Admissible multipointed Heegaard data for knot Floer complexes

Work in the smooth category inherited from the oriented-link convention. In particular, an embedded bridge witness disk means the image of a smooth rank-two embedding of a planar disk, with ordinary smooth boundary or smooth transverse corners at its two marked endpoints; smoothness at a boundary or corner means local smooth extension in its charts. Witness parametrizations can be chosen for the finite disk families. This category convention does not require the knot arcs to meet the Heegaard surface transversely: tangential smooth crossings remain permitted.

Let $K\subset S^3$ be a smooth oriented knot. Let $\mathrm{\Sigma }$ be a closed oriented surface of genus $h$, let $k\ge 1$, and put $m=h+k-1$. A handlebody is a compact three-manifold that is a regular neighborhood of a finite connected graph. A $2k$-pointed Heegaard diagram for $S^3$ consists of a splitting $S^3=U_{\alpha }{\cup }_{\mathrm{\Sigma }}{U}_{\beta }$ into handlebodies, two systems $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_m)$ and $\mathit{\beta }=({\beta }_1,\dots ,{\beta }_m)$ of pairwise disjoint simple closed curves on $\mathrm{\Sigma }$ that bound pairwise disjoint properly embedded disks in $U_{\alpha }$ and $U_{\beta }$, respectively, cutting each handlebody into exactly $k$ three-balls, and two ordered sets of distinct points $\mathbf{w}=(w_1,\dots ,w_k)$ and $\mathbf{z}=(z_1,\dots ,z_k)$ disjoint from the curves. Curves from different systems intersect transversely in finitely many points. Each component of $\mathrm{\Sigma }\setminus \mathit{\alpha }$ and of $\mathrm{\Sigma }\setminus \mathit{\beta }$ contains exactly one $w$-point and exactly one $z$-point. The diagram also includes trivial arc systems in the two handlebodies: the $\alpha$-arcs join $w_i$ to $z_i$, and the $\beta$-arcs join $z_i$ to $w_{i-1}$, with indices modulo $k$. Each system is disjoint from its compression disks and is simultaneously boundary parallel: its arcs cobound pairwise disjoint embedded disks with arcs in the corresponding components of the attaching-system complement on $\mathrm{\Sigma }$. Thus each ball obtained by compression contains one unknotted arc. Together the arcs form an oriented knot, oriented from $w_i$ to $z_i$ in $U_{\alpha }$ and from $z_i$ to $w_{i-1}$ in $U_{\beta }$. The diagram is compatible with $K$ when this knot is orientation-preservingly ambient isotopic to $K$. For a grid diagram, compatibility uses the row and column orientation fixed in the grid definition, with $w$-points at $O$ and $z$-points at $X$. For a grid representing one knot, the basepoint pairs can be numbered cyclically in reverse order of travel along the oriented knot so that $w_i,z_i$ lie in the same component $A_i$ of $\mathrm{\Sigma }\setminus \mathit{\alpha }$ and $w_i,z_{i+1}$ lie in the same component $B_i$ of $\mathrm{\Sigma }\setminus \mathit{\beta }$.

Comparisons of these embedded diagrams allow orientation-preserving diagram isomorphisms: an orientation-preserving diffeomorphism of $S^3$ carrying the two handlebodies, their oriented Heegaard surface, the two attaching systems, the ordered $w,z$ points, and the two simultaneously boundary-parallel arc systems to the corresponding data. Equivalently one may specify the surface isomorphism together with its compatible collared extensions over both handlebodies carrying those arcs. The attaching systems are carried as systems; reordering their curve factors induces the corresponding product identification. Comparisons also allow arc normalizations in the fixed splitting: a smooth isotopy of the represented oriented knot, supported near the ordered marks and fixing each mark, during which every open connecting arc stays in its original handlebody and the two simultaneous boundary-parallel witness systems are retained or replaced by embedded witnesses for those same arcs. The surface, attaching curves, marked points and compression disks stay fixed. Such a normalization may replace a tangential marked crossing by a transverse one; the original data are not required to have neat arcs. This is a change of knot representative relative to the fixed surface, not a diagram isomorphism carrying both objects. Analytic and assigned grading data depending on the fixed surface diagram can be kept identical; the corresponding generator and disk-count identification is then the identity. Existence of normalizations with the required witnesses is a separate geometric assertion, not part of this definition. Such a comparison transports the represented oriented knot and any specified Seifert surface as well. For analytic data, it transports $j$, the surface area form, all chosen divisor neighborhoods and $J_t$ by the induced symmetric-product diffeomorphism. The comparison maps on generators and disk classes are their pushforwards under this map; compatibility with the count differential is a mathematical property to be proved. No identity action is part of this convention, even when the ambient diffeomorphism is isotopic to the identity.

The $m$-fold symmetric product ${\mathrm{Sym}}^m(\mathrm{\Sigma })={\mathrm{\Sigma }}^m\mathrm{/}{S}_m$ is the space of unordered $m$-tuples of points of $\mathrm{\Sigma }$, where $S_m$ permutes coordinates. Let $q:{\mathrm{\Sigma }}^m\to {\mathrm{Sym}}^m(\mathrm{\Sigma })$ be the quotient map. The product tori are the images ${\mathbb{T}}_{\alpha }=q({\alpha }_1\times \cdots \times {\alpha }_m)$ and ${\mathbb{T}}_{\beta }=q({\beta }_1\times \cdots \times {\beta }_m)$. Their intersection points are the generators. A Whitney disk class $\varphi \in {\pi }_2(\mathbf{x},\mathbf{y})$ is a relative homotopy class of maps from a disk to ${\mathrm{Sym}}^m(\mathrm{\Sigma })$ whose two boundary arcs lie on ${\mathbb{T}}_{\alpha }$ and ${\mathbb{T}}_{\beta }$ and whose endpoints are $\mathbf{x}$ and $\mathbf{y}$. Its domain is the integral two-chain of complementary regions with coefficients equal to local multiplicities. Write $n_{w_i}(\varphi )$ and $n_{z_i}(\varphi )$ for its local multiplicities at the marked points. An almost complex structure $J$ on ${\mathrm{Sym}}^m(\mathrm{\Sigma })$ is a smooth endomorphism of the tangent bundle with $J^2=-1$. Fix a complex structure $j$ and an area form on $\mathrm{\Sigma }$. For $m>0$, choose a smooth path $\mathbb{J}=(J_t{)}_{t\in [0,1]}$ of nearly symmetric almost complex structures, constant near $t=0,1$: each $J_t$ agrees with the induced integrable structure ${\mathrm{Sym}}^m(j)$ on a fixed neighborhood of the diagonal and of every divisor $\{p\}+{\mathrm{Sym}}^{m-1}(\mathrm{\Sigma })$ for a chosen interior point $p$ of every complementary region, including every marked point, and tames the form descended from the product area form off that neighborhood. The path is a sufficiently small perturbation in this class. For $m=0$ the symmetric product is a point. Local multiplicities are intersection numbers with these divisors; positivity is a mathematical property to be proved, not an additional choice of disk counts. A $\mathbb{J}$-holomorphic strip representative is a smooth map $u:\mathbb{R}\times [0,1]\to {\mathrm{Sym}}^m(\mathrm{\Sigma })$ satisfying ${\mathrm{\partial }}_{\tau }u+J_t(u){\mathrm{\partial }}_{t}u=0$, with $u(\tau ,0)\in {\mathbb{T}}_{\alpha }$, $u(\tau ,1)\in {\mathbb{T}}_{\beta }$, and limits $\mathbf{x},\mathbf{y}$ as $\tau \to -\mathrm{\infty },+\mathrm{\infty }$. Translation in $\tau$ defines an $\mathbb{R}$-action. The Maslov index $\mu (\varphi )$ is the Fredholm index of the linearized Cauchy–Riemann operator at a representative of $\varphi$; the relative Maslov grading convention is $M(\mathbf{x})-M(\mathbf{y})=\mu (\varphi )-2{\sum }_{i=1}^k{n}_{w_i}(\varphi )$, up to one overall additive constant.

A periodic domain is an integral two-chain in the complementary regions whose boundary is an integral sum of full $\alpha$- and $\beta$-curves; its local multiplicity at a marked point is the coefficient of the containing region. The pointed diagram is admissible when every nonzero periodic domain with $n_{w_i}=0$ for all $i$ has at least one positive and at least one negative region coefficient. A path $\mathbb{J}$ is regular for a set of disk classes when the linearized Cauchy–Riemann operator at every holomorphic-strip representative in those classes is surjective. For a path regular on the classes used, let ${\mathcal{M}}_{\mathbb{J}}(\varphi )$ be the space of strip representatives. If, for each generator $\mathbf{x}$, only finitely many pairs $(\mathbf{y},\varphi )$ with ${\mathcal{M}}_{\mathbb{J}}(\varphi )\mathrm{\ne }\mathrm{\varnothing }$, $\mu (\varphi )=1$, and $n_{w_i}(\varphi )\ge 0$ for every $i$ occur, and each quotient is finite, the multipointed full filtered minus module is the free module over ${\mathbb{F}}_2[U_1,\dots ,U_k]$ generated by ${\mathbb{T}}_{\alpha }\cap {\mathbb{T}}_{\beta }$, with differential ${\mathrm{\partial }}^{-}\mathbf{x}={\sum }_{\mathbf{y}}{\sum }_{\begin{array}{c}\varphi \in {\pi }_2(\mathbf{x},\mathbf{y}),{\mathcal{M}}_{\mathbb{J}}(\varphi )\mathrm{\ne }\mathrm{\varnothing }\\ \mu (\varphi )=1,n_{w_i}(\varphi )\ge 0\mathrm{\forall }i\end{array}}{\mathrm{\#}}_2({\mathcal{M}}_{\mathbb{J}}(\varphi )\mathrm{/}\mathbb{R})U_1^{n_{w_1}(\varphi )}\cdots U_k^{n_{w_k}(\varphi )}\mathbf{y}\mathrm{.}$ Here ${\mathrm{\#}}_2$ is the count modulo two of the zero-dimensional quotient when it is finite. Quotienting by all $U_i$ leaves the fully blocked multipointed chain complex (the hat chain complex when $k=1$), with terms only from classes satisfying $n_{w_i}=0$ for every $i$; its Alexander-filtered associated-graded complex further retains only classes with every $n_{z_i}=0$. The relative Alexander grading convention is $A(\mathbf{x})-A(\mathbf{y})={\sum }_i{n}_{z_i}(\varphi )-{\sum }_i{n}_{w_i}(\varphi )$, with an overall additive constant. Each $U_i$ lowers Maslov grading by two and Alexander filtration by one. In a grid diagram, $\mathrm{\Sigma }=T^2$, $h=1$, $k=n$, and $m=n$; the horizontal circles are the $\alpha$-curves, the vertical circles are the $\beta$-curves, and the $w$-points and $z$-points are the $O$- and $X$-markings respectively.

Smooth oriented knots and ambient isotopy in the standard three-sphere

Manolescu, Ozsvath, and Sarkar, “A combinatorial description of knot Floer homology,” §2

Manolescu, An Introduction to Knot Floer Homology, §§3.3–3.5 and 4.2–4.3