Algebraic variations of the full knot Floer complex

Question

In Hom's paper (arXiv link), p.20, Section 3.3 ends with

"There are other algebraic modifications one may consider, such as setting $U^n = 0$ or $UV = 0$",

referring to the knot Floer homology with $\mathbb{F}[U,V]$ coefficients. That is, we consider pseudoholomorphic disks in the Heegaard diagram (or rather, in the symmetric product of the Heegaard surface), including those passing through both basepoints $w$ and $z$.

However, there is no reference to works that do consider these different variations. My first question is, which modifications have already been considered/published, and for each what is a good reference?

The second question is similar to the first: in the specific example that we set say $U^2=0$, is there a good geometric interpretation of what kind of information we're losing?

Answer 1

A version of knot Floer homology defined over the ring $\mathbb{F}[U,V]/(UV)$ is roughly equivalent to the information which is called, in Holomorphic discs and knot invariants, in the statement of Theorem 4.1, $CFK^0(Y, K)$. It's also called the 'bent complex' in Floer Simple Manifolds and L-Space Intervals. (It's also implicitly used in earlier work of J. Rasmussen, including his Ph.D. thesis.)

More modern references are more explicit about the use of this ring; I think, as Mike points out in the comments, More concordance homomorphisms from knot Floer homology is one of the earlier places to do this.

I don't know of an explicit reference for the quotients by $U^n$. (This is more likely to be a gap in my knowledge than the literature.)

With regards to the second question: essentially, you're limiting which holomorphic curves you count in the differential - in the case of $\mathbb{F}[U, V]/(UV)$ you only allow curves which cross one or other base-point, but not both. More philosophically, the resulting invariant is something like `the simplest version of knot Floer homology needed to compute the hat invariant of surgeries on the knot'. You throw out just enough holomorphic curves to still be able to compute this - simplifying things, but you pay the price of not being able to compute any of the fancier invariants of closed three-manifolds.