Decorated knot cobordisms and the two-color link Floer target

Let $Y_0,Y_1$ be closed smooth oriented three-manifolds, let $W$ be a compact smooth oriented four-manifold with boundary $-Y_0\bigsqcup Y_1$, where $-Y_0$ has the reversed orientation, and let $K_i\subset Y_i$ be images of smooth oriented embeddings of the circle. Every component of $Y_i$ is required to meet $K_i$; the boundary parametrizations and compatible product collars are part of the data. A decorated knot cobordism from $(Y_0,K_0)$ to $(Y_1,K_1)$ is a properly embedded smooth oriented surface $\mathrm{\Sigma }\subset W$ with boundary $(-K_0)\bigsqcup K_1$, together with one $w$-basepoint and one $z$-basepoint on each end knot and a compact properly embedded one-manifold $\mathcal{A}\subset \mathrm{\Sigma }$, which may have arc and closed-circle components. Every component of $W$ meets $\mathrm{\Sigma }$, and every component of $\mathrm{\Sigma }$ meets $\mathcal{A}$. The full dividing set $\mathcal{A}$ decomposes $\mathrm{\Sigma }$ into subsurfaces ${\mathrm{\Sigma }}_w,{\mathrm{\Sigma }}_z$, with each $w$-basepoint in the closure of ${\mathrm{\Sigma }}_w$, each $z$-basepoint in the closure of ${\mathrm{\Sigma }}_z$, and $\mathcal{A}$ meeting the boundary transversely away from the marked points. Every component of $K_i\setminus \mathcal{A}$ contains exactly one of the two marked points; in particular each boundary knot meets the dividing set in two points, separating its $w$- and $z$-intervals. The basepoint labels, dividing set, two regions and their two-color assignment are part of the data.

For an end pair $(Y,K)$, a doubly pointed Heegaard diagram consists of a closed genus-$g$ surface $S$ oriented as the boundary of the $\alpha$-handlebody, two $g$-tuples $\mathit{\alpha },\mathit{\beta }$ of pairwise disjoint simple closed curves giving a Heegaard splitting of $Y$, and points $w,z$ off the curves. The curves are complete meridian systems: each bounds $g$ pairwise disjoint compressing disks in its handlebody, and cutting along those disks gives one ball. Require $K\cap S=\{w,z\}$, transversely, with the oriented knot meeting $S$ negatively at $w$ and positively at $z$. The portions of $K$ in the two cut handlebodies are specified properly embedded arcs, one in each ball, and are boundary-parallel there: each is isotopic within that ball, relative to its endpoints, into the matching component of $S\setminus \mathit{\alpha }$ or $S\setminus \mathit{\beta }$. Their union, with the orientations inherited from $K$, is $K$. Curves from different attaching systems meet transversely in finitely many points. Each intersection point $\mathbf{x}\in {\mathbb{T}}_{\mathit{\alpha }}\cap {\mathbb{T}}_{\mathit{\beta }}$ determines a Spin-c structure ${\mathfrak{s}}_w(\mathbf{x})$ on $Y$ from the homology class of the nonvanishing gradient vector field specified by $\mathbf{x}$ and $w$. For $\mathfrak{t}\in {\mathrm{Spin}}^c(Y)$, let $CF{L}_{UV}^{-}(Y,K,\mathfrak{t})$ be generated over $R={\mathbb{F}}_2[U,V]$ by those intersection points with ${\mathfrak{s}}_w(\mathbf{x})=\mathfrak{t}$. A periodic domain is an integral two-chain in the regions of $S\setminus (\mathit{\alpha }\cup \mathit{\beta })$ whose boundary is an integral sum of the attaching curves; its capped homology class in $Y$ is denoted $[P]$, and $n_w(P)$ is its multiplicity at $w$. Restrict to diagrams strongly admissible for $\mathfrak{t}$: there is a generator $\mathbf{x}$ with ${\mathfrak{s}}_w(\mathbf{x})=\mathfrak{t}$, and for every nonzero periodic domain $P$ with $n_w(P)=0$ and $⟨c_1(\mathfrak{t}),[P]⟩=2N\ge 0$, some region of $P$ has multiplicity strictly greater than $N$. Here $c_1(\mathfrak{t})$ is the first Chern class of its determinant line bundle. The symmetric product ${\mathrm{Sym}}^g(S)$ is the quotient of $S^g$ by permutations of the factors. The associated product tori ${\mathbb{T}}_{\mathit{\alpha }},{\mathbb{T}}_{\mathit{\beta }}\subset {\mathrm{Sym}}^g(S)$ determine the differential on each Spin-c summand by

Here ${\pi }_2(\mathbf{x},\mathbf{y})$ is the set of homotopy classes of Whitney disks, $\mu$ is the Maslov index, $\mathcal{M}(\varphi )\mathrm{/}\mathbb{R}$ is the moduli space of index-one disks modulo translation for chosen generic almost-complex data, $n_w,n_z$ are their local multiplicities at the marked points, and ${\mathrm{\#}}_2$ is the finite count modulo two. This is the two-variable specialization of the colored link Floer complex. Its curvature vanishes: for a knot with one $w$- and one $z$-basepoint the curvature formula is $UV+VU=0$ in ${\mathbb{F}}_2[U,V]$, so ${\mathrm{\partial }}^2=0$. The complex has the two-coordinate filtration induced by the $w$- and $z$-multiplicities.

For a chosen $\mathfrak{s}\in {\mathrm{Spin}}^c(W)$, the Floer map indexed by $\mathfrak{s}$ has the complexes for $(Y_0,K_0,\mathfrak{s}{\mathrm{\mid }}_{Y_0})$ and $(Y_1,K_1,\mathfrak{s}{\mathrm{\mid }}_{Y_1})$ as source and target; ${\mathrm{Spin}}^c(W)$ denotes the set of Spin-c structures on $W$. The following local schematic shows the two regions separated by dividing arcs; it records decoration data and does not represent the cobordism map.

Admissible pointed Heegaard data and basepoint multiplicities

Proper surface and collar conventions

Smooth oriented knots and links in $S^3$

Spin-c structures

Zemke, Link cobordisms and functoriality in link Floer homology, §§1–3

Zemke, Quasi-stabilization and basepoint moving maps in link Floer homology, §2

Juhasz, Cobordisms of sutured manifolds and the functoriality of link Floer homology, §§4 and 8

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

The general-$Y$ two-color identity ${\mathrm{\partial }}^2=0$ is supplied by the external input [F1], under that source's closed-three-manifold, multi-based-link, Spin-c, and Heegaard-diagram hypotheses. No independent local proof of the general-$Y$ disk analysis is asserted here; the narrower local suppliers in [F2] must not be used as if they covered arbitrary $Y$.