The absolute Alexander grading of knot Floer homology

Let $Y$ be a closed connected oriented $3$-manifold, let $K\subset Y$ be a smooth oriented null-homologous knot, and let $F\subset Y$ be a compact connected oriented embedded surface with oriented boundary $K$. Choose a closed tubular neighborhood $N(K)$, put $M=Y\setminus \mathrm{int}N(K)$, and let ${\gamma }_{\mu }$ be the union of two disjoint annuli on $\mathrm{\partial }M$ whose core sutures are parallel meridians of $K$ with opposite orientations. Delete a collar of $\mathrm{\partial }F$ to regard $F$ as a properly embedded surface in $(M,{\gamma }_{\mu })$, with relative class $[F]\in H_2(M,\mathrm{\partial }M;\mathbb{Z})$. Let $t_{\mu }$ be the meridional boundary trivialization of the oriented plane field used in the relative Spin-c definition. For $\mathfrak{s}\in {\mathrm{Spin}}^c(M,{\gamma }_{\mu })$, its Alexander grading is

$A_F(\mathfrak{s})=\frac{1}{2}⟨c_1(\mathfrak{s},t_{\mu }),[F]⟩,$

with the standard sign convention that gives the usual Alexander grading for knots in $S^3$. For a balanced admissible sutured Heegaard diagram of $(M,{\gamma }_{\mu })$, the Spin-c summand $SFH(M,{\gamma }_{\mu },\mathfrak{s};\mathbb{Z})$ is the homology of the integral sutured Floer disk-count complex generated by the Heegaard generators assigned to $\mathfrak{s}$. Define the integral knot Floer group in Alexander grading $i$ by

$\widehat{\mathrm{H}\mathrm{F}\mathrm{K}}(Y,K,[F],i;\mathbb{Z})={⨁}_{\begin{array}{c}\mathfrak{s}\in {\mathrm{Spin}}^c(M,{\gamma }_{\mu })\\ A_F(\mathfrak{s})=i\end{array}}SFH(M,{\gamma }_{\mu },\mathfrak{s};\mathbb{Z})\mathrm{.}$

This fixes the integral coefficient convention. The equivalent associated-graded closed Heegaard-complex presentation, invariance under changing diagrams, and comparison with the mod-two grid complex are separate theorems; this definition supplies none of their proofs or a coefficient-change map.

Pointed Heegaard diagrams, disk classes, and the mod-two knot Floer complex

The grid Alexander grading and orientation convention

Ni, Knot Floer homology detects fibred knots, §1

Ozsvath and Szabo, Holomorphic disks and knot invariants, §3.4

Manolescu, An Introduction to Knot Floer Homology, §§3.3–4.3