Skein Exact Triangles in Floer Homology

Updated 10 July 2026

Skein exact triangles are local constructs that relate three homological objects differing by a prescribed replacement in links, tangles, or graphs.
They employ mapping cones, holomorphic polygon counts, and cobordism maps to build exact long sequences in various Floer and instanton homology theories.
Their applications span bordered–sutured Floer, grid, and gauge theories, enabling recursive analysis, grading refinements, and extension to spatial graphs and web configurations.

Skein exact triangles are exact-triangle or long exact sequence relations in which three homological objects are attached to links, tangles, webs, spatial graphs, or surgery data that agree outside a ball and differ only by a prescribed local replacement. In the literature represented here, the local models range from a crossing tangle together with its two unoriented resolutions, to oriented crossing-change triples, trivalent web replacements, vertex moves in spatial graphs, and rational-tangle substitutions; exactness is realized by mapping cones, holomorphic polygon counts, standard foam or saddle cobordism maps, and family-metric arguments in gauge theory (Vela-Vick et al., 2018, Kronheimer et al., 2015, Zhuang, 2021, Eftekhary, 10 Sep 2025).

1. Local models and the formal meaning of skein exactness

In Floer and knot-homological settings, a skein exact triangle typically relates three objects assigned to three links or tangles differing only in a small ball, and exactness means that they fit into a long exact sequence or an exact triangle in an appropriate homological category (Karan, 2019). The fundamental unoriented local picture is the triple $(L,L_0,L_1)$ , where $L_0$ and $L_1$ are the two unoriented resolutions of a chosen crossing of $L$ ; the oriented analogue is $(L_+,L_-,L_o)$ , where $L_o$ is the canonically oriented resolution compatible with the crossing orientation (Karan, 2019).

A particularly flexible formulation appears in bordered–sutured Floer homology. There one fixes elementary tangles

$\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$

with $\elT_\infty$ the crossing tangle and $\elT_0,\elT_1$ its two resolutions, and then considers tangles

$\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$

in an arbitrary compact oriented $L_0$ 0-manifold $L_0$ 1, required to agree with $L_0$ 2 inside an embedded $L_0$ 3 and to coincide outside that ball (Vela-Vick et al., 2018). In this formulation no orientation on the tangle strands is required, which is precisely why the triangle is unoriented (Vela-Vick et al., 2018).

Later extensions replace the classical crossing triple by more general local data. In tangle Floer homology one has both the unoriented triple $L_0$ 4 and the oriented triple $L_0$ 5 at the level of type $L_0$ 6 bimodules (Petkova et al., 2016). In link Floer homology one can instead start from rational tangles $L_0$ 7, with $L_0$ 8 the trivial tangle and $L_0$ 9 the positive half-twist, and ask for exact triangles for all skein triples of that local tangle type (Eftekhary, 10 Sep 2025).

Setting	Local triple	Exactness form
Knot or tangle Floer	$L_1$ 0, $L_1$ 1	mapping cone or long exact sequence
Oriented skein theory	$L_1$ 2, $L_1$ 3	exact triangle with graded shifts
Web instanton homology	$L_1$ 4	$L_1$ 5-periodic exact sequence
Rational-tangle Floer	$L_1$ 6	exact triangle with auxiliary graded modules

The terminology is not completely uniform across the literature. Some papers study the same three-link local geometry but only at the level of recursive closure or exact sequences of algebras, rather than distinguished triangles of chain complexes; those distinctions are substantial and recur throughout the subject.

2. Chain-level mechanisms: cones, polygons, and homotopies

A dominant mechanism is the mapping-cone description. In bordered–sutured Floer homology, if $L_1$ 7 and

$L_1$ 8

satisfies the minimal compatibility conditions imposed away from the skein ball, then the main theorem states

$L_1$ 9

equivalently

$L$ 0

The proof is first local, at the level of the elementary complements $L$ 1, and is then globalized by the bordered–sutured pairing theorem (Vela-Vick et al., 2018).

At chain level, this exactness is usually verified by the same homological-algebra pattern. One constructs chain maps $L$ 2, null-homotopies $L$ 3 for the composites $L$ 4, and higher homotopies $L$ 5 satisfying an identity that makes the relevant mapping cone quasi-isomorphic to the remaining vertex. In Wong’s grid-diagram proof of Manolescu’s triangle, the maps are defined by counting empty pentagons and triangles,

$L$ 6

the null-homotopies by empty hexagons and quadrilaterals,

$L$ 7

and the final homotopies by heptagons,

$L$ 8

with the key identities

$L$ 9

and

$(L_+,L_-,L_o)$ 0

(Wong, 2013).

The same architecture appears in local Heegaard-Floer models with twisted complexes. In unoriented link Floer homology $(L_+,L_-,L_o)$ 1, the local theorem identifies a twisted complex

$(L_+,L_-,L_o)$ 2

with an object $(L_+,L_-,L_o)$ 3 carrying a rank-two local system $(L_+,L_-,L_o)$ 4, and the exact triangle follows from a mapping-cone argument using the associated band maps (Nahm, 2 Jan 2025). This chain-level viewpoint is pervasive: exactness is not an afterthought on homology, but is built from concrete polygon counts, cobordism counts, or explicit algebraic cones.

3. Heegaard Floer, knot Floer, and tangle Floer realizations

For links in $(L_+,L_-,L_o)$ 5, Manolescu’s unoriented skein triangle for $(L_+,L_-,L_o)$ 6 was rederived combinatorially by Wong using grid diagrams. If $(L_+,L_-,L_o)$ 7 is a link and $(L_+,L_-,L_o)$ 8 are the two resolutions of a chosen crossing, with $(L_+,L_-,L_o)$ 9 the numbers of components and $L_o$ 0, then one has an exact triangle

$L_o$ 1

and Wong extends the combinatorial proof from $L_o$ 2 to $L_o$ 3 by sign refinements, then iterates the triangle to obtain a cube-of-resolutions complex converging to knot Floer homology (Wong, 2013).

Petkova–Wong’s tangle Floer construction moves the triangle to the local tangle level. For an unoriented skein triple of tangles $L_o$ 4, they prove

$L_o$ 5

as ungraded type $L_o$ 6 structures, and they also prove an oriented skein relation in which

$L_o$ 7

in the blocked bigraded theory (Petkova et al., 2016). The essential point is that the skein triangle is first established locally, at the level of tangle bimodules, and only afterward recovered for $L_o$ 8 by gluing and box tensor products (Petkova et al., 2016).

Bordered–sutured Floer homology pushes this locality further. The triangle is proved first for the complements of the three elementary tangles in a $L_o$ 9-ball and then promoted, via the bordered–sutured pairing theorem, to tangles in arbitrary ambient $\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$0-manifolds with bordered boundary and mild hypotheses on the sutures (Vela-Vick et al., 2018). In the closed case this reduces to a triangle for $\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$1, and for $\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$2 the result is shown to agree with Manolescu’s theorem (Vela-Vick et al., 2018).

Two later link-Floer developments enlarge the local repertoire. Nahm defines band maps in unoriented link Floer homology $\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$3, where $\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$4- and $\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$5-basepoints are treated symmetrically, and proves an $\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$6-linear unoriented skein exact triangle

$\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$7

for the three band maps in cyclic order, together with analogous $\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$8- and hat-version triangles (Nahm, 2 Jan 2025). Eftekhary constructs a new family of skein exact triangles in link Floer homology for skein triples of rational tangles $\elTan_k=(\overline{B^3},\elT_k), \qquad k\in\{\infty,0,1\},$9, and in particular for

$\elT_\infty$0

where $\elT_\infty$1 is obtained from the trivial tangle by $\elT_\infty$2 positive half-twists. In that case

$\elT_\infty$3

and the exact triangle takes the form

$\elT_\infty$4

with maps of degrees $\elT_\infty$5, $\elT_\infty$6, and $\elT_\infty$7 in the general framework (Eftekhary, 10 Sep 2025).

4. Gauge-theoretic, web, and graph extensions

In $\elT_\infty$8 instanton theory for webs, Kronheimer–Mrowka prove a genuine skein exact triangle for the web homology $\elT_\infty$9. If $\elT_0,\elT_1$0 are identical outside a ball and differ by the three standard local trivalent configurations, then the foam cobordism maps fit into the $\elT_0,\elT_1$1-periodic exact sequence

$\elT_0,\elT_1$2

and in $\elT_0,\elT_1$3 this specializes to

$\elT_0,\elT_1$4

The same paper also realizes an octahedral axiom by organizing four exact triangles into an octahedral diagram (Kronheimer et al., 2015).

Real monopole Floer homology provides another branched-cover refinement. For an unoriented skein triple $\elT_0,\elT_1$5, Li proves that the real monopole groups

$\elT_0,\elT_1$6

fit into exact triangles for each flavor, with maps induced by the elementary saddle cobordisms after passing to double branched covers and restricting Seiberg–Witten theory to the fixed locus of an anti-linear involution (Li, 2023).

Equivariant singular instanton Floer theory yields a more delicate three-case picture. For a nonzero-determinant skein triple $\elT_0,\elT_1$7 in an integer homology $\elT_0,\elT_1$8-sphere, the $\elT_0,\elT_1$9-complexes

$\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$0

fit into exact triangles whose form depends on

$\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$1

When $\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$2, one vertex must be suspended, and the suspension correction matches the extra term in the Euler-characteristic skein relation for the Murasugi signature (Daemi et al., 2024).

Spatial graphs support a vertex-based analogue. Zhuang’s grid homology for oriented spatial graphs extends the singular-knot skein exact sequence to a local replacement at a vertex of valence at least $\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$3, producing exact sequences such as

$\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$4

and analogous hat and $\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$5 versions (Zhuang, 2021). The local move is at a vertex rather than a crossing, but the proof again passes through mapping-cone identifications.

A nearby but distinct development is Bhat’s surgery exact triangle in instanton theory. It relates

$\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$6

is built from orbifold rational surgery traces and singular instanton cobordism maps, and is explicitly presented as a surgery triangle rather than a crossing-resolution skein triangle (Bhat, 2023). Its role in the present context is structural adjacency rather than direct inclusion in the skein-triangle canon.

A recurrent theme is that skein exact triangles are fundamentally local. The bordered–sutured result is explicit on this point: the triangle is first proved for the complements of the three elementary tangles inside a $\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$7-ball and only then promoted to arbitrary $\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$8-manifolds by gluing (Vela-Vick et al., 2018). Tangle Floer homology and the rational-tangle constructions exhibit the same principle: the local bimodule or local characteristic-link data determines the global exact triangle after tensoring or gluing (Petkova et al., 2016, Eftekhary, 10 Sep 2025).

Theories differ sharply in their grading behavior. Wong shows that the unoriented skein triangle in grid homology does not preserve the full Maslov/Alexander bigrading, but it does admit a $\Tan_k=(Y,T_k), \qquad k\in\{\infty,0,1\},$9-graded refinement and a signature-normalized form

$L_0$ 00

(Wong, 2013). In tangle Floer homology, the unoriented triangle has explicit $L_0$ 01-degree shifts determined by the local crossing counts $L_0$ 02, while the oriented triangle is stated in $L_0$ 03-bigraded form (Petkova et al., 2016). Eftekhary’s rational-tangle triangles fix the map degrees at $L_0$ 04, $L_0$ 05, and $L_0$ 06 (Eftekhary, 10 Sep 2025).

Bordered–sutured Floer homology develops a specifically local grading reduction. For the $L_0$ 07-punctured-sphere arc diagram, the refined grading group is

$L_0$ 08

and the skein reduction

$L_0$ 09

is introduced precisely so that all three local modules lie in a common grading set and the triangle maps become homogeneous (Vela-Vick et al., 2018).

Coefficients are equally varied. Wong’s grid proof extends from $L_0$ 10 to $L_0$ 11 via sign assignments on rectangles and induced signs on higher polygons (Wong, 2013). Nahm’s unoriented link Floer theory uses

$L_0$ 12

and a differential in which $L_0$ 13- and $L_0$ 14-basepoints contribute symmetrically as $L_0$ 15, reflecting the unoriented nature of the theory (Nahm, 2 Jan 2025). The equivariant singular instanton $L_0$ 16-complexes support local coefficients and a suspension operator $L_0$ 17 with

$L_0$ 18

which is exactly the correction needed when $L_0$ 19 in the signature-governed skein triangle (Daemi et al., 2024).

These grading and coefficient refinements are not peripheral. They control whether exactness is literal, suspended, stabilized by tensor factors such as $L_0$ 20 and $L_0$ 21, or only visible after passing to a reduced grading or a completed coefficient ring.

6. Analogues, nonexamples, and broader significance

Not every skein-triangle result is an exact triangle in the homological sense. Greene’s generating-set theorem for nonzero determinant links studies the same local triples $L_0$ 22 and $L_0$ 23, but only as a closure principle on sets of links. The main result is that closure under the unoriented skein triangle plus the unknot forces a set to contain all nonzero determinant links, while in the oriented setting one needs the unknot together with all orientations of the Hopf link (Karan, 2019). The paper explicitly emphasizes that it does not prove distinguished triangles, long exact homology sequences, or chain-level maps (Karan, 2019).

Similarly, Higgins’s $L_0$ 24 stated skein-algebra work studies triangular decomposition in the geometric sense of ideal triangles in a surface decomposition. It proves an exact sequence

$L_0$ 25

but explicitly does not construct exact triangles in the triangulated-category or Floer-homological sense (Higgins, 2020).

Sahamie’s comparison of a Dehn-twist exact sequence with a surgery exact triangle in knot Floer homology occupies an intermediate position. It produces the exact sequence

$L_0$ 26

and then shows that the older Dehn-twist sequence fits into a commutative diagram with this surgery triangle, yielding coherent orientations and $L_0$ 27-refinements (Sahamie, 2010). This is not a crossing skein triangle, but it clarifies how surgery exactness and skein-type exactness can encode the same phenomenon in different forms (Sahamie, 2010).

Taken together, these developments suggest that skein exact triangles are best understood not as a single theorem but as a family of local calculi. In some settings they are literal mapping-cone identities; in others they are surgery triangles on branched covers, exact sequences for graph or web theories, or propagation principles that imitate exactness without chain complexes. This suggests that the central invariant content lies in locality, functoriality under controlled replacement, and the ability to recover one corner of a three-term configuration from the other two.