The Alexander filtration and tau level

Let $K$ be a smooth oriented knot in the standard oriented three-sphere. Let $F$ be a compact connected oriented Seifert surface with boundary $K$, and let $\mathcal{H}=(\mathrm{\Sigma },\mathit{\alpha },\mathit{\beta },w,z)$ be an admissible doubly pointed Heegaard diagram representing $(K,F)$. The associated one-pointed hat chain complex $\widehat{CF}(\mathcal{H})$ is the finite-dimensional chain complex over ${\mathbb{F}}_2$ obtained from the minus complex by setting its basepoint variable to zero; its differential counts index-one disks with zero multiplicity at $w$. Assume for this specified data that the differential is nonincreasing in the Alexander filtration, that the absolute Alexander formula below takes integer values, and that $H(\widehat{CF}(\mathcal{H}))\cong {\mathbb{F}}_2$. Give a generator $\mathbf{x}$ its absolute Alexander filtration $A_F(\mathbf{x})=\frac{1}{2}⟨c_1({\mathfrak{s}}_{w,z}(\mathbf{x})),[\widehat{F}]⟩,$ where ${\mathfrak{s}}_{w,z}(\mathbf{x})$ is the homology class of the nowhere-vanishing vector field assigned to $\mathbf{x}$ by the pointed Morse construction, with its prescribed boundary data near the two basepoints; $c_1$ is the first Chern class of the oriented plane field perpendicular to that vector field. To form the zero-framed surgery, remove a tubular neighborhood of $K$ and glue in a solid torus whose meridian is the longitude induced by $F$. The surface $\widehat{F}$ is obtained by deleting a boundary collar of $F$ and capping its longitude by the meridian disk of that solid torus. This is the standard integer normalization, with the one-generator diagram of the standard unknot at filtration zero. The subcomplex $F_s\widehat{CF}(\mathcal{H})$ is spanned by generators of Alexander filtration at most $s$.

More generally, a finite integer-filtered chain complex is a finite-dimensional chain complex $C$ over ${\mathbb{F}}_2$ with subcomplexes $F_{s}C$ such that $F_{s}C=0$ for all sufficiently small integers $s$ and $F_{s}C=C$ for all sufficiently large integers $s$. If $H(C)\cong {\mathbb{F}}_2$, define $\tau (C):=\min \{s\in \mathbb{Z}:\mathrm{im}(H(F_{s}C)⟶H(C))\mathrm{\ne }0\}\mathrm{.}$ For the fixed diagram and surface above, write ${\tau }_{\mathcal{H},F}(K)=\tau (\widehat{CF}(\mathcal{H}),F_{\ast }\widehat{CF}(\mathcal{H}))$. Once diagram and surface independence is proved, their common value is the knot invariant $\tau (K)$. Until that comparison is proved, ${\tau }_{\mathcal{H},F}(K)$ denotes only the value for the specified data.

Pointed diagrams, the hat complex, and the Alexander filtration convention

Oriented Seifert surfaces for knots

Spin-c structures as lifts of the oriented frame bundle and their associated plane-field data

Ozsvath and Szabo, Knot Floer homology and the four-ball genus, §2.1

Hom, A survey on Heegaard Floer homology and concordance, §§2.2 and 3.1

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links