3$, Khovanov homology $Kh(L)$ is a bigraded vector space which categorifies the Jones polynomial.  Annular Khovanov homology, $AKh$, is a *triply* graded invariant of oriented links in $D^2 \times S^1$, developed by Asaeda-Przytycki-Sikora.  The third grading is called the annular  grading, and we write $AKh(L,f)$ to mean the summand in annular grading $f$.  It's an important point that this is an invariant of the link up to isotopy in $D^2 \times S^1$ and not up to diffeomorphism.  In particular, you can cut along a meridional disk and reglue by ...

In algebraic topology, the excision theorem is an extremely useful tool for computing homology.  In case your algebraic topology class is ancient history, let's recall the statement: if $U \subset V \subset X$ are a sequence of spaces with $\overline{U} \subset V^\circ$, then $$H_*(X - U, A - U) \cong H_*(X, A).$$In other words, relative homology is unaffected by "cutting" out $U$.  In Floer homology, we also have excision, which involves a different kind of cutting.  In this post I want to discuss some aspects of Floer homology in excision.  In the next post, I'll give proof of a similar statement in annular Khovanov homology.     Simplest version of excision:  First I want to state and prove an easy version of Floer excision.  I'm going to write this in a way that applies in most Floer theories, so I'll use $F$ to denote an arbitrary Floer homology.  Let $Y$ be a closed connected oriented three-manifold.  Suppose that $x$ is...

There is an upcoming conference celebrating Cameron Gordon's 80th birthday.  Cameron has had a big influence on my career (both directly and through his mathematics), so I wanted to write some posts giving quick summaries of his influential work.  These will be short samples instead of giving full details on anything.  This post was inspired by a comment of Josh Wang. I thought it might be kind of cliche to start with the most famous results, like the Knot Complement Theorem, so I'll instead start with one of my personal favorites: "Only integral surgeries can yield reducible manifolds" by Gordon-Luecke.  While the title is the result, we'll see some more background and at the end, I'll give a really nice application to the unknotting number of knots.   First, we need to have a little chat about Dehn surgeries.  (As an aside, Cameron is also a historian of topology.  Since Dehn came up, I'll point out Cameron wrote a historical arti...

I don't necessarily think the various areas I discuss below are more or less important than the others, just what I view as their popularity.  Others likely have a different view.  Recently, I was listening to The Beths' "Expert in a Dying Field".  It's a break-up song about all the minutia you learn about a person which no longer feels relevant when the relationship ends.  (Don't worry, I'm not going through a breakup/angst phase.  I just like the song.)  When I listen to it, I often times think about Heegaard Floer homology.   Heegaard Floer homology, developed by Ozsvath and Szabo, exploded in the early 2000s.  It seemed to pack most of the punch of gauge theory while being super computable.  When I started grad school there was this incredible flurry of activity, both in theory building (e.g. bordered Floer homology) and applications (e.g. structure of the knot concordance group, classifying fillable contact structures).  Hot fi...

Here's a short post on a construction I really enjoy.   Theorem: (Kawauchi)  Let $K \subset S^3$ be a strongly negative amphichiral knot (SNACK).  Then $K$ bounds a smoothly embedded disk in a rational homology ball.   Let me first explain the statement and then give a proof.  A knot $K$ is strongly negative amphichiral if there is an orientation-reversing diffeomorphism of $(S^3,K)$ which is also orientation-reversing on $K$ and the fixed point set is two points which sit on $K$.  Sometimes $K$ can be slice, e.g. the square knot is a slice SNACK, but often it is not, e.g. the figure-eight knot.  (The Fox-Milnor condition shows it is not even topologically slice.)   Here is the proof, which is very quick.  First, $X_0(K)$ denote a 0-framed 2-handle attachment to $B^4$ along $K$.  Then $K$ is smoothly slice in this four-manifold - the slice disk is just the core of the 2-handle!  (This applies even if $K$ is not SNACK...