Heegaard-Link Diagrams

• Heegaard-link diagrams are link-adapted Heegaard structures that combine surfaces with embedded link data to encode links, tangles, or sutured complements in 3-manifolds.
• They are constructed from planar projections, grid diagrams, or Kauffman-state models, yielding explicit combinatorial representations that support algorithmic Floer homology computations.
• These diagrams extend naturally to sutured, tangle, and bordered-sutured variants, facilitating functorial invariants and TQFT applications in low-dimensional topology.

Heegaard-link diagrams are link-adapted Heegaard descriptions in which a Heegaard surface is augmented by embedded link data, by basepoints, or by boundary and suture structures so that the diagram encodes not only a $3$-manifold but also a link, tangle, or sutured complement. In the literature, the term appears both in the narrow form $(\Sigma,\alpha,\beta,D)$, where $D$ is a link diagram on the splitting surface, and in the multi-pointed Floer-theoretic form $$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$, together with bordered-sutured and tangle analogues. Across these variants, the common theme is that combinatorial surface data are made to carry link-theoretic, sutured, or cobordism information in a form compatible with Heegaard Floer, sutured Floer, bordered-sutured, and instanton constructions (Armond et al., 2014, Juhász et al., 2012, Zibrowius, 2016).

1. Definitions and formal variants

A Heegaard diagram for a closed $3$-manifold consists of a surface and two attaching sets. In the link-adapted setting, the additional data record an embedded link or its complement. One explicit formulation is the “Heegaard-link diagram” $(\Sigma,\alpha,\beta,D)$ for $S^3$, where $(\Sigma,\alpha,\beta)$ is a Heegaard diagram and $D$ is a link diagram embedded on $\Sigma$ with controlled incidence relative to the attaching curves. In the formulation associated to Turaev surfaces, $(\Sigma,\alpha,\beta,D)$0 is transverse to $(\Sigma,\alpha,\beta,D)$1 and $(\Sigma,\alpha,\beta,D)$2, satisfies

$(\Sigma,\alpha,\beta,D)$3

and these points are not crossings of $(\Sigma,\alpha,\beta,D)$4; moreover, there is a partition

$(\Sigma,\alpha,\beta,D)$5

with

$(\Sigma,\alpha,\beta,D)$6

This rigid version makes the link diagram part of the Heegaard data itself (Armond et al., 2014).

A second, and in Floer theory more common, variant is the multi-pointed Heegaard diagram

$(\Sigma,\alpha,\beta,D)$7

used for links in closed $(\Sigma,\alpha,\beta,D)$8-manifolds. Here the basepoints encode the link, and the generators of the associated chain complex are intersection points in the symmetric product. Naturality results are stated precisely for such multi-pointed diagrams and their based diffeomorphisms (Juhász et al., 2012). For tangles, one further passes to surfaces with boundary and combines $(\Sigma,\alpha,\beta,D)$9-circles, $D$0-arcs, $D$1-circles, and boundary data $D$2 in a Heegaard diagram $D$3 for the tangle complement (Zibrowius, 2016). For sutured manifolds, the basic datum is the sutured diagram $D$4, where both attaching sets are interpreted in compression bodies with sutures along $D$5 (Qin, 6 Aug 2025).

Variant	Data	Typical role
Heegaard-link diagram	$D$6	Link diagrams on Turaev surfaces
Multi-pointed diagram	$D$7	Link Floer homology and functoriality
Tangle diagram	$D$8 with circles, arcs, $D$9	Tangle complements and $$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$0
Sutured diagram	$$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$1	Sutured manifolds and handleslide complexes

A recurrent misconception is that there is a single canonical formal definition of “Heegaard-link diagram.” The current literature instead exhibits a family of closely related models, each optimized for a different categorical or Floer-theoretic target. This suggests that the notion is best understood structurally: a Heegaard diagram becomes a Heegaard-link diagram once it is equipped with enough auxiliary data to recover an embedded link, tangle, or sutured complement.

2. Constructions from link projections and states

One major source of Heegaard-link diagrams is the passage from planar link diagrams to splitting surfaces. For connected link diagrams on $$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$2, there is a $$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$3-to-$$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$4 correspondence between Turaev surfaces and special Heegaard-link diagrams $$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$5 satisfying the conditions above. In this correspondence, the original diagram becomes alternating on $$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$6, cuts $$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$7 into disks, and the Turaev surface is simultaneously a link-carrying surface and a Heegaard surface for $$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$8 (Armond et al., 2014). The construction proceeds by pushing the all-A and all-B states to opposite sides of $$(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{w},\mathbf{z})$$9, inserting saddles near crossings, capping the resulting state circles, and extracting attaching circles from the checkerboard structure.

Grid diagrams furnish a different systematic construction. An $3$0 grid with exactly one $3$1 and one $3$2 in each row and column becomes a toroidal Heegaard link diagram after identifying opposite sides of the square: the horizontal lines become $3$3-curves, the vertical lines become $3$4-curves, and the $3$5 markings become the basepoints. The generators are the $3$6 placements with one point in each row and column, and the differential in the combinatorial link Floer complex is given by counts of empty rectangles. This converts the holomorphic-disk problem into an algorithmic enumeration problem on the torus (Manolescu, 2012).

A third diagrammatic route is through the Kauffman-states and planar Heegaard diagrams used in knot Floer homology. These are two systematic multi-pointed constructions from a knot or link projection, and there is a completely explicit sequence of Heegaard moves transforming the Kauffman-states diagram into the planar diagram. The moves are local at regular or singular crossings, minima, and maxima, and are assembled globally through gluing. The result is a local-to-global interpolation between two diagrammatic models that had previously been used in different algebraic contexts (Manion, 2018).

Taken together, these constructions show that Heegaard-link diagrams are not merely abstract Heegaard splittings decorated after the fact. They can be built directly from planar combinatorics, and in several important instances the combinatorics are explicit enough to support algorithmic Floer calculations.

3. Tangles, sutures, and bordered-sutured extensions

For tangles, the Heegaard surface acquires boundary, and the diagram must encode the complement of intervals and circles in a $3$7-ball or more general $3$8-manifold. A Heegaard diagram for a tangle $3$9 uses an oriented surface $(\Sigma,\alpha,\beta,D)$0 with boundary, $(\Sigma,\alpha,\beta,D)$1 disjoint $(\Sigma,\alpha,\beta,D)$2-circles, $(\Sigma,\alpha,\beta,D)$3 $(\Sigma,\alpha,\beta,D)$4-arcs, and $(\Sigma,\alpha,\beta,D)$5 $(\Sigma,\alpha,\beta,D)$6-circles, arranged so that the standard handle attachments recover the tangle complement. In parallel, the combinatorics of the tangle diagram are organized by Kauffman states partitioned by sites, and the polynomial invariants $(\Sigma,\alpha,\beta,D)$7 defined by Kauffman states and Alexander codes admit a natural Heegaard interpretation. Their categorification is the bordered-sutured Heegaard Floer invariant $(\Sigma,\alpha,\beta,D)$8 for tangles (Zibrowius, 2016).

This bordered-sutured perspective extends to framed link complements. For a framed link $(\Sigma,\alpha,\beta,D)$9, holomorphic polygon counts in splayed Heegaard multi-diagrams define an $S^3$0 multi-module $S^3$1 over the torus algebra $S^3$2, and this multi-module is quasi-isomorphic to the bordered-sutured invariant $S^3$3. The structure maps are defined by zero-dimensional moduli spaces of holomorphic polygons with asymptotics prescribed by Reeb-chord data, so the Heegaard-link diagram simultaneously controls both boundary parameterization and higher algebra (Hockenhull, 2018).

A related construction connects doubly-pointed Heegaard diagrams for knots to sutured instanton theory. Starting from $S^3$4 for $S^3$5, one attaches a $S^3$6-dimensional $S^3$7-handle along the basepoints, increases the genus, introduces a new pair of curves $S^3$8 and $S^3$9, and produces an explicit balanced sutured handlebody $(\Sigma,\alpha,\beta)$0. This turns Heegaard data for a knot into sutured data accessible to instanton methods (Li et al., 2020).

These generalizations show that the link-adapted Heegaard formalism is stable under passage from closed links to tangles, from closed manifolds to sutured manifolds, and from ordinary Heegaard Floer complexes to bordered-sutured algebraic structures. The same surface-level data can therefore be repackaged to target different flavors of Floer theory.

4. Floer-theoretic and instanton applications

In sutured Heegaard Floer theory, an admissible sutured Heegaard diagram

$(\Sigma,\alpha,\beta)$1

gives tori

$(\Sigma,\alpha,\beta)$2

inside $(\Sigma,\alpha,\beta)$3, and the generator set

$(\Sigma,\alpha,\beta)$4

The principal inequality established for such diagrams is

$(\Sigma,\alpha,\beta)$5

Thus the number of generators of the sutured Heegaard Floer complex bounds the dimension of sutured instanton homology from above. The same work defines the simultaneous trajectory number

$(\Sigma,\alpha,\beta)$6

so that $(\Sigma,\alpha,\beta)$7, and deduces in particular that strong L-spaces are instanton L-spaces (Baldwin et al., 2020).

For knots, the explicit sutured handlebody construction from doubly-pointed Heegaard data yields a sequence of inequalities

$(\Sigma,\alpha,\beta)$8

When a genus $(\Sigma,\alpha,\beta)$9 diagram has total $D$0-$D$1 intersection number $D$2, this leads to

$D$3

The same framework proves

$D$4

for any rationally null-homologous knot and gives computations for families of $D$5-knots, including torus knots in $D$6 (Li et al., 2020).

Closed-manifold Heegaard diagrams enter the link-adapted picture through branched covering constructions. A strong Heegaard diagram is characterized by the fact that all generators have the same sign, equivalently

$D$7

with $D$8 a Pólya matrix. In genus two, every strong L-space admitting a strong Heegaard diagram is the double branched cover of an alternating link:

$D$9

for $\Sigma$0 of the same sign (Greene et al., 2014). This provides a direct link between Heegaard diagram combinatorics and alternating link geometry.

A notable structural consequence is that Heegaard-link diagrams can control analytically defined invariants through purely diagrammatic counts. The upper bounds above do not identify Floer groups canonically with generator sets, but they show that the surface combinatorics already impose nontrivial restrictions on instanton and Heegaard-Floer-type homologies.

5. Naturality, mapping classes, and cobordisms

One of the decisive advances in the subject is the proof that Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural: they assign concrete groups to based $\Sigma$1-manifolds, based links, and balanced sutured manifolds, functorially in based diffeomorphisms, with isotopic diffeomorphisms inducing identical maps. The proof analyzes the fundamental group of the space of Heegaard diagrams and shows that there is no monodromy around a generating set of loops. In the multi-pointed link setting, this turns the Heegaard-link diagram from a presentation tool into a functorial input object (Juhász et al., 2012).

For sutured diagrams, this naturality problem has been reformulated in handleslide-complex terms. The cut-system complex of a sutured compression body is connected and simply connected once appropriate $\Sigma$2-cells are attached, and the corresponding handleslide complex becomes simply connected after adding six kinds of handleslide loops: the slide triangle, four square types, and the slide pentagon. On this basis, tight Heegaard invariants are defined and shown to admit unique extensions to strong Heegaard invariants. The result provides a new framework for proving naturality statements for Floer theories associated to sutured manifolds (Qin, 6 Aug 2025).

Heegaard-link diagrams also support a TQFT for link cobordisms. There is a Heegaard-Floer homology functor from the category of oriented links in closed $\Sigma$3-manifolds and oriented surface cobordisms in $\Sigma$4-manifolds to the category of $\Sigma$5-modules and $\Sigma$6-homomorphisms. In contrast with earlier decorated-link TQFTs, the construction is independent from the decoration. Diagrammatically, the objects are represented by pointed Heegaard diagrams, while morphisms are represented by trisection diagrams or Heegaard quadruples encoding the relevant surface cobordisms (Eftekhary, 2024).

The $\Sigma$7-dimensional analogue is the relative trisection diagram

$\Sigma$8

where $\Sigma$9 is a compact oriented surface with boundary and each of $(\Sigma,\alpha,\beta,D)$00 has $(\Sigma,\alpha,\beta,D)$01 simple closed curves, subject to pairwise standardness. This is described explicitly as the $(\Sigma,\alpha,\beta,D)$02-dimensional analogue of a sutured Heegaard diagram for a sutured $(\Sigma,\alpha,\beta,D)$03-manifold (Castro et al., 2016). In this sense, Heegaard-link diagrams sit inside a larger hierarchy of surface-diagrammatic structures extending from links in $(\Sigma,\alpha,\beta,D)$04-manifolds to link cobordisms in $(\Sigma,\alpha,\beta,D)$05-manifolds.

6. Computational, algebraic, and emerging directions

Recent work has emphasized that Heegaard diagrams can be stored and manipulated as effective computational objects. A compressed data structure represents Heegaard diagrams as normal curves with respect to a surface triangulation, with complexity measured by the bit-length of the normal-coordinate vector:

$(\Sigma,\alpha,\beta,D)$06

Within this model one has polynomial time algorithms for checking diagrams, isotoping curves to efficient position, disk slides, stabilization and destabilization, detecting reductions, and computing $(\Sigma,\alpha,\beta,D)$07 and $(\Sigma,\alpha,\beta,D)$08 (Ennes et al., 15 Jul 2025). Although this work is phrased for Heegaard splittings of closed $(\Sigma,\alpha,\beta,D)$09-manifolds, the same emphasis on succinct surface-level representations is directly relevant to link-adapted settings.

Heegaard diagrams have also become a vehicle for extracting classical algebraic invariants. For a closed, connected, oriented $(\Sigma,\alpha,\beta,D)$10-manifold $(\Sigma,\alpha,\beta,D)$11 with Heegaard diagram $(\Sigma,\alpha,\beta,D)$12, the Lagrangian subgroups

$(\Sigma,\alpha,\beta,D)$13

satisfy

$(\Sigma,\alpha,\beta,D)$14

and the triple-cup product form $(\Sigma,\alpha,\beta,D)$15 can be computed explicitly from the diagram. If $(\Sigma,\alpha,\beta,D)$16 are represented by appropriate multicurves, then

$(\Sigma,\alpha,\beta,D)$17

is the algebraic intersection number of a diagrammatically constructed $(\Sigma,\alpha,\beta,D)$18-cycle $(\Sigma,\alpha,\beta,D)$19 with $(\Sigma,\alpha,\beta,D)$20 (Kayali, 15 Apr 2026). This shows that Heegaard diagrams determine not only Floer complexes but also classical ring-theoretic data.

An emerging variant is the quantum Heegaard diagram

$(\Sigma,\alpha,\beta,D)$21

where $(\Sigma,\alpha,\beta,D)$22 carries additional quantum basepoints. The associated grading data are

$(\Sigma,\alpha,\beta,D)$23

and a two-variable Lagrangian intersection polynomial on the symmetric power of the quantum Heegaard surface specializes to both the Alexander and Jones polynomials (Anghel et al., 13 Jan 2026). This suggests a further extension of the Heegaard-link diagram paradigm in which additional puncture data refine the classical Alexander grading.

Adjacent developments reinforce the breadth of the diagrammatic viewpoint. Heegaard presentation length has been used to relate entropy of pseudo-Anosov monodromy to hyperbolic volume (Liu, 2023), and alternating or weakly alternating genus-two Heegaard diagrams have been classified and used to realize branched-cover descriptions and Williams solenoid dynamics in many $(\Sigma,\alpha,\beta,D)$24-manifolds (Wang et al., 2015). While these directions are not all link-adapted in the strict sense, they confirm a general pattern: Heegaard-link diagrams are not merely auxiliary pictures but a versatile surface-level calculus connecting link projections, Floer theories, branched coverings, mapping class actions, and effective computation.