Some thoughts on mathematics

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...