Link-surgery filtered complex for a double branched cover

Let $D$ be a finite regular diagram of a nonempty oriented link $L\subset S^3$ with ordered crossing set $\{1,\dots ,n\}$. The resolution cube has vertices $I\in \{0,1{\}}^n$, and $D(I)$ is obtained by applying the indicated local smoothing at every crossing. The link-surgery filtered module associated to $D$ is the direct sum

$$C_D=\underset{I\in \{0,1{\}}^n}{⨁}\widehat{CF}(\mathrm{\Sigma }(D(I));{\mathbb{F}}_2),$$

with the summand for $I$ in increasing filtration degree $n-\mathrm{\mid }I\mathrm{\mid }$. Choose the subordinate Heegaard multi-diagram, compatible perturbation data and top-generator cycles for successive attaching-curve systems. The diagonal component of the candidate differential is the ordinary hat disk differential. For $I<J$ coordinatewise, sum over every ordered path $I=I_0<I_1<\cdots <I_r=J$ changing one coordinate at each step. Its summand counts the zero-dimensional holomorphic $(r+2)$-gon spaces with input at the $I$ vertex, the selected top-generator cycles between each pair of successive attaching systems, and output at the $J$ vertex. The count includes the allowed source conformal parameters and imposes zero basepoint multiplicity. These are the surgery polygon maps; the component is zero if $I\nleqq J$. Thus changing several crossings is a sum over paths and polygon counts, not an arbitrary direct edge map. The increasing filtration is $F_p{C}_D={⨁}_{\mathrm{\mid }I\mathrm{\mid }\ge n-p}\widehat{CF}(\mathrm{\Sigma }(D(I));{\mathbb{F}}_2)$, extended by zero below $0$ and by $C_D$ at and above $n$. Thus a component increasing $\mathrm{\mid }I\mathrm{\mid }$ lowers its filtration degree and preserves every $F_p$. The cube direction and the reduced Khovanov cochain degree remain separate indices. The polygon-count finiteness, square-zero, and total-homology identifications are justified by the surgery lemma, which labels its actual Ozsváth–Szabó construction and top-input/filling calculations as external inputs, separately from the retained local analytic theorem and finite filtration algebra.

The local smoothing labels are those of Ozsváth–Szabó Figure1: in the chart with northwest–southeast over-strand, 0 joins the two left and two right endpoints, and 1 joins the two top and two bottom endpoints. These become A and B on the mirror chart, respectively. This fixes the convention for the consumers without a diagram-invariance assertion. The general surgery lemma also applies to links in arbitrary closed connected oriented smooth three-manifolds; this definition is only its branched-cover instance.

Double branched covers of links

Hat Floer diagram-level chain module

Increasing filtration preserved by a differential

Required well-definedness and surgery identities ↗

Ozsváth and Szabó, On the Heegaard Floer homology of branched double-covers, §§2 and 4

Lipshitz, Heegaard Floer Homologies, §5