Link Floer Homology

• Link Floer homology is a framework that enhances classical link invariants, like the Alexander polynomial, with multi-graded algebraic and categorical structures.
• It utilizes holomorphic disk counts and combinatorial grid techniques with precise sign conventions to construct bi- and multi-graded homology theories.
• The theory connects to quantum invariants, TQFT structures, and spectral sequences, enabling applications in link detection and low-dimensional topology.

Link Floer homology is a package of powerful link invariants that lift classical topological data—including the Alexander polynomial, the Thurston norm, and the fiberedness of links—into algebraic and categorical structures with deep connections to symplectic topology, gauge theory, and quantum invariants. Originally constructed by Ozsváth and Szabó via holomorphic disk counts in Heegaard diagrams, and further developed in combinatorial, algebraic, and Fukaya-category frameworks, link Floer homology has seen a rich spectrum of refinements, detection results, and structural unifications. Its extensions relate directly to Khovanov and Khovanov–Rozansky homologies and provide a fertile setting for functorial and TQFT-type constructions, as well as categorifications of the Alexander and Jones polynomials.

1. Grading Structures, Sign Conventions, and Combinatorial Foundations

Link Floer homology is defined for multi-component oriented links $L \subset S^3$ (or more generally in $3$-manifolds) as bi- or multi-graded groups. The multi-grading encodes one overall Maslov grading $M$ and Alexander gradings $A_1,\dots,A_\ell$, one for each component:

• The differential decreases the Maslov grading by $1$ and preserves Alexander gradings.
• In the analytic theory, these gradings arise from intersection data in the symmetric product of the Heegaard surface and counts of holomorphic disks passing through marked basepoints.

A subtle and crucial point is the role of sign conventions in lifting the theory from $\mathbb{F}_2$ to $\mathbb{Z}$. For a link with $\ell$ components, there are $2^{\ell-1}$ bi-graded $\mathbb{Z}$-valued "versions" of the hat flavor, corresponding to choices of orientation systems or sign assignments compatible with decompositions of domains $D = D_1 + D_2$:

$o(D_1) \wedge o(D_2) = o(D)$

Two orientation systems are "weakly equivalent" if they agree on all empty periodic domains; modulo $2$, all these theories coincide with the same $\mathbb{F}_2$ theory (Sarkar, 2010).

In combinatorial settings—most notably, grid diagrams—generators correspond to permutations in $S_N$, and the differential counts "empty rectangles", with an explicit sign assignment $s: \{\text{rectangles}\} \to \{\pm 1\}$ satisfying a key relation for Maslov index two domains:

$s(R_1)s(R_2) = -s(R_1')s(R_2')$

for two decompositions $D = R_1+R_2 = R_1'+R_2'$. This ensures the combinatorial chain complexes over $\mathbb{Z}$ (with $2^{\ell-1}$ choices of signs) correspond to those arising from holomorphic orientation data.

2. Link Floer Homology as Categorification and Quantum Categorifications

A fundamental result is that the Euler characteristic of the multi-graded hat version of link Floer homology recovers the multivariable Alexander polynomial, up to normalization:

$$\chi(\widehat{\mathrm{HFL}}(L); t) = \prod_{k=1}^\ell (t_k^{1/2} - t_k^{-1/2}) \cdot \nabla_L(t^{1/2})$$

Here, $\nabla_L$ is the Conway function—a canonical representative of the multivariable Alexander polynomial, fixing the ambiguity up to sign and monomials (Benheddi et al., 2014). This identification, achieved by modeling the Conway function via Fox calculus on rectangular diagrams and relating it explicitly to the Euler characteristic of combinatorial link Floer homology, provides a full categorification of the multivariable Alexander polynomial.

Recent advances generalize this categorification: the framework of (Aganagic et al., 2023) provides "Floer-type" link homologies categorifying quantum group invariants for arbitrary Lie (super)algebras $\ ^L\mathfrak{g}$, constructed from explicit A-brane categories on symmetric products of Riemann surfaces equipped with Landau–Ginzburg potentials. For $\mathfrak{gl}_{1|1}$, the theory recovers standard link Floer homology; for $\mathfrak{su}_2$, it provides a robust categorification of the Jones polynomial, with bi-gradings and skein formulas governed by the chosen algebra and representation data.

3. Functoriality, Cobordism Maps, and TQFT Structures

Link Floer homology admits cobordism maps associated to decorated link cobordisms $(W, F^\sigma)$ in 4-manifolds. These maps are defined on "curved" chain complexes (where $\partial^2$ equals a curvature term $\omega_{L,\sigma}$), and satisfy invariance under change of auxiliary choices and composition laws for decomposed cobordisms (Zemke, 2016). Explicitly:

$$F_{(W, F^\sigma, \mathfrak{s})}: \mathcal{C}^-(Y_1, L_1^\sigma, \mathfrak{s}_1) \to \mathcal{C}^-(Y_2, L_2^\sigma, \mathfrak{s}_2)$$

The construction encompasses quasi-stabilization (adding/removing basepoints), basepoint moving maps, and band-saddle cobordisms, with the triangle maps for bands modeled via explicit holomorphic polygons in subordinate Heegaard triple-diagrams. These maps are compatible with natural gradings (Zemke, 2017), and the grading change formula enables applications to adjunction inequalities and genus bounds.

The resulting structure defines a (3+1)-dimensional TQFT, with compositions and algebraic reductions matching the "graph TQFT" of Heegaard Floer homology. Applications include the paper of ribbon homology concordances via split injection results on $\mathcal{HFL}^{-}$, and bounding the number of critical points of surfaces in a ribbon cobordism in terms of torsion orders in link Floer homology (Guth, 2021).

4. Spectral Sequences and Connections to Khovanov Theories

Through a broad framework of "Khovanov–Floer theories", one constructs spectral sequences from diagrammatic Khovanov homology (or its variants) to various flavors of Floer homology (Heegaard, symplectic, instanton) (Baldwin et al., 2015). The setup assigns to each link diagram $D$ a filtered chain complex $C(D)$ such that:

• $E_2(C)$ is canonically identified with Khovanov homology,
• $E_\infty(C)$ is a Floer-theoretic invariant (e.g., Heegaard Floer homology of the branched double cover).

Functoriality is rigorously established: every page $E_r$ ($r\geq2$) of the spectral sequence is a link invariant, and the functor extends to (smooth) link cobordisms, reproducing the correct Khovanov functoriality at the $E_2$ page.

Recent work in the unoriented setting constructs spectral sequences from (reduced) Khovanov homology of the mirror to (hat or minus versions of) (unoriented) link Floer homology, via an iterative process that refines the unoriented skein exact triangle with "modified band maps" (Nahm, 3 May 2025, Nahm, 2 Jan 2025). For knots in $S^3$, the induced spectral sequence refines knot concordance invariants and relates directly to concordance lifts such as the $t$- and $v$-invariants.

Key formulas:

• The spectral sequence is constructed from a cube of resolutions:

$$E_1 \cong \bigoplus_{f: X \to \{0,1\}} \mathrm{Kh}^-(m(L(f)))$$

• The differentials $d_k$ have $q$- and $h$-bidegrees $(2k-2, k)$, matching established Khovanov–Floer spectral sequences.

5. Detection and Structural Results

Link Floer homology and its module structure are sharp enough to detect split links: for a two-component link $L\subset S^3$, the module structure

$X: (\mathrm{HFL}(L), X^2=0)$

over $\mathbb{F}_2[X]/(X^2)$ is free if and only if $L$ is split (Wang, 2020). Sutured Floer homology provides a parallel detection result, tied to the existence of embedded spheres with odd intersection number against chosen arcs.

Rank bounds in link Floer homology, especially in the next-to-top grading, allow one to classify links with low rank (e.g., rank $\leq 8$) and show, for instance, that certain torus links (e.g., $T(2,8)$, $T(2,10)$) are detected by knot Floer homology (Binns et al., 2022).

Spectral sequence and Floer lasagna module constructions—such as in (Chen, 2022)—are used to relate cabled link Floer homology and 4-manifold topology, giving (graded) module invariants for 4-manifolds obtained by 2-handle attachments and reflecting subtle features like the vanishing of the module for links intersecting capping disks.

6. Algebraic and Categorical Models

Recent progress describes link Floer homology in terms of explicit algebraic and categorical structures:

• Every link Floer complex $CFL(Y, L)$ over $\mathbb{F}[U,V]/(UV)$ splits canonically as a direct sum of snake complexes and local systems (Popović, 2023), generalizing and extending earlier results using combinatorial or immersed-curve-based invariants.
• The arrow-sliding algorithm employed there is analogous to the simplification techniques in the bordered Floer setting (Lipshitz–Ozsváth–Thurston; Hanselman–Rasmussen–Watson), but over the full $U,V$-module structure, and works algebraically over arbitrary fields.

In addition, bordered algebra techniques allow for fast computation of link Floer homology and the associated Thurston polytope via decomposition into elementary type D or DA modules and gluing via tensor products, thus reducing analytic complexity to a tractable algebraic problem (Ozsvath et al., 2020).

7. Interplays with Hypergraphs, Quantum Invariants, and Homological Mirror Symmetry

Connections to hypergraph theory arise in the identification of the support of sutured Floer homology for Seifert surfaces of special alternating links with the hypertree lattice in a natural associated hypergraph (Juhász et al., 2011). The combinatorial polytope model not only facilitates explicit computation of Floer-theoretic invariants but also encodes the leading term coefficients of the HOMFLY polynomial via the interior polynomial of hypergraphs.

Category-theoretic generalizations to Fukaya–Seidel categories of Landau–Ginzburg models enable the realization of link Floer-type theories categorifying arbitrary Reshetikhin–Turaev quantum invariants via explicit projective resolutions and grading structures (Aganagic et al., 2023). This connects Floer homology directly with homological mirror symmetry and further unifies the toolbox for categorification, with features such as explicit equivariant gradings and coefficient categories.

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Summary Table: Key Structures and Correspondences

Construction/Feature	Algebraic/Combinatorial Input	Output or Invariant
Sign conventions (orientation system)	Grid/Holomorphic domains; $2^{\ell-1}$ choices	Bi-graded $\mathbb{Z}$-homology (Sarkar, 2010)
Link Floer homology (standard)	Grid diagrams; rectangular diagrams	Multi-graded or bi-graded groups categorifying Alexander polynomial
Bordered algebra approach	DA and D bimodules; bridge/Heegaard position	Efficient computation; Thurston polytope extraction (Ozsvath et al., 2020)
Cobordism maps, TQFT	Decomposed cobordism; Heegaard triples	Functorial invariants, absolute and relative gradings (Zemke, 2016)
Spectral sequences (Khovanov–Floer)	Cube of resolutions; modified band maps	Filtration from Khovanov $\to$ Floer; concordance invariants (Nahm, 3 May 2025)
Splitting decomposition	Algebraic; $CFL(Y, L)$ over $\mathbb{F}[U,V]/(UV)$	Direct sum of snake complexes and local systems (Popović, 2023)
A-brane/fukaya-seidel categorical	Derived A-branes; Landau–Ginzburg potential	Floer categorification of quantum group invariants (Aganagic et al., 2023)

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Link Floer homology occupies a central position in low-dimensional topology, categorifying classical link invariants, providing sharp detection criteria for topological types (split, fibered, torus links, etc.), and underpinning functorial, quantum, and categorical extensions. Its development and unification with bordered algebras, spectral sequences, and homological mirror symmetry show that it is not only an invariant of knots and links but also an organizing principle for interactions between topology, symplectic geometry, and algebraic geometry.