Some thoughts on mathematics

This is a series of posts outlining a proof of the surgery exact triangle in Floer homology. One of the most important features of Floer homology theories for 3-manifolds is the surgery exact triangle.  Let me state this loosely for an arbitrary Floer homology theory $F$.  Suppose that $K$ is a knot in a 3-manifold $Y$ with framing $\lambda$.  Then, there is an exact triangle  $$ \ldots \to F(Y) \to F(Y_\lambda(K)) \to F(Y_{\lambda + \mu}(K)) \to F(Y) \to \ldots   $$ Here, $Y_{\lambda}(K)$ means surgery on $K$ corresponding to the framing $\lambda$, and $\lambda + \mu$ is the framing on $K$ which wraps one more time meridionally than $\lambda$.  (This exact triangle exists as stated, for example, in Heegaard Floer homology, monopole Floer homology, and framed instanton Floer homology.  There are lots of variants of the surgery exact triangle in a bunch of Floer theories.)  The surgery exact triangle has a bazillion applications i...

Recently I was at a conference for graduate students that featured a panel on collaboration.  This was super cool - I really like the idea of having some professional development panels at a conference, especially ones that I haven't seen at conferences in the past.  There was a lot to take in, but I think there was some really good advice on collaboration.  I had some tangential thoughts from the panel experience I wanted to share.  Warning:  I don't think what's written below may be that helpful or even cohesive, but I'm still going to post it in case it's useful for sparking some conversation.     I asked a graduate student after the collaboration panel what they thought about it.  This student explained to me that they just started research recently - collaboration was not on their mind at all.  Why am I pointing it out?  Often, we feel we need to be put on high alert for all things: make sure to seek out any possible ...

Back for more!  Recall that we are interested in proving the following proposition:  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.   In the last post, we showed that it suffices to show that there is a prime homology 3-sphere that bounds a contractible 4-manifold that is *not* surgery on a knot in $S^2 \times S^1$.  That was because being surgery on a knot in $S^2 \times S^1$ is equivalent to bounding a Mazur manifold, and those are the only contractible 4-manifolds with a handle decomposition with fewer than five handles.  That's our goal.  If you don't care about a proof of the proposition, here's a surgery picture for an example (and if you do care, use this picture as reference for the constructive proof below):   We'll start the proof by giving an obstruction to a homolo...

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 ...