Grid homology and combinatorial knot Floer theory

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

Let $K\subset S^3$ be a smooth oriented knot, meaning one oriented component of a finite smooth oriented link in the standard oriented three-sphere $S^3={\mathbb{R}}^3\cup \{\mathrm{\infty }\}$. The grid orientation below fixes the sign convention for its Alexander grading.

For an integer $n\ge 2$, put $T_n={\mathbb{R}}^2\mathrm{/}(n\mathbb{Z}{)}^2$, with horizontal circles ${\alpha }_i=\{y=i\}$ and vertical circles ${\beta }_j=\{x=j\}$ for $i,j\in \mathbb{Z}\mathrm{/}n\mathbb{Z}$. A toroidal grid diagram of grid number $n$ is a pair of disjoint sets $\mathbb{O},\mathbb{X}$ of cell centers $(j+\frac{1}{2},i+\frac{1}{2})$, read modulo $n$, such that every row and every column contains exactly one point of each set. In the fundamental square $[0,n)\times [0,n)$, join the two marks in each row and each column by the straight segment between their centers. At each crossing of the projected link, the vertical segment is over and the horizontal segment is under. Orient each horizontal segment from its $O$-marking to its $X$-marking and each vertical segment from its $X$-marking to its $O$-marking. The markings and these segments determine an oriented link. Following each component from an $O$ along its row to an $X$ and then along that column to the next $O$ defines a permutation of the $n$ $O$-markings; its cycles are the link components. The grid represents a knot precisely when this permutation is one $n$-cycle.

A grid state is a subset $\mathbf{x}\subset {\bigcup }_{i,j}({\alpha }_i\cap {\beta }_j)$ containing exactly one intersection on each horizontal and each vertical circle. After numbering the circles, grid states correspond bijectively to permutations of $\{1,\dots ,n\}$, so there are $n!$ states. For states $\mathbf{x},\mathbf{y}$, a rectangle from $\mathbf{x}$ to $\mathbf{y}$ is an embedded torus rectangle whose sides lie in grid circles, whose northeast and southwest corners lie in $\mathbf{x}$, whose northwest and southeast corners lie in $\mathbf{y}$, and whose remaining state points agree. Rectangles are subsets of the torus, so their lifts may cross a chosen fundamental-domain edge. Such a rectangle is empty when its interior contains no point of $\mathbf{x}$; write ${\mathrm{Rect}}^{\circ }(\mathbf{x},\mathbf{y})$ for the empty rectangles from $\mathbf{x}$ to $\mathbf{y}$.

To define the gradings, fix the fundamental square $[0,n)\times [0,n)$. For finite signed point sets $P,Q$ in this square, set $I(P,Q)=\mathrm{\#}\{(p,q):p_1<q_1,p_2<q_2\}$ and $J(P,Q)=(I(P,Q)+I(Q,P))\mathrm{/}2$, extended bilinearly to signed point sets. Regard $\mathbf{x}$, $\mathbb{O}$, and $\mathbb{X}$ as signed point sets and define

$M_{\mathbb{O}}(\mathbf{x})=J(\mathbf{x}-\mathbb{O},\mathbf{x}-\mathbb{O})+1,M_{\mathbb{X}}(\mathbf{x})=J(\mathbf{x}-\mathbb{X},\mathbf{x}-\mathbb{X})+1.$

For a knot grid, define the rational Alexander value

$A_G(\mathbf{x})=\frac{M_{\mathbb{O}}(\mathbf{x})-M_{\mathbb{X}}(\mathbf{x})}{2}-\frac{n-1}{2}\mathrm{.}$

The lemma establishing integrality and the rectangle-change formula is a well-definedness supplier for this grading. For any rectangle $r$, let $O_i(r)$ and $X_i(r)$ be $1$ if the interior of $r$ contains the corresponding marking and $0$ otherwise; let $n_{\mathbb{O}}(r)={\sum }_i{O}_i(r)$ and $n_{\mathbb{X}}(r)={\sum }_i{X}_i(r)$.

For illustration, this $3\times 3$ marking pattern has one marking of each type in every row and column; its cell centers are read modulo three on the torus.

Oriented links, knots, and ambient isotopy in the standard three-sphere

Integrality and rectangle-change formula for the grid Alexander grading ↗

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, §§3.1–3.2 and 4.1–4.3

Manolescu, Ozsvath, and Sarkar, “A combinatorial description of knot Floer homology,” §§1 and 3

Manolescu, An Introduction to Knot Floer Homology, §4.1

Let $Y$ be a closed connected oriented smooth three-manifold. A pointed genus-$g$ Heegaard diagram for $Y$ consists of an embedded splitting $Y=U_{\alpha }{\cup }_{\mathrm{\Sigma }}{U}_{\beta }$, where $\mathrm{\Sigma }=\mathrm{\partial }{U}_{\alpha }$ is a closed oriented genus-$g$ surface and each $U$ is a handlebody, meaning a regular neighborhood of a finite connected graph; ordered systems $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_g)$ and $\mathit{\beta }=({\beta }_1,\dots ,{\beta }_g)$ of pairwise disjoint smooth simple closed curves bounding specified disjoint proper compression disks in the respective handlebodies, each disk system cutting its handlebody into one ball; and one point $z$ off the curves. Opposite-system curves meet transversely in finitely many points. This is closed-manifold data; it includes neither a knot nor a second marked point.

Write $q:{\mathrm{\Sigma }}^g\to {\mathrm{Sym}}^g(\mathrm{\Sigma })={\mathrm{\Sigma }}^g\mathrm{/}{S}_g$ for the permutation quotient. The product tori are the images $T_{\alpha }=q({\alpha }_1\times \cdots \times {\alpha }_g),T_{\beta }=q({\beta }_1\times \cdots \times {\beta }_g)\mathrm{.}$ The group $S_g$ acts on the ambient product, rather than on a fixed ordered product of distinct attaching circles. A generator is an intersection $x\in T_{\alpha }\cap T_{\beta }$; equivalently it selects one intersection point on each curve of each system. There are finitely many generators. For $g=0$ the symmetric product and both empty products are one point, with the empty generator.

A Whitney class $\varphi \in {\pi }_2(x,y)$ is a relative homotopy class of maps of a closed disk with two marked boundary points into the symmetric product, taking its two boundary arcs into $T_{\alpha },T_{\beta }$ and its marked points to $x,y$. For $g>0$, its domain is the integral two-chain of complementary regions; its coefficient at an interior point $p$ is the intersection multiplicity $n_p(\varphi )$ with $\{p\}+{\mathrm{Sym}}^{g-1}(\mathrm{\Sigma })$. In particular $n_z(\varphi )$ is the basepoint coefficient. A periodic domain is an integral region two-chain whose boundary is an integral sum of full attaching circles. The domain of a class $\psi \in {\pi }_2(x,x)$ is periodic. Such a class is normalized when $n_z(\psi )=0$. Write $\mu (\varphi )$ for the real Fredholm index of the disk linearization, before quotienting longitudinal translation; well-definedness and its topological index formula are properties to be proved in the analytic and index theory.

For $g=0$, the target is a point and the only Whitney class is constant: define its domain and every $n_p(\varphi )$ to be zero and its index to be zero. Formal periodic domains are $m\mathrm{\Sigma }$, with coefficient $n_p(m\mathrm{\Sigma })=m$ at every point. Thus the only basepoint-normalized periodic domain is zero, and every sector weak-admissibility test is vacuous. No negative symmetric power or point-divisor intersection is used in this rank.

For a periodic domain P, cap its full attaching-circle boundary with the specified compression disks, using the boundary multiplicities and orientations, to obtain the integral class $H(P)\in H_2(Y;\mathbb{Z})$. A Spin-c structure $\mathfrak{s}$ is a lift of the oriented tangent frame bundle through ${\mathrm{Spin}}^c(3)\to SO(3)$; $c_1(\mathfrak{s})$ is the first Chern class of its determinant line, with determinant character $[s,u]\mapsto u^2$. Weak admissibility in the specified Spin-c sector $\mathfrak{s}$ means that EVERY nonzero integral periodic domain P with $n_z(P)=0$ and $⟨c_1(\mathfrak{s}),H(P)⟩=0$ has both positive and negative region coefficients. This condition is imposed even when the sector has no diagram generators. It is a condition on domains and the intended sector, not only on periodic classes based at existing generators.

The ordinary sector module $\widehat{CF}(\mathcal{H},\mathfrak{s})$ uses generators with Morse Spin-c label ${\mathfrak{s}}_z(x)=\mathfrak{s}$. The label is defined by compatible handle Morse data: replace the gradient in disjoint tubes around the generator's index-one/index-two trajectories and the basepoint's index-zero/index-three trajectory by nonsingular fields, and take the Spin-c lift of their oriented perpendicular planes. Its actual construction, independence properties and periodic-index formula are obligations of the Spin-c and index theory. In particular the equality between the index of a REALIZED normalized periodic class based at x and the Chern evaluation in ${\mathfrak{s}}_z(x)$ is not assumed as its own proof. A realized-generator test alone is not substituted for the stated domain condition.

An admissible diagram for the full single-diagram hat module satisfies this sector condition for EVERY Spin-c structure on Y, including the sectors with no generators. The full module below uses all generators and hence all intended sectors; an empty sector contributes a zero summand but is not omitted from the admissibility test. Under Countable Choice, this full-diagram condition is equivalent to global weak admissibility: every nonzero periodic domain with $n_z=0$ has both signs. Global weakness implies every sector test by restriction; conversely, the zero-Chern Spin-c sector supplied under Countable Choice by Complete Heegaard data have a zero-Chern Spin-c sector tests every normalized periodic domain. No generator in that sector is needed. For a SINGLE specified sector, ordinary weak admissibility remains the stated Chern-kernel condition and need not imply global weakness. No assertion of the full group from a diagram tested only at its represented sectors is made.

For $g>0$ choose a complex structure $j$ and positive area form on $\mathrm{\Sigma }$. Analytic data include a smooth path $J_t$ on the symmetric product, constant on actual neighborhoods of $t=0,1$. On a fixed neighborhood of the diagonal and the point-divisors for a chosen interior point of every complementary region, including $z$, it agrees with the integrable structure ${\mathrm{Sym}}^g(j)$; away from that neighborhood it is tame for the form descended from the product surface area form. The path is chosen in a sufficiently small nearly symmetric class. Product-integrable normal forms near the finitely many transverse generators may be included among its specified auxiliary choices. Existence of such choices and of the regular perturbations needed below remains a mathematical obligation. A holomorphic strip is an actual smooth finite-Dirichlet-energy map $u:\mathbb{R}\times [0,1]\to {\mathrm{Sym}}^g(\mathrm{\Sigma })$ satisfying $u_s+J_t(u)u_t=0,u(s,0)\in T_{\alpha },u(s,1)\in T_{\beta },$ with uniform generator limits at both longitudinal ends. The space in class $\varphi$ is ${\mathcal{M}}_J(\varphi )$, and translation in $s$ acts on it. Regular means that its actual linearized operator is surjective at the solutions being counted. Local multiplicity positivity, Fredholmness, the class index and compactness are results to be proved, not choices of which maps to count.

The hat diagram-level module $\widehat{CF}(\mathcal{H};{\mathbb{F}}_2)$ is free over ${\mathbb{F}}_2$ on the generators. For regular data for which the displayed quotient sets are finite and, for each $x$, only finitely many nonempty pairs $(y,\varphi )$ occur, its candidate count operator is $\widehat{\mathrm{\partial }}x={\sum }_y{\sum }_{\begin{array}{c}\varphi \in {\pi }_2(x,y)\\ \mu (\varphi )=1,n_z(\varphi )=0\end{array}}{\mathrm{\#}}_2({\mathcal{M}}_J(\varphi )\mathrm{/}\mathbb{R})y\mathrm{.}$ Here ${\mathrm{\#}}_2$ is parity of an actual finite zero-dimensional quotient. For $g=0$ the operator is zero on the empty generator. That the required regular finite counts exist, that ${\widehat{\mathrm{\partial }}}^2=0$, and that its homology is invariant under changing pointed admissible diagrams and auxiliary choices are separate analytic and move theorems. Once these hold, the common homology is denoted $\widehat{HF}(Y;{\mathbb{F}}_2)$. No filtered chain-homotopy invariant, integral orientations or absolute grading is asserted by this definition.

Comparisons of embedded diagrams permit orientation-preserving isomorphisms carrying $Y$, its two handlebodies, their oriented surface, the attaching systems and $z$ to the specified counterparts, equivalently compatible collared extensions of the surface map. Curve-factor reordering gives the corresponding product identification. Analytic data are transported by the induced symmetric-product map, including $j$, the area form, protected neighborhoods and $J_t$. The intended count identification is the actual pushforward of generators and disk classes with these transported data; its compatibility with the operator requires proof. An ambient isotopy is not assumed to act as the identity. Comparison of independently chosen perturbations likewise requires the continuation theorem.

Countable Choice

Given Spin and Spin-c structures and their spinor bundles

Complete Heegaard data have a zero-Chern Spin-c sector

Ozsvath and Szabo, Holomorphic disks and topological invariants for closed three-manifolds, §§2–3, Definition4.10, Theorem4.9 and Theorem7.5

Lipshitz, Heegaard Floer Homologies, §§2–5

The multipointed knot construction has additional knot-arc and marked-pair data and is specialized to knots in the three-sphere. Its terminology is contextual here; no closed-manifold analytic conclusion is deduced from it.

The knot-specialized pointed convention is defined independently with additional knot-arc and marked-pair data; it does not alter this closed-manifold definition.

The separate knot-specialized pointed convention

Let $G$ be a toroidal grid of grid number $n$ representing one oriented knot, and let $M_{\mathbb{O}}$, $M_{\mathbb{X}}$, and $A_G$ be the functions defined from its finite point sets and the pair-counting function $J$. Then $M_{\mathbb{O}}$ and $M_{\mathbb{X}}$ are integer-valued, and every rectangle $r$ from a grid state $\mathbf{x}$ to a grid state $\mathbf{y}$ satisfies $M_{\mathbb{O}}(\mathbf{x})-M_{\mathbb{O}}(\mathbf{y})=1-2n_{\mathbb{O}}(r)+2\mathrm{\#}(\mathbf{x}\cap \mathrm{Int}r)$ and $M_{\mathbb{X}}(\mathbf{x})-M_{\mathbb{X}}(\mathbf{y})=1-2n_{\mathbb{X}}(r)+2\mathrm{\#}(\mathbf{x}\cap \mathrm{Int}r)$. Consequently $A_G$ is integer-valued and $A_G(\mathbf{x})-A_G(\mathbf{y})=n_{\mathbb{X}}(r)-n_{\mathbb{O}}(r)$.

The toroidal grid, its states, and the Maslov/Alexander formulas using $J$

Every toroidal grid diagram representing a knot is an admissible $2n$-pointed Heegaard diagram in the stated knot Floer convention: each nonzero periodic domain with zero local multiplicity at every $O$-marking has both positive and negative region coefficients.

The toroidal grid, its one-component condition, and the $O$/$X$ markings

The definition of multipointed Heegaard admissibility

Let $U$ be the standard oriented unknot and take its genus-zero one-pointed Heegaard diagram: $\mathrm{\Sigma }=S^2$, each handlebody is a three-ball, there are no $\alpha$- or $\beta$-curves, and one $w$-point and one $z$-point are joined by trivial arcs in the two balls. The associated-graded hat chain complex has one generator, zero differential, and homology of rank one over ${\mathbb{F}}_2$. This is a computation for the stated diagram; it makes no invariance claim for other diagrams.

The pointed Heegaard diagram and associated-graded hat complex

The oriented unknot in the standard three-sphere

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

Manolescu, An Introduction to Knot Floer Homology, §3.5

Assume countable choice. Let $G_0,G_1$ be finite planar grid diagrams, with vertical segments over horizontal segments and orientation from $O$ to $X$ in rows and $X$ to $O$ in columns, representing the same oriented knot in the standard oriented three-sphere. A cyclic permutation moves the first row or column to the last position. A row commutation exchanges adjacent rows when the closed intervals between their respective $O$- and $X$-mark abscissae are disjoint or one lies in the interior of the other; a column commutation is defined by the same condition on the intervals between marking heights. A stabilization chooses a marked cell and temporarily removes its marking together with the other markings in its row and column. It splits that row and column into two, places the three removed markings, and adds one new $O$-mark and one new $X$-mark so that every row and column again has one of each. In the four-cell subdivision of the chosen cell, exactly three cells are marked: the chosen marking type appears twice, the other type once, and any one corner may be the empty cell; the two other markings removed from the original row and column are placed outside this block in the new row and column. A destabilization is the inverse operation. Then a finite sequence of cyclic permutations, commutations, stabilizations, and destabilizations carries $G_0$ to $G_1$.

Planar and toroidal grid conventions and oriented knot diagrams

Finite Reidemeister sequences for ambient-isotopic oriented knots

Countable choice

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, Theorem B.4.1

Assume countable choice. Let $\mathcal{H}$ be any admissible $2k$-pointed Heegaard diagram for an oriented knot in $S^3$, with transverse attaching curves, product tori, and marked points as specified in the pointed-data definition. Admissibility means every nonzero periodic domain with all $w$-multiplicities zero has both positive and negative region coefficients. There is a smooth path $\mathbb{J}=(J_t{)}_{t\in [0,1]}$, constant near its endpoints and chosen among perturbations for which all holomorphic-strip representatives used in the differential have nonnegative local multiplicity at every marked point, such that the linearized Cauchy–Riemann operator is surjective for every Maslov-index-$1$ strip class with all $w$-multiplicities nonnegative. Each corresponding moduli space modulo translation is finite. For every Maslov-index-$2$ class with all $w$-multiplicities nonnegative, the compactification of the one-dimensional moduli space modulo translation has as codimension-one boundary broken pairs of index-one strips meeting at an intermediate generator and disk bubbles whose entire boundary lies on one of the product tori (the boundary-degeneration strata). Gluing gives a collar at every broken end, and the boundary-degeneration contributions cancel modulo two in the differential-square count. No sphere bubbling or escape through a nonzero periodic domain gives an additional codimension-one end. Consequently the disk-count differential in the pointed-data definition squares to zero and preserves its Alexander filtration.

Admissibility, disk classes, basepoint multiplicities, and the Maslov index

Ozsvath and Szabo, Holomorphic disks and topological invariants for closed three-manifolds

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, Chapter 16

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

Assume ACω. Let a finite pointed Heegaard multidiagram have actual complete handlebody disk systems for closed oriented three-manifolds, with a basepoint in every one-system cut component. Ordered polygons use distinct labeled, pairwise transverse attaching systems; when an operation repeats a geometric system, use its separately labeled small transverse auxiliary copies. For the full hat operations impose global weak multidiagram admissibility: every nonzero full-curve multiperiodic domain vanishing at all basepoints has both signs. For operations in a specified total Spin-c family, ordinary weak admissibility is also allowed: every nonzero formal periodic domain difference in the capped Spin-c kernel, normalized at every mark and with Chern evaluation zero, has both signs. The same formal sector condition is imposed on each relevant boundary pair, including intended sectors with no generators. In this ordinary weak branch all basepoint multiplicities are fixed to zero, the required arities and compact auxiliary parameter families are finite, and counts use the corresponding fixed actual index bands. Strict all-mark normalized strong admissibility is a further allowed branch; it is not substituted for ordinary weakness. For a single diagram the full-module admissibility convention tests all intended Spin-c sectors, including empty sectors, and implies the required global weak domain condition.

In positive symmetric rank choose, among the auxiliary data, an arbitrarily small pointed Hamiltonian representative with no triple product-torus intersection, following its pair generators and transporting its domains, actual Spin-c labels and admissibility. Fix that representative. Compatible generic source-dependent perturbations, coherent over the required conformal faces and auxiliary parameter families, then make the disk and polygon spaces entering the hat differential, continuation, handleslide, stabilization and surgery operations regular with their actual expected dimensions. The common protected tensor and arbitrary constant product-angle end charts can be prescribed, and coherent perturbations can be arbitrarily small about a supplied compatible reference model. Auxiliary polygon and anchored-continuation parameter families here keep the autonomous end paths fixed and regular on the required energy budget; regular compatible parameter-boundary data may be retained. No slice-regularity assertion is made for arbitrary parameter-varying autonomous end paths. This does not assert regularity by almost-complex variations alone on an arbitrary fixed triple, or invariance under the preliminary curve choice.

Dimensions include every allowed conformal and auxiliary family parameter. Quotient only actual source automorphisms: longitudinal translation acts freely on autonomous nonconstant two-ended strips; an anchored source-dependent continuation strip and a three-marked triangle have no translation quotient. Every total zero-dimensional space used in a mod-two count is finite, across all its literal realized classes and with its specified labels. The one-dimensional spaces entering the chain-map, composition and homotopy identities compactify to actual compact one-manifolds. At fixed data their boundaries are precisely the matching rigid polygon–polygon conformal splittings and strip–polygon breakings. In an auxiliary family the broken configurations are the rigid joint fiber configurations at one common parameter, together with the prescribed parameter-boundary strata; they need not be products of individually rigid parameterized polygon spaces. Every such broken end has an exhaustive gluing collar. For the zero-basepoint operations no nonconstant sphere or one-torus boundary disk survives; thus the specified boundary-degeneration terms vanish, and there is no further escape or uncontrolled periodic-domain end. Boundary parity gives the differential square-zero identity and the polygon chain-map, composition and homotopy identities.

At symmetric rank zero retain the original empty systems and use the point target directly: the differential is zero, the matching constant triangle is counted once, higher rigid operations vanish, and the conformal and parameter boundary identities are the actual source-space identities. No triple-torus exclusion or free translation quotient is imposed on that point map.

This is the analytic count theorem. It does not construct a subordinate surgery diagram, prove that a named top generator is a cycle, identify the homology of a surgery total complex with a filling, or prove Heegaard-move invariance. Those remain duties of the respective operation constructions.

Hat Heegaard Floer complex of a closed three-manifold

Complete Heegaard data have a zero-Chern Spin-c sector

Pointed transverse multi-diagrams admit small triple-free curve representatives

Globally weak zero-marked polygon counts have the actual A-infinity boundaries

Strictly admissible triple-free polygons have full-marked low-dimensional counts