Grid triples, link cobordisms in \CPbar and the computation of Heegaard-Floer TQFT

Introduction

A recurring theme in Heegaard–Floer theory is the passage from invariants of individual three-manifolds or knots to maps associated with cobordisms. Ozsváth and Szabó’s original construction associated Floer groups to closed three-manifolds and, through knot Floer homology, to knots in three-manifolds [27, 26, 25]; Rasmussen’s early work provided a closely related viewpoint on the knot filtration [29] and Eftekhary extended the definition to knots that are not null-homologous [Eft05]. The theory rapidly developed into a collection of invariants sensitive to surfaces, concordance, the Thurston norm, and four-dimensional topology; see, for example, [24, 22]. From this perspective, understanding the maps induced by cobordisms is not secondary to the theory: it is part of the mechanism by which the invariants interact with topology.

A second fundamental development concerned naturality. Heegaard Floer groups are originally defined from auxiliary Heegaard diagrams, and constructing a TQFT requires canonical control over the identifications associated with changes of diagram. Juhász established functoriality for sutured cobordisms [15], while the naturality problem for Heegaard Floer homology and the associated mapping class group actions was developed systematically by Juhász, Thurston and Zemke [14] and independently by Alishahi and Eftekhary [3]. More recently, a decoration-independent version of the theory was constructed in [9], producing a Heegaard–Floer TQFT for oriented link cobordisms in which the auxiliary decoration is no longer part of the final invariant. These results made it possible to regard the Floer groups as genuinely functorial objects rather than groups defined only up to unspecified isomorphism. These theories have since been used in a variety of four-dimensional applications, including concordance maps, slice disks, ribbon concordance, bridge-type questions and different notions of unknotting bounds [12, 16, 34, 13, 2, 11, 10].

Geometric Floer constructions, which are by nature deeply analytic, become truly computational only after the analytic data have been translated into sufficiently rigid combinatorial structures. Grid diagrams provide perhaps the most successful example of such a translation. Their use in knot theory goes back well before Floer homology; see, for instance, Cromwell’s work on arc presentations [6]. In Heegaard Floer theory they became the basis of the combinatorial descriptions of knot and link Floer homology developed by Manolescu, Ozsváth, Sarkar, Szabó and Thurston [20, 21], and later organized systematically in [28]. In a grid diagram the generators become permutations, the relevant holomorphic disks become empty rectangles, and Maslov and Alexander gradings become finite counting functions. This replacement of analysis by planar combinatorics was strong enough not only to compute the homology but also to support further constructions involving skein triangles, cobordism maps and related operations; see, among others, [18, 19, 31, 32]. Sarkar’s polygon index formulas [30] provide another bridge between the analytic and combinatorial sides of the theory, and Eftekhary’s comparison between sign assignments extends the equivalence to the theory with integer coefficients [8].

The question motivating this paper is whether a comparable phenomenon occurs for link-cobordism maps arising from Heegaard triples. Ordinary grids are adapted to pairs of attaching-curve systems and therefore to differentials. Triangle maps require three systems of curves simultaneously. Triple grid diagrams supply precisely such a configuration. Blackwell, Gay and Lambert-Cole used them to construct Lagrangian surfaces in $\mathbb{CP}^{2}$ [4]; see also Blackwell’s thesis [5]. Related lattice and hexagonal descriptions of curves and surfaces in $\mathbb{CP}^{2}$ appear, for example, in [36]. A triple grid consists of three families of curves of slopes $0$, $\infty$ and $-1$ on a torus. Any two families form an ordinary grid diagram, while the three families determine the oppositely oriented projective-plane cobordism

Thus the same elementary diagram simultaneously records three boundary links, an embedded surface joining them, and the Heegaard triple from which the Floer cobordism map is defined. This paper studies the Heegaard–Floer meaning of that diagram.

The basic regions of a triple grid are triangles $L_{i,j}$ and $U_{i,j}$ of two orientation types, as well as hexagons $H_{i,j}$ for $i,j=1,\ldots,n$. A nice domain is a compatible sum of $n$ Euclidean triangular sheets satisfying a geometric separation condition: their graphs over a common source triangle are pairwise disjoint. Each embedded triangle may of course be constructed as a union of basic regions. We give a completely combinatorial computation of the triangle map associated with the two-variable link Floer complex and its blocked specializations.

The first statement follows from the Maslov-index and fat-diagonal formula of Sarkar [30]: we show that an index-zero contributing domain has zero intersection with the fat diagonal, so a holomorphic representative separates into $n$ genuine triangular sheets. The separation condition is itself combinatorial. If the lifted sheets are $A_i+\lambda_iT$, where $T$ is the unit right triangle and $\lambda_i$ is the signed leg length, it is equivalent to

Proposition 4.4 also computes the Maslov index by counting lattice points in these difference triangles.

The $\operatorname{Spin}^{c}$ structure associated with a triangle class can also be read directly from its domain. If the formal domain associated with a triangle class $\psi$ is

we define

Combining this topological information with the absolute grading formula for link-cobordism maps [33] gives an explicit bidegree for every $\operatorname{Spin}^{c}$-summand.

For a knot $K\subset S^3$, let $\mathcal{I}(K)$ denote the torsion-free quotient of its doubly based two-variable knot Floer homology over $\mathbb{A}=\mathbb{F}[u,v]$. Choose a monomial-ideal realization $\mathcal{I}(K)\cong\mathcal{J}(K)[\eta(K)]$, where the shift $\eta(K)$ is added to the usual monomial bidegree of the monomial ideal $\mathcal{J}(K)\subset\mathbb{A}$. The grading computation gives the following consequence of Theorem 4.6.

We also study a concrete arithmetic family. For integers $n,p,s$ we define affine grid links $K(n;p,s)$ by placing the two markings in each column according to affine permutations. Their elementary symmetries, component numbers and distinguished grid gradings can be computed explicitly. Under suitable invertibility assumptions the three boundary diagrams of the associated triple grid are again affine and take the form

respectively, where $r=(p-1)^{-1}\pmod n$. The subfamily $K(n;p,pq)=T_{q,n-pq}$ recovers torus links and produces explicit cobordisms between families of torus knots and links. These examples provide a testing ground in which the surface topology, the combinatorial triangle enumeration, the $\operatorname{Spin}^{c}$ labels and the grading constraints can all be calculated simultaneously.

The paper is organized as follows. We first recall the multi-based link Floer complexes, decorated link cobordisms and their triangle maps. We then introduce pointed triple grids and compute the topology of the associated surfaces. The next sections develop nice domains and the difference-number description of $\operatorname{Spin}^{c}$ structures, and prove the combinatorial formula for the triangle maps, including the geometric separation test and finiteness in each fixed $\operatorname{Spin}^{c}$ summand. We then derive the absolute grading formula and its consequences for nonzero torsion-free summands. Finally, we study affine grids and the resulting torus-link families.

Acknowledgements. I am deeply grateful to my PhD advisor, Professor Eaman Eftekhary, for his kind and generous support throughout my doctoral research. His insight and guidance were essential to the results of this work, and his invaluable help in writing and editing the manuscript greatly improved its presentation.

Heegaard-Floer TQFT for link cobordisms

Pointed links, link cobordisms and trisection diagrams

Let $L$ be an oriented link in a closed $3$-manifold $Y$ together with a finite collection of points on it, with at least one point on every connected component of $L$. Considering $L$ as a $1$-manifold, we can choose neighborhoods around each point on $L$. These neighborhoods give us a collection of disjoint oriented arcs on $L$ with the orientations induced by $L$. Let $w_i$ and $z_i$ for $i=1,2,\ldots,k$ be respectively the initial and terminal points of these arcs. By a pointed link $\mathcal{L}$ we mean a triple

defined in this way. To every pointed link $\mathcal{L}\subset Y$ one can associate a pointed Heegaard diagram $H=(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{w},\boldsymbol{z})$ where $\Sigma$ is a separating closed surface in $Y$, and $\boldsymbol{\alpha},\boldsymbol{\beta}$ are collections of disjoint simple closed curves bounding disks on the two sides of $\Sigma$ which, together with $\Sigma$, cut $Y$ into disjoint $3$-balls and $L\cap\Sigma=\boldsymbol{w}\cup\boldsymbol{z}$ where each of $\boldsymbol{w}$ and $\boldsymbol{z}$ contains precisely one basepoint in each connected component of $\Sigma-\boldsymbol{\alpha}$ and $\Sigma-\boldsymbol{\beta}$. Also, given a pointed Heegaard diagram $H=(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{w},\boldsymbol{z})$ with $\#\boldsymbol{w}=\#\boldsymbol{z}$, one can construct a sutured manifold that determines an oriented pointed link $\mathcal{L}_H$ in a closed $3$-manifold $Y_H$. For the multi-based Heegaard-diagram convention, see [35 Definition 3.1]; for the sutured-manifold viewpoint, see [3 Section 2.1] and [1].

Heegaard-Floer TQFT for link cobordisms

Given a pointed trisection diagram $\mathcal{H}=(\Sigma,\alpha,\beta,\gamma,\mathbf{w},\mathbf{z})$, define

then each $\mathcal{H}_i$ is a Heegaard diagram for a pointed link $L_i$ in a 3-manifold $Y_i$. In the rest of this paper, we sometimes refer to pointed trisection diagrams simply as trisection diagrams.

Given a trisection diagram $\mathcal{H}$ one can construct a smooth cobordism $X_{\mathcal{H}}$ which is a compact oriented 4-manifold with boundary $-Y_0\sqcup-Y_1\sqcup Y_2$ and a smooth surface $S_{\mathcal{H}}\subset X_{\mathcal{H}}$ with boundary $-L_0\sqcup-L_1\sqcup L_2$ in the following way. Let $U_0$, $U_1$ and $U_2$ denote the compression bodies determined by the surface $\Sigma$ and $\gamma$, $\alpha$ and $\beta$ respectively. The basepoints $\mathbf{w}$ and $\mathbf{z}$ determine a union of oriented arcs in $U_i$, which is denoted by $T_i$. We denote the end-points of $T_i$ on $\partial U_i=\Sigma$ by $\mathbf{w}_i$ and $\mathbf{z}_i$. Let $\triangle$ denote a triangle with vertices $v_0$, $v_1$, $v_2$ and edges $e_0$, $e_1$, $e_2$ so that $e_i$ connects $v_{i+1}$ to $v_{i-1}$ for every $i\in\mathbb{Z}/3$. Define

We use the standard Heegaard-triple orientation $\partial X_{\alpha\beta\gamma}=-Y_{\alpha\beta}\sqcup-Y_{\beta\gamma}\sqcup Y_{\alpha\gamma}$: over the interior of the parameter triangle its sheets are $\mathbf{z}\times\triangle$ and $-\mathbf{w}\times\triangle$.

Then $X_{\mathcal{H}}$ will be a cobordism from $Y_0\sqcup Y_1$ to $Y_2$ and $S_{\mathcal{H}}$ will be a cobordism from $L_0\sqcup L_1$ to $L_2$. We denote these cobordisms by

Following [35 Definitions 2.1–2.4], a decoration of a link cobordism $(X,S)$ with multi-based boundary links consists of a properly embedded 1-manifold $A\subset S$ (the dividing set) and a partition of the regions of $S\setminus A$ into $S_w$ and $S_z$, which meet along $A$. On each boundary link, the points of $A\cap\partial S$ lie between consecutive basepoints, so that every component of $L_i\setminus A$ contains exactly one basepoint; the $w$-basepoints lie on $S_w$ and the $z$-basepoints on $S_z$. Zemke also specifies a compatible coloring of the basepoints and the components of $S\setminus A$. Here we use the two-color specialization that identifies all $w$-variables with $u$ and all $z$-variables with $v$. We write $\mathcal{F}=(S,A)$ for this decorated surface, with its coloring understood.

The formula above for $S_{\mathcal{H}}$ specifies the underlying surface and its boundary basepoints. In order to specify the dividing set $A_{\mathcal{H}}$, let $p_i$ denote the union of the midpoints of the arcs in $T_i$ and set

The triangle formulas below apply to a fixed decorated trisection for which the pointed triple diagram represents the corresponding map. Changing the dividing set can change the link-cobordism map; compare the distinct band decorations in [35 Section 6.1]. Nevertheless, it is shown in [9] that a substantial part of the information encoded in such cobordism maps remains invariant under decoration change.

For a fixed underlying pair $(X,S)$, the pointed trisection diagram is not unique. Stabilization, destabilization, perturbation and deperturbation are among the moves used to compare such diagrams; see [3 Sections 3–4] for the related diagram and Cerf moves in tangle Floer theory. Diagram moves preserving the decorated, multi-based cobordism induce the usual identifications of its map. Moves changing the number of basepoints require the quasi-stabilization maps discussed below; they are not silently identified with isomorphisms of the collapsed complexes. When comparing decorated cobordisms one must also keep track of the dividing set and the boundary basepoints. We make no identification of maps associated with different dividing sets in this paper, and refer the reader to [9] for a complete discussion.

Heegaard-Floer TQFT

We use the two-variable specialization of the multi-based link complex in [35 Remark 2.14 and Section 3.3]. Let $\mathcal{L}=(L,w,z)$ be a pointed link in a 3-manifold $Y$, and $\mathfrak{s}\in\operatorname{Spin}^{c}(Y)$ be a $\mathfrak{s}$ structure on $Y$. Let $H=(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},w,z)$ be a corresponding strongly $\mathfrak{s}$-admissible Heegaard diagram and $C^{-}_{H,\mathfrak{s}}=\operatorname{CFL}^{-}(Y,\mathcal{L},\mathfrak{s})$ denote the link Floer chain complex associated with $H$ and $\mathfrak{s}$, which is freely generated by the set $\mathcal{S}(H,\mathfrak{s})\subset\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}$ of generators representing the $\operatorname{Spin}^{c}$ structure $\mathfrak{s}$, over the polynomial ring $\mathbb{F}[u,v]$. Throughout, $\mathbb{F}=\mathbb{F}_{2}$. We retain the superscript $-$ for this fully unblocked two-variable complex, and reserve $\circ$ for its $z$-blocked specialization. The differential $d^{-}_{H,\mathfrak{s}}$ is defined as follows. Given a pair of generators $\mathbf{x},\mathbf{y}\in\mathcal{S}(H,\mathfrak{s})$ let $\pi_{2}^{j}(\mathbf{x},\mathbf{y})$ denote the set of homotopy classes of Whitney disks connecting $\mathbf{x}$ to $\mathbf{y}$ with Maslov index $j$. Define

where $\mathcal{M}(\phi)$ denotes the moduli space of $J$-holomorphic representatives of $\phi$ for a generic path of almost complex structures $J$ which is suppressed in the notation, and

where $n_{w}(\phi)$ is the coefficient of the region containing $w$ in $D(\phi)$. Define $n_{z}(\phi)$ similarly. We denote the homology of this chain complex by $\operatorname{HFL}^{-}(Y,\mathcal{L},\mathfrak{s})$ which is an $\mathbb{F}[u,v]$-module. The latter homology group is an invariant for the pointed link $(Y,\mathcal{L},\mathfrak{s})$. Let $\operatorname{HFL}^{-}(Y,\mathcal{L})$ denote the direct sum of these homology groups for all $\operatorname{Spin}^{c}$ structures:

Note that different admissible Heegaard diagrams may be needed for computing the latter group.

Following the relative grading formulas in [35 Equation (2)], we can define relative homological gradings using the basepoints in $w$ and $z$ in the following way. The homological bidegrees of $u$ and $v$ are defined to be $(-2,0)$ and $(0,-2)$ respectively. Together with the bigradings of the generators, this gives a bigrading on the chain complex $(C^{-}_{H,\mathfrak{s}},d^{-}_{H,\mathfrak{s}})$. The differential drops the bigrading by $(1,1)$. Note that the bigrading on the link Floer homology is usually defined in a slightly different (but equivalent) way. All grids considered below represent links in $S^{3}$, where we use the absolute normalizations specified in § 6. Define

and denote the homology of the chain complex $\operatorname{CFL}^{\circ}(Y,\mathcal{L},\mathfrak{s})$ by $\operatorname{HFL}^{\circ}(Y,\mathcal{L},\mathfrak{s})$. Setting $v=0$ is a chain-level specialization: precisely the disks with $n_{z}=0$ remain. It is not, in general, interchangeable with taking homology and then tensoring. The analogous $w$-blocked complex is obtained by setting $u=0$. A cycle or a nonzero homology class in one of these specializations need not have the same property in the fully unblocked complex. All polynomial-valued triangle maps are taken in a fixed $\operatorname{Spin}^{c}$ summand; an unrestricted sum over $\operatorname{Spin}^{c}$ structures is not asserted to be polynomial.

For the link Floer complex and its naturality, see [35 Sections 3.3–3.4]; compare the tangle Floer naturality result [3 Theorem 1.1].

Let $\mathcal{H}=(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\gamma},\mathbf{w},\mathbf{z})$ be a trisection diagram equipped with compatible dividing data $\mathcal{A}_{\mathcal{H}}$, and let $\mathfrak{t}\in\operatorname{Spin}^{c}(X_{\mathcal{H}})$ be such that $\mathcal{H}$ is $\mathfrak{t}$-admissible. Let $(Y_i,L_i)$ denote its boundary pointed links and put $\mathfrak{t}_i=\mathfrak{t}|_{Y_i}$. The holomorphic polygon construction is given in [3 Section 2.5], and the $\operatorname{Spin}^{c}$-refined triangle map in [27 Section 8.3]. Under the fixed decorated identification assumed here, the associated triangle maps on the link Floer complexes are

where tensor products and maps are taken over $\mathbb{F}[u,v]$ for $\star=-$ and over $\mathbb{F}[u]$ for $\star=\circ$. For generators $\mathbf{x}_0$ and $\mathbf{x}_1$, the formulas, with the output generator also summed over, are

Here $\pi_2^0(\mathbf{x}_0,\mathbf{x}_1,\mathbf{x}_2)$ is the set of Maslov-index-zero Whitney triangle classes, $\mathfrak{s}(\Delta)=\mathfrak{s}_w(\Delta)$ is the $w$-based $\operatorname{Spin}^{c}$ structure represented by $\Delta$, and $\mathcal{M}(\Delta)$ is the moduli space of holomorphic representatives. Composing the natural map from the tensor product of homologies to the homology of the tensor-product complex with the induced triangle map gives the bilinear homology map

All these maps retain the choice of dividing data in their notation; our later combinatorial formula concerns the $\circ$-triangle map for the fixed decorated trisection.

Admissible domains for pointed trisection diagrams

Let $\mathcal{H}=(\Sigma,\boldsymbol{\lambda}_0,\boldsymbol{\lambda}_1,\boldsymbol{\lambda}_2,\mathbf{w},\mathbf{z})$ be a pointed trisection diagram. Fix generators $\mathbf{x}_i\in\mathbb{T}_{\lambda_i}\cap\mathbb{T}_{\lambda_{i+1}}$ for $i\in\mathbb{Z}/3$ and a Whitney triangle class $\Delta\in\pi_2^0(\mathbf{x}_0,\mathbf{x}_1,\mathbf{x}_2)$. Then, there is a domain $D(\Delta)$ which is a linear combination of the connected components in $\Sigma-\lambda_0-\lambda_1-\lambda_2$ (called basic domains) representing the image of $\Delta$ on the surface in the cylindrical reformulation of Heegaard Floer theory [17]. By positivity of holomorphic domains, in order for $\Delta$ to correspond to a non-empty moduli space $\mathcal{M}(\Delta)$ of $J$-holomorphic representatives, the coefficients of $D(\Delta)$ must be non-negative (provided that the path $J$ of almost complex structures is standard near the marked points). Additionally, the coefficients around every intersection must satisfy a specific equation, defined as follows (see [30 Section 4]).

Let $(\lambda,\lambda')\in\lambda_i\times\lambda_{i+1}$ with $i\in\mathbb{Z}/3$ and fix $P\in\lambda\cap\lambda'$. Let $U_P$ be an open neighborhood of $P$ in $\Sigma$ such that $V_P=U_P\cap\lambda$ and $W_P=U_P\cap\lambda'$ are homeomorphic to $\mathbb{R}$ and $U_P\cap\lambda_{i+2}=\varnothing$. We can define a Cartesian coordinate system using an orientation preserving diffeomorphism $\eta:U_P\to\mathbb{R}^2$ such that $\eta(V_P)$ and $\eta(W_P)$ are $x$-axis and $y$-axis respectively. Then, basic domains around $P$ are in correspondence with four quadrants of $\mathbb{R}^2$. Denote the coefficient of the domain corresponding to the $j$-th quadrant by $c_j(P)$. The point measure below is Sarkar’s average of the four local multiplicities [30 Section 2]. We define

The point measure and $E(P)$ are independent of these choices.

The alternating corner condition is $E(P)=0$ away from the generators and has value $+1$ at every intersection point which belongs to one of the three corner generators.

The above use of “admissible domain” is distinct from weak or strong admissibility of a Heegaard diagram.

Grid triples for cobordisms in $\overline{\mathbb{CP}}^2

Let $\Sigma=\mathbb{R}^2/\mathbb{Z}^2$, with its standard complex orientation, and let $\alpha,\beta,\gamma$ be horizontal, vertical, and slope $-1$ curves. We use the ordered triangle map

Each pairwise three-manifold is $S^3$. With the standard cobordism orientation, the associated four-manifold is

Indeed, after a cyclic relabeling this is the genus-one triple subordinate to a $-1$-framed unknot, as in [23 Section 6]. Here $\overline{\mathbb{CP}}^2$ denotes the projective plane with the opposite of its complex orientation. Reversing the four-manifold orientation gives the positive projective-plane trisection picture, but also changes the ordered boundary and Floer conventions; it is not the triangle map displayed above.

We replace each of $\alpha,\beta,\gamma$ by $n$ parallel copies. These redundant compressing disks do not change the underlying handlebodies or the four-manifold. They provide the additional bridge arcs and basepoints needed for the grid links. Thus the same oriented identification of $X_H$ applies to every triple grid below.

Triple grid diagrams

Using a Cartesian coordinate system on the universal cover of $\Sigma$, we represent the aforementioned diagrams in a more concrete way. Let $L_0$, $L_1$ and $L_2$ be three families of lines on the plane defined as follows:

Consider the torus $\Sigma=\mathbb{R}^2/(n\mathbb{Z})^2$ and its covering map $\eta:\mathbb{R}^2\longrightarrow\Sigma$. We can define $\alpha$, $\beta$ and $\gamma$ using sets $L_i'=\eta(L_i)$ with $i\in\mathbb{Z}/3$ in the following way. Using the square $(0,n]\times(0,n]$ as a fundamental domain for $\Sigma$, we define coordinates for points of $\Sigma$. In this square $L_0'$ and $L_1'$ form an $n\times n$ grid with $n^2$ unit squares and $L_2'$ forms $2n$ diagonal segments dividing each unit square into three parts, a hexagon and two right angled triangles (see Figure 1). In what follows, whenever we define an object in $\Sigma$ using this coordinate system, we extend the same definition to any object in the preimage of the original object under the map $\eta$.

In this construction all the horizontal, vertical and diagonal lines appearing in $L'_0$, $L'_1$ and $L'_2$ are closed loops on $\Sigma$, and there are $n$ curves of each kind. We call the horizontal, vertical and diagonal curves $\alpha$-curves, $\beta$-curves and $\gamma$-curves, respectively, and color them blue, red and black respectively. Denote the horizontal curve with the equation $y=i$ by $\alpha_i$, and the vertical curve with the equation $x=i$ by $\beta_i$. Note that any $\gamma$-curve on $\Sigma$ is represented by a pair of diagonal segments on the planar diagram. Therefore, we denote the union of diagonal segments with equations

by $\gamma_i$. Therefore, given a triple grid diagram, one can construct $(\Sigma,\alpha,\beta,\gamma)$ where

as above. All indices for curves in $\alpha,\beta$ and $\gamma$ will be understood modulo $n$, i.e. as elements in $\mathbb{Z}/n\mathbb{Z}$.

For $1\leq i,j\leq n$, the $(i,j)$-square of $\Sigma$ is defined as the open square $(i-1,i)\times(j-1,j)$ within $\Sigma$ and $U_{i,j}$, $H_{i,j}$ and $L_{i,j}$ are respectively the upper triangle, the hexagon and the lower triangle inside the $(i,j)$-square of $\Sigma$. Define the set of fundamental domains by

The diagonal value of the $(i,j)$-square is defined to be the remainder of $i+j-1$ modulo $n$. We also define the $i$th column of $\Sigma$ as the set of $(i,j)$-squares with $1\leq j\leq n$ and similarly, the $j$th row of $\Sigma$ as the set of $(i,j)$-squares with $1\leq i\leq n$. We also denote the intersection points of the curves in the following way:

Thus $A_{i,j}\in\alpha_j\cap\beta_i$, $B_{i,j}$ is the intersection of the vertical curve $\beta_i$ with $\gamma_{i+j-1}$, and $C_{i,j}$ is the intersection of the horizontal curve $\alpha_j$ with $\gamma_{i+j-1}$. All indices are understood modulo $n$.

Let $S_n$ be the symmetric group of the set $\{1,2,\ldots,n\}$. Given permutations $\sigma_w,\sigma_z\in S_n$, we define two sets of marked points on $\Sigma$:

We require $\sigma_w(i) \ne\sigma_z(i)$ for every $i$, so that the two sets of basepoints are disjoint. We place each $w_i$ at the center of $(i,\sigma_w(i))$-square and each $z_i$ at the center of $(i,\sigma_z(i))$-square, for $1 \le i \le n$. Moreover, we define a diagonal value function

that assigns to each marked point the diagonal value of the square containing it. Therefore, we obtain

Denote the grid diagrams obtained by removing the diagonal curves, the horizontal curves and the vertical curves by $\mathcal{H}_0$, $\mathcal{H}_1$ and $\mathcal{H}_2$ respectively. Therefore, these grid diagrams are formed by the pairs $(\alpha,\beta)$, $(\beta,\gamma)$ and $(\alpha,\gamma)$, respectively.

Equivalently, we need $(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\gamma},\mathbf{w},\mathbf{z})$ to be a pointed trisection diagram, or that each connected component of $\Sigma-\boldsymbol{\alpha}$, $\Sigma-\boldsymbol{\beta}$ and $\Sigma-\boldsymbol{\gamma}$ contains precisely one marking from $\mathbf{w}$ and one marking from $\mathbf{z}$.

Appropriateness is equivalent to requiring that both maps $i \mapsto i+\sigma_w(i)$ and $i \mapsto i+\sigma_z(i)$ are permutations modulo $n$. In particular $n$ is odd: summing either permutation condition gives $n \mid n(n-1)/2$. Since disjoint central markings exclude $n=1$, the centrally marked grids below have $n \ge3$.

We can construct the diagrams $\mathcal{H}_1$ and $\mathcal{H}_2$ by changing the fundamental domain of $\Sigma$. Let $\mathbb{D}_1$ be the parallelogram whose sides are located on lines with equations

Similarly, define $\mathbb{D}_2$ as the parallelogram whose sides are segments located on lines with equations

Then $\mathbb{D}_1$ and $\mathbb{D}_2$ are fundamental domains for $\Sigma$ and diagonal curves are not divided in these diagrams as shown in parts (c) and (d) of Figure 2.

Every triple grid diagram with appropriate marked points gives a pointed trisection diagram for a link cobordism in $\mathbb{CP}^2$:

We abuse the notation and identify $\mathcal{H}$ with the corresponding triple grid diagram.

The genus and the homology class of the decorated surface

We next record the homology class and the Euler characteristic of the decorated surface $S_{\mathcal{H}}$ determined by a triple grid $\mathcal{H}$.

Define a triply periodic domain $\mathcal{P}_0$ by setting

This triply periodic domain, illustrated in Figure 3, is in fact the primitive periodic domain in the genus-one $-1$-framed-unknot triple once the redundant $\alpha$-, $\beta$- and $\gamma$-curves are removed and just $\alpha_n$, $\beta_n$ and $\gamma_{n-1}$ are preserved. We fix $H=[P_0]\in H_2(X_{\mathcal{H}};\mathbb{Z})$ as the homology class obtained by attaching disks to the boundary of $P_0$ (i.e. to $\alpha_n$, $\beta_n$ and $\gamma_{n-1}$) in $X_{\mathcal{H}}$, so that $H^2=-1$. Let $H_{\mathrm{rel}}$ denote the image of $H$ in relative homology.

Proof. In the standard orientation the surface sheets over the parameter triangle are positive at $z$ and negative at $w$. The caps for $P_0$ may be chosen disjoint from the bridge arcs. Therefore

The left side is $-d(\mathcal{H})$, since $H\cdot H_{\mathrm{rel}}=-1$. This proves the formula, and agrees with [33 Equation (3.2)].

Proof. The periodic domain $P_0$ has hexagonal coefficients

Every marking is in a hexagon. For either marking permutation $\sigma$, let $r_i\in\{1,\ldots,n\}$ be congruent to $i+\sigma(i)$. Appropriateness makes the $r_i$ a permutation, and hence

It follows from the above computation, which is the same for $\sigma_w$ and $\sigma_z$, and the preceding proposition, that $d(\mathcal{H})=0$.

Proof. The surface has one disk $\{v\} \times\Delta$ for each of the $2n$ basepoints and one band for each bridge arc in each of the three curve families, hence $3n$ bands. Therefore $\chi(S_{\mathcal{H}}) = 2n - 3n = -n$. The formula $\chi= 2c - 2g_{\mathrm{tot}} - b$ gives the genus statements. If any boundary link is a knot, its boundary circle meets every disk in the corresponding two-colored graph, so the whole surface is connected.

Nice domains for a triple grid diagram

Let $\mathcal{H}$ be a triple grid of grid number $n$ with appropriate markings.

The theorem is proved in § 5.2. For a nice domain, following the boundary of a sheet encounters one $\alpha\beta$-corner, one $\beta\gamma$-corner, and one $\alpha\gamma$-corner before returning to the starting point. This sheetwise rule remains meaningful when projected domains overlap; such overlap is not the same as intersection with the symmetric-product diagonal.

$\operatorname{Spin}^{c}$ structures for triple grid diagrams

Write $X = X_{\mathcal{H}}$. For generators $\mathbf{x}_0,\mathbf{x}_1,\mathbf{x}_2$ in the three pairwise diagrams, the Whitney-triangle construction associates the $\operatorname{Spin}^{c}$ structures $\mathfrak{s}_w(\psi)$ and $\mathfrak{s}_z(\psi)$ in $\operatorname{Spin}^{c}(X)$; see [27, 3]. We label cobordism maps by $\mathfrak{s}_w$. The identity

is [33 Lemma 3.9]. By Proposition 3.4 these two structures agree for the grids in this paper; we write their common value as $\mathfrak{s}(\psi)$. The boundary structures on $S^3$ are unique, and $\operatorname{Spin}^{c}(X)$ is an affine space over $H^2(X;\mathbb{Z}) \cong\mathbb{Z}$.

For later reference, let $\psi_0$ be the product of the small upper triangles $U_{i,j}$ in the squares containing the $z$-markings. Appropriateness ensures that its corners are three grid states. The triangles are translates of one another, so the product of their conformal parametrizations avoids the fat diagonal and has Maslov index zero. We denote the $\operatorname{Spin}^{c}$ class represented by $\psi_0$ by $\mathfrak{v}_0$.

Proof. The displayed coefficients give

The functional $\delta$ vanishes on every domain supported on a pairwise grid: each elementary pairwise parallelogram contains equal total numbers of upper and lower triangles. This includes pairwise Whitney-disk domains and doubly periodic domains. It also vanishes on $[\Sigma]$. Modulo the pairwise periodic relations and the whole-surface domain, the triply periodic domains identify with $H_2(X;\mathbb{Z})$, generated by $\mathcal{P}_0$. Thus $\mathcal{P}-m\mathcal{P}_0$ is a sum of domains annihilated by $\delta$, proving the formula. If basepoint normalization is required, subtract row or column domains to make the individual basepoint multiplicities zero. Such normalization does not change $\delta$ or the represented class.

Let $\operatorname{Tri}(\mathcal{H})$ denote the set of all Whitney-triangle classes, with their corners allowed to vary, and put $h=-\operatorname{PD}[H]\in H^2(X;\mathbb{Z})$; in particular $\langle h,H\rangle=1$.

Proof. Triangle classes representing the same $\operatorname{Spin}^{c}$ structure differ by splicing pairwise Whitney disks; their domains have equal $\delta$, by the preceding proof. After corners have been connected by such disks, the remaining difference is a triply periodic domain $\mathcal{P}$, and

If $[\mathcal{P}]=mH$, then $\operatorname{PD}[\mathcal{P}]=-mh$, while Proposition 3.10 gives $\widetilde{\delta}(\psi+\mathcal{P})-\widetilde{\delta}(\psi)=-m$. This proves the asserted difference formula. The reference triangle class consists of $n$ upper triangles, so $\delta(\psi_0)=-n$. This proves half-integrality. Finally

and evaluation on $H$ changes the Chern number by $2m$.

An absolute calibration of this affine coordinate is given in Corollary 4.8 below.

Computation of the link cobordism map

In this section, we prove that the holomorphic triangle map associated with a grid triple may be computed combinatorially. Let us start with the index computations.

Euler measure

Let

be the domain of a Whitney triangle

In particular, its corner function is 1 at the coordinates of the relevant generator and 0 at every other pairwise intersection. For later use, we also set

Then $a_{i,j}=1$ exactly when $A_{i,j}\in\mathbf{x}_0$, $b_{i,j}=1$ exactly when $B_{i,j}\in\mathbf{x}_1$, and $c_{i,j}=1$ exactly when $C_{i,j}\in\mathbf{x}_2$; all other values are 0.

We use the Euler measure convention of [30]. The upper and lower triangles have Euler measure $1/4$, while a hexagonal region has Euler measure $-1/2$. Therefore

On the other hand,

and hence, after reindexing the sums,

By the defining corner condition, $E(A_{i,j})=1$ at the $n$ coordinates of $\mathbf{x}_0$ and is 0 elsewhere. We have proved:

Holomorphic representatives

Equip $\Sigma=\mathbb{C}/(n\mathbb{Z}+in\mathbb{Z})$ with its flat complex structure, and its symmetric product with the induced complex structure. Away from the fat diagonal, the ordered configuration cover has the product complex structure. We compute index-zero triangle counts in this split model. The representatives below are regular, so these counts may be used in the Floer construction and transported by continuation through the standard admissible almost-complex structures preserving the diagonal. Indeed the index formula and positivity prohibit negative-index holomorphic triangles; disk or sphere bubbling therefore do not give an extra end in the zero-dimensional triangle count.

Proof. Sarkar’s index formula [30 Theorem 4.1] is

where $\iota$ is intersection with the fat diagonal. Lemma 4.1 therefore gives $\iota(\psi)=0$. The diagonal is a complex divisor and the boundary tori are disjoint from it. Positivity of intersections implies that a holomorphic representative avoids the diagonal. The source triangle is simply connected, so the map lifts from $\operatorname{Sym}^n(\Sigma)$ to $\Sigma^n$ and thus splits into $n$ holomorphic maps to $\Sigma$. Each has one boundary arc on a curve of each family.

Lift a component to $\mathbb{C}$. Its boundary arcs lie on three fixed straight lines. For a point off these lines, the winding number of its boundary equals that of the polygon obtained by replacing each boundary arc by the straight segment with the same endpoints: excursions along a line do not change that winding number. The argument principle implies that the component maps with degree one onto the bounded triangle and has no image outside it. It is therefore a conformal parametrization of that Euclidean triangle. The generator condition supplies distinct attaching curves for the $n$ sheets, The product avoids the diagonal, so Proposition 4.4 gives the separation condition, proving niceness.

Suppose now $n_z(D)=0$. A right isosceles triangle with the prescribed directions and leg length $L<n$ projects injectively, since both coordinate ranges have length less than $n$. If instead $L\ge n$, its half-integral length is at least $n+\frac{1}{2}$. In the row of square centers one half-unit from its horizontal leg, its interior contains representatives of all $n$ columns. The unique $z$-marking in that row is therefore in the triangle, a contradiction. Thus $L\le n-\frac{1}{2}$, and every sheet is empty and embedded. The same argument, with $w$ in place of $z$, applies to a $w$-blocked representative. □

Proof. For each triangular summand choose its lift to $\mathbb{C}$. The Riemann mapping theorem gives a unique conformal map from the common source triangle to that lift with the three ordered vertices prescribed. Their product, followed by the symmetric-product quotient, is holomorphic and has the prescribed domain.

The linearized problem consequently splits into the problems for the individual planar triangles. Each has index zero and trivial kernel: an infinitesimal holomorphic deformation tangent to all three straight sides, with fixed vertices, pulls back to a holomorphic vector field on the disk tangent to its boundary and vanishing at three boundary points, and hence is zero. Equivalently this is the standard three-convex-corner Riemann–Hilbert problem. Index zero now gives surjectivity. Thus the product representative is regular and isolated.

Each attaching curve occurs on only one sheet. Its oriented boundary one-chain therefore fixes the corresponding lifted side, including any winding; hence the domain determines its triangular sheet matching. Any other representative avoids the diagonal and splits into those same triangles by the preceding theorem. Uniqueness of their conformal maps proves that the moduli space has one point. □

Given a Spin$^{c}$ class of triangles and having fixed the corners $x_0,x_1,x_2$, there are only finitely many candidate sheets which have index 0. The separation test (4.1) provides us with a more friendly test for computing the Maslov index, and checking whether it is zero for a domain which is a sum of triangles. Therefore, it gives a more concrete combinatorial algorithm for finding the contributing triangles.

Separation of triangular sheets and finiteness

We now give a geometric test for the index-zero condition which uses only the vertices and side lengths of the triangular sheets. Put

For a compatible collection of $n$ nice triangles, choose lifts $A_i=(a_i,b_i)\in\mathbb{Z}^2$ of their right-angle vertices and signed leg lengths $\lambda_i\in\frac{1}{2}+\mathbb{Z}$, so that their lifted triangles are $A_i+\lambda_iT$. Positive lengths correspond to triangles extending upwards and rightwards, and negative lengths to those extending downwards and leftwards. The affine maps $f_i(q)=\eta(A_i+\lambda_iq)$ parametrize the sheets with the same ordered source corners. Compatibility means that the sheets use distinct attaching curves in each family and have the prescribed corner generators. Their product defines a holomorphic Whitney triangle $\psi$ in the split model, without any initial assumption on its index.

Proof. Two sheets coincide at $q$ precisely when $A_i-A_j+(\lambda_i-\lambda_j)q\in(n\mathbb{Z})^2$. Compatibility excludes coincidences on the boundary: on each source edge the sheets lie on distinct curves of the corresponding family. If $\lambda_i=\lambda_j$, they are distinct translates and never coincide. Otherwise each interior lattice point of $R_{ij}$ gives one simple zero of their difference. Locally the discriminant of the symmetric-product quotient is the product of the squared pairwise differences. Each such zero therefore contributes two to $\iota$, with contributions adding also at simultaneous collisions of several sheets. This proves the formula for $\iota$. Lemma 4.1 and Sarkar’s index formula give $\mu=\iota$, proving all the assertions.

The test can alternatively be expressed without enumerating lattice points. For $d=\lambda_i-\lambda_j\ne0$, set

using representatives in $\{1,\ldots,n-1\}$; these are nonzero by distinctness of the vertical and horizontal curves. A collision occurs exactly when $r_x+r_y<|d|$. Equality is excluded by distinctness of the diagonal curves. Thus the pair satisfies (4.1) exactly when $|d|<r_x+r_y$. This is a finite test for niceness, independent of a prior Maslov-index computation. Together with Lemma 4.3, it determines precisely which compatible triangular collections contribute.

Proof. The separation test bounds every difference $|\lambda_i-\lambda_j|<2n$. Counting elementary upper and lower triangles in a sheet gives $\delta(\mathcal{T}_i)=2\lambda_i$, so $\delta(\psi)=2\sum_i\lambda_i$. Theorem 3.11 fixes this sum in a fixed $\operatorname{Spin}^{c}$ structure. Each signed length is therefore bounded. There are only finitely many half-integral choices and finitely many right-angle vertices modulo $(n\mathbb{Z})^2$, proving the assertion.

Individual sheets need not project injectively to $\Sigma$. For example, enlarge all the small upper triangles of $\psi_0$ to the common leg length $kn+\frac{1}{2}$, with $k\ge1$. Their corners are unchanged, and the sheets are still distinct translates, so they satisfy (4.1) and have index zero. Their individual projections are noninjective, but their difference number is $-n(2kn+1)$, so their $\operatorname{Spin}^{c}$ structures vary with $k$. In contrast, enlarging an upper and a lower sheet to this common length gives a signed-length difference $2kn+1>2n-2$, forcing a collision and positive index. Thus an increase in domain multiplicities with fixed corners and fixed $\operatorname{Spin}^{c}$ structure does not preserve index zero: for such classes the periodic-domain index formula gives

see [33 Lemma 6.10]. The fixed-summand finiteness above justifies polynomial-valued maps over $\mathbb{F}[u,v]$; it does not justify an unrestricted sum over all $\operatorname{Spin}^{c}$ structures without a suitable completion.

Torsion-free Floer ideals and grid diagrams

Set $A=\mathbb{F}[u,v]$, with $\deg u=(-2,0)$ and $\deg v=(0,-2)$. For a knot $K\subset Y$ and a $\operatorname{Spin}^{c}$ structure $\mathfrak{s}\in\operatorname{Spin}^{c}(Y)$, let $C^{(n)}(Y,K,\mathfrak{s})$ denote the two-variable knot Floer chain complex associated with $K$ decorated with $n$ pairs of basepoints, constructed from a compatible Heegaard diagram and freely generated over $A$ by the corresponding grid or Heegaard generators. Let $C(Y,K,\mathfrak{s})=C^{(1)}(Y,K,\mathfrak{s})$ denote the doubly based chain complex and define

By [2 Proposition 3.4], when $Y=S^3$ this is a rank-one torsion-free module represented by a monomial ideal after specifying its grading shift.

Given a multi-based Heegaard diagram representing a knot $K\subset Y$, a sequence of diagram identifications and quasi-stabilizations gives a chain homotopy equivalence

In the cobordism grading convention, $\deg\theta^+=\left(\frac{1}{2},-\frac{1}{2}\right)$ and $\deg\theta^-=\left(-\frac{1}{2},\frac{1}{2}\right)$. In particular, for $Y=S^3$, the torsion-free rank before reduction is $2^{n-1}$; in general it is $2^{n-1}\operatorname{rank}_A\mathcal{I}(Y,K,\mathfrak{s})$.

Let us now assume that $\mathcal{H}=(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{w},\boldsymbol{z})$ is a Heegaard triple with $|z|=|w|=n$, which is $\mathfrak{t}$-admissible for some $\mathfrak{t}\in\operatorname{Spin}^{c}(X_{\mathcal{H}})$. Let $K_0$, $K_1$ and $K_2$ denote the corresponding boundary links in the three-manifolds $Y_0$, $Y_1$ and $Y_2$, respectively, and $\mathfrak{s}_i=\mathfrak{t}|_{Y_i}$. Further assume that each $K_i$ is a knot. Then the associated cobordism map

induces a corresponding cobordism map

which takes torsion classes to torsion classes, see [9 Theorem 3.7]. Here the reduced homology groups are identified with the common kernels of the $z$-basepoint actions in the multi-based homologies. Equivalently, in the tensor identification above we use the factor $(\theta^+)^{\otimes(n-1)}$ on each boundary. The analogous reduction uses the $w$-actions and $(\theta^-)^{\otimes(n-1)}$; we retain the $\theta^+$ choice. The compatibility with basepoint actions in the cited theorem is what allows the cobordism map to restrict to these reduced groups.

The decoration-independence argument of [9 Theorem 3.11 and Section 4], with $w$ and $z$ interchanged for this choice of reduction, gives equality on the $u$-multiple of the reduced source. More explicitly, if $F$ and $F'$ are the maps associated with two decorations, then for every reduced input $x$,

Their difference is therefore $u$-torsion in the target and vanishes after passage to its torsion-free quotient. Moreover, bilinearity sends an input with a torsion factor to torsion, so the quotient map is well-defined. Thus the induced map

is well-defined and only depends on $\mathfrak{t}$ and the surface $S_{\mathcal{H}}$ determined by $\mathcal{H}$ (and not its decoration).

When $\mathcal{H}$ is a grid triple, $Y_0,Y_1,Y_2\simeq S^3$ and there is only one $\operatorname{Spin}^c$ structure on each boundary component. Moreover, each $\mathcal{I}(K_i)=\mathcal{I}(Y_i,K_i,\mathfrak{s}_i)$ can be represented by a monomial ideal. This gives the maps

The absolute grading formula for the grid diagrams uses the formulas

For a grid of size $n$ these are related to the symmetric cobordism normalization of [33] by

We keep the conventional grid Alexander grading

in § 7.2; the symmetric multi-based Alexander grading is

For a term from an index-zero triangle $\psi$, its raw grid bidegree is

The cobordism degree, as computed by Zemke, is obtained by subtracting $(n-1)/2$ in each coordinate, since there are two inputs and one output. Write $c_t=\langle c_1(t),H\rangle$, an odd integer. Since $H^2=-1$, we have $c_1(t)^2=-c_t^2$. For the midpoint decoration each colored subsurface is a union of $n$ disks with collars. Thus

Together with $\chi(X)=0$, $\sigma(X)=-1$, and $[S_H]=0$, the grading-change theorem of Zemke [33 Theorem 1.4] gives

The same shift applies to a nonzero $z$-blocked summand with these absolute gradings. If $t=t_0+mh$,

For the reduced map, the bidegree is $(d_t+\rho_w,d_t+\rho_z)$; in the tensor choices above $\rho_w=-\rho_z=\frac{n-1}{2}$. With these choices, we obtain the map

With the above degree normalization, we also have the monomial-ideal identifications

where $\mathcal{J}(K)\subset\mathbb{A}$ is a bigraded ideal with bigrading induced from $\mathbb{A}$ and $\mathcal{J}(K)[i,j]$ means that the degree of a monomial is increased by the bidegree $(i,j)$.

Proof. We have already proved most of the claims, and only the last part needs a justification. After tensoring with $\mathbb{F}(u,v)$, each module is one-dimensional. A nonzero map is multiplication by an element of $\mathbb{F}(u,v)$. Homogeneity for the two independent variable degrees makes that element a Laurent monomial (with scalar coefficient one over $\mathbb{F}_2$). Let us assume that the monomial is $u^av^b$. It then follows that $G_{H,t}=u^av^b$ as a bigraded map. Accounting for the three ideal shifts shows that the bidegree of $G_{H,t}$ is

This completes the proof.

Negative exponents are permitted; the containment can equivalently be written after clearing denominators. No containment is asserted for a zero reduced map.

The distinguished corner generators

Every ordinary grid diagram has canonical corner states determined by its marked squares. Let $\mathbf{x}^{NE}$, $\mathbf{x}^{SW}$, $\mathbf{x}^{NW}$, and $\mathbf{x}^{SE}$ denote the states occupying the northeast, southwest, northwest and southeast corners of all $z$-marked squares, respectively.

In the $z$-blocked complex, $\mathbf{x}^{NE}$ and $\mathbf{x}^{SW}$ are cycles; see [28 Proposition 6.3.1 and Remark 6.3.2]. When the distinguished class realizes the extremal non-torsion Alexander grading in the appropriate knot complex, that grading determines $\tau$.

To identify corners in all three ordered grids, use the orientation-preserving coordinates

The first curve family is horizontal and the second vertical in each chart. For a marking $z$ in the $(i,j)$-square, the triangle

has vertices

They are respectively the northeast, northeast, and northwest corners of the marked cells in these three charts. Its leg length is $3/2$. The triangles $T_z$ give a class

The sheets are distinct translates with equal signed lengths, so Proposition 4.4 and Lemma 4.3 show that $\psi_z$ is a nice contributing class. Put $t_z=\mathfrak{s}(\psi_z)$.

Proof. The preceding construction and Lemma 4.3 give the contribution $v^n$. If $\psi$ is another contributing index-zero class with the same corners and $\operatorname{Spin}^{c}$ structure, homogeneity in (5) gives $n_w(\psi)=0$ and $n_z(\psi)=n$. The $w$-blocked version of the embedding argument in § 5.2 makes all its sheets embedded, with leg lengths less than $n$.

At each $A_{i,j}$ in the first corner state, the other two vertices are fixed by its unique vertical and horizontal boundary curves: they must be $B_{i,j-1}$ and $C_{i-1,j}$. Let $t$ denote the signed displacement from this $A$-corner to its $B$-corner in the vertical direction. Then $t\equiv-3/2\pmod n$. Since $\lvert t\rvert<n$, the only choices are $-3/2$ and $n-3/2$. Their difference numbers are $-3$ and $2n-3$, respectively: a triangle of signed leg $t$ has $\delta=2t$, as follows by counting its upper and lower elementary triangles. If $k$ sheets use the second choice,

Theorem 3.11 and the fixed $\operatorname{Spin}^{c}$ structure force $k=0$. Thus every sheet is the specified $T_z$, proving uniqueness of the contributing class and the coefficient formula.

The corner $J$-formulas give $M_z(\mathbf{x}^{NE})=1-n$ and $M_z(\mathbf{x}^{NW})=0$. Hence its raw $z$-degree is $0-2(1-n)-2n=-2$. Equation (6), using the nullhomology of the surface, makes the $w$-degree equal to it. The normalization change subtracts $(n-1)/2$ from each raw degree.

Proof. In the charts above, the small-upper reference $\psi_0$ has corners $x_0^{NE}$, $x_1^{NW}$, $x_2^{NE}$ and $n z=0$. Its raw $z$-degree is zero. Thus its cobordism degree is $-(n-1)/2$, and (4.5) gives $c_0^2=1$. The theorem gives $c_{t_z}^2=9$. Since $\delta(\psi_z)-\delta(\psi_0)=-2n$, Theorem 3.11 gives $c_{t_z}=c_0+2$. These equations force $c_0=1$ and $c_{t_z}=3$. The general formula follows from the affine difference formula.

Remark 4.9

The theorem is a statement about a chain coefficient. It does not assert that its input states are unblocked cycles or that any selected basepoint reduction is nonzero on torsion-free homology. In particular, no bound on the least power of $u$ occurring at $x^{NW}$ in a non-torsion $z$-blocked cycle follows from this $v^n$-term alone. The nonvanishing hypothesis of Theorem 4.6 remains necessary.

A family of examples: affine grid triples

Affine grid knots and links

We begin with a family of ordinary grids which will be used throughout the examples.

The elementary grid symmetries will be useful below. Interchanging the $z$- and $w$-markings reverses the orientation of every component and replaces $s$ by $-s$. Reflection in a horizontal axis takes the mirror and replaces $(p,s)$ by $(-p,-s)$. Thus

up to the harmless cyclic row and column translations needed to restore the chosen constant term $n-1$.

Two further elementary symmetries make the ordered grid charts below explicit. Rotation through $180^\circ$, followed by translations, takes $G(n;p,s)$ to $G(n;p,-s)$ without changing its oriented link type. Thus these affine links are isotopic to their simultaneous orientation reversals. Transposing a planar grid exchanges the crossing convention as well as the coordinate axes; the combined operation preserves the ambient orientation and reverses all link orientations. Solving for its row permutation gives

Here simultaneous link reversal has been absorbed using the preceding rotation symmetry. These identities concern link types, not equality of particular corner states.

Proof. Put $m=n-pq>0$. On the toroidal grid choose each vertical arc from $z_i$ to $w_i$ to run upwards a distance $m$, and each horizontal arc from $w_i$ to $z_{i+q}$ to run rightwards a distance $q$. These are valid bridge representatives: changing to the complementary arc in a row or column uses its compressing disk and does not change the grid link. We check that their union on the torus is embedded.

A possible interior intersection of the horizontal arc beginning in column $i$ with a vertical arc must occur in column $i+k$, where $1\leq k<q$. The height of that horizontal arc relative to the bottom $z_{i+k}$ of the vertical arc is, modulo $n$,

This lies strictly between $m$ and $n$, so it is outside that vertical arc. Parallel arcs also have disjoint interiors. Consequently the entire staircase link is embedded on the Heegaard torus, and the usual small pushes of its vertical and horizontal arcs can be undone without crossings.

Its total horizontal and vertical winding numbers are $q$ and $m$: the $n$ horizontal arcs have total length $nq$, and the vertical arcs have total length $nm$. Its components are disjoint essential curves and therefore parallel. If $g=\gcd(q,m)=\gcd(n,pq)$, each has primitive slope $(q/g,m/g)$. Thus the link is the standard $T_{q,m}$, with the orientation prescribed in Definition 5.1. $\square$

Remark 5.4 Proposition 5.3 is deliberately an integer statement, rather than merely a congruence claim. A congruence $pq\equiv s\pmod n$ does not in general identify $K(n;p,s)$ with $T_{q,n-s}$; wrapping in the planar grid may change the knot type. The symmetries above nevertheless give you many further torus identifications by first replacing $(p,s)$ by one of the equivalent parameter pairs and then applying Proposition 5.3.

The distinguished bigradings of affine knots

The affine form makes an otherwise separate calculation completely explicit: the bigrading of the distinguished northeast-$z$ state. This calculation remains useful independently of the triangle-map discussion.

Assume $1 \le p < n$, $\gcd(p,n)=1$, and $0 < s < n$. Let $x^+_{n;p,s}$ denote the state occupying the northeast corner of every $z$-marked square of $G(n;p,s)$. As before, set $b=[p^{-1}s]_n \in\{1,\ldots,n-1\}$ and define

We may then extend the definition and set

for arbitrary integers $p$ and $s$, where $\gcd(n,p)=1$ and $s$ is not a multiple of $n$. The apparently asymmetric formula is invariant under cyclic row and column translations of the affine grid.

Proof. The universal northeast-corner calculation gives

For the first Maslov grading, evaluate the standard $J$-formula

on the affine permutations.

For clarity, put $r_i=[n-1-pi]_n$ and $t_i=[r_i-s]_n$, using $0\le i<n$. In the fundamental square $(0,n]^2$, the northeast state has coordinates $(i+1,r_i+1)$. Expanding the three $J$-terms gives

The single sum is $n-s$. For the pair sum, use $t_i=r_i+b$, with indices read cyclically, where $b=[p^{-1}s]_n$. Cancel the pairs common to the two cyclic orders. The remaining pairs give

Thus the explicit $J$-expansion yields

It remains only to rewrite the last count. Write $pb=s+n\kappa$, for a non-negative integer $\kappa$. Then, for $1\leq j<b$ one has

which is $\Lambda(n;p,s)$. Finally, the grid identity

reduces here to $A_{\mathrm{grid}}(\mathbf{x}^{+})=M_w(\mathbf{x}^{+})/2$.

Affine triple grids

We now return to the construction of triple grids. Start with the markings of $G(n;p,s)$ in the $\alpha\beta$ diagram and assume in addition that $p-1$ is a unit modulo $n$. Put

The order of the two attaching-curve families matters when the other two pairwise diagrams are read as ordinary grids: for an ordered pair $(\lambda,\mu)$ we draw the first family horizontally, the second vertically, and let the second family pass over the first.

Proof. Use the three orientation-preserving charts in § 6.1. In the $\beta\gamma$ chart $(X,Y)=(\frac{1}{2}-x-y,x)$, the centers of the $z$-marked cells satisfy

Solving for the row as a function of the new column gives the parameters $(-r,rs)$, up to a cyclic row translation. By the transpose symmetry,

In the $\alpha\gamma$ chart $(X,Y)=(x+y-\frac{1}{2},y)$, the new column is $(1-p)i-1$ modulo $n$. Solving the row equation gives slope parameter $-pr$. The $w$-row at a fixed new column differs from the $z$-row by $rs$, so its separation parameter is $-sr$. This proves the three link identifications.

The first cobordism is the ordered triangle map

Cyclically ordering the same triple as $(\gamma,\alpha,\beta)$ gives

Reversing an ordered pair reverses the ambient three-manifold orientation, so its link is mirrored (any simultaneous component reversal is harmless for this affine family). The mirror identities above yield the second displayed cobordism. Both are in $\overline{\mathbb{CP}}^{2}$ with three balls removed, with their indicated standard incoming and outgoing boundaries.

Proof. Here $r = 1$ modulo $n$. Proposition 5.3 gives $K(n;2,2q) = T_{q,m}$. Moreover

Now apply Theorem 5.6.

A second useful family shows that $p = 2$ is not the only case in which all three boundary knots can be identified explicitly.

Proof. The first knot is $K(n;3,6) = T_{2,n-6}$. The second is

Since $r = (2)^{*} = (1-n)/2$ modulo $n$, we may write

The output of the native cobordism has parameters

and $n - 3 = 2((n-3)/2) < n$. Proposition 5.3 therefore identifies the output with $T_{2,3}$.

For the affine family put $g = \gcd(n,s)$. Each boundary link has $g$ components. The two-step moves between $z$-markings in the bridge graph translate the column by $p^{-1}s$ or $rs$. Their orbits are precisely the residue classes modulo $g$, so the surface has $g$ components. Each has three boundary circles and $2n/g$ disks and $3n/g$ bands, hence genus $(n/g-1)/2$. In the knot cases of Corollaries 5.7 and 5.8, the surface is connected of genus $(n-1)/2$. All these surfaces are relatively nullhomologous by Proposition 3.4.

Tool Use

Assistance from GPT-6 has been crucial in certain parts of the paper. Most importantly, the current proof of Theorem 4.2 was suggested to replace a much longer proof by the author. The separation criteria of § 5.3 also grew from discussions with GPT-6. It also helped in editing the paper for better flow and proofreading, which resulted in finding several earlier mistakes in bigrading computations.