Framed Instanton Homology

• Framed instanton homology is a family of Floer‐theoretic invariants defined via gauge stabilization that controls reducible connections in 3‐manifold topology.
• It combines techniques from SU(2) and SO(3) instanton gauge theory, sutured instanton homology, and equivariant Morse theory to yield exact triangles and spectral sequences.
• Its computational frameworks—using surgery triangles, mapping cones, and spectral sequences—connect the theory to Heegaard Floer and Khovanov homologies for manifolds and knots.

Framed instanton homology is a family of Floer-theoretic invariants arising from the Chern–Simons functional after a framing or stabilization that eliminates, controls, or exploits reducible flat connections. In the closed 3-manifold literature it is usually denoted $I^\#(Y,\lambda)$ or $I^\#(Y)$, while in singular knot gauge theory one also encounters a framed singular theory $\widetilde I_*(Y,K)$, identified with $I^\natural(Y,K)$. Across these variants, the subject combines $SU(2)$ or $SO(3)$ instanton gauge theory, sutured instanton homology, equivariant Morse theory, and surgery exact triangles, and it now serves as a point of contact with Heegaard Floer homology, Khovanov-type theories, concordance invariants, and representation-theoretic constraints on Dehn surgeries [(Scaduto, 2014); (Daemi et al., 2019)].

1. Definitions and formal variants

For a closed, connected, oriented 3-manifold $Y$ and an oriented multicurve $\lambda\subset Y$, one standard definition sets

$$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$

where $\gamma$ is a fiber of $I^\#(Y)$0. This is a finite-dimensional complex vector space with a relative $I^\#(Y)$1-grading and a canonical absolute $I^\#(Y)$2-grading, and its isomorphism type depends only on $I^\#(Y)$3 and the mod 2 homology class of $I^\#(Y)$4 (Alfieri et al., 2020). In the sutured formulation, one removes a ball from $I^\#(Y)$5, writes $I^\#(Y)$6, chooses a simple closed curve $I^\#(Y)$7, and identifies $I^\#(Y)$8 with $I^\#(Y)$9 (Li et al., 2020).

A second, older description views $\widetilde I_*(Y,K)$0 as a $\widetilde I_*(Y,K)$1-graded abelian group defined from the framed configuration space of an $\widetilde I_*(Y,K)$2-bundle $\widetilde I_*(Y,K)$3 with $\widetilde I_*(Y,K)$4 Poincaré dual to $\widetilde I_*(Y,K)$5, and as four consecutive gradings of Floer’s relatively $\widetilde I_*(Y,K)$6-graded instanton homology for a non-trivial admissible bundle over $\widetilde I_*(Y,K)$7 restricting to $\widetilde I_*(Y,K)$8 over $\widetilde I_*(Y,K)$9 and a non-trivial bundle over $I^\natural(Y,K)$0 (Scaduto et al., 2016). Scaduto’s formulation emphasizes counting ASD instantons on $I^\natural(Y,K)$1 modulo a framed gauge group, and equips $I^\natural(Y,K)$2 with an absolute $I^\natural(Y,K)$3-grading (Scaduto, 2014).

For knots, Daemi–Scaduto’s equivariant singular framework introduces a different object: the framed singular instanton complex $I^\natural(Y,K)$4, whose homology is $I^\natural(Y,K)$5. This theory is built from singular $I^\natural(Y,K)$6 connections with prescribed meridional holonomy and a basepoint framing on the knot, and it is naturally chain homotopy equivalent to Kronheimer–Mrowka’s $I^\natural(Y,K)$7 (Daemi et al., 2019).

Variant	Input	Defining feature
$I^\natural(Y,K)$8	closed 3-manifold and multicurve $I^\natural(Y,K)$9	$SU(2)$0
$SU(2)$1	punctured 3-manifold with a suture $SU(2)$2	sutured instanton realization
$SU(2)$3	based knot in an integer homology 3-sphere	homology of the $SU(2)$4-complex $SU(2)$5
$SU(2)$6	based knot	naturally chain homotopy equivalent to $SU(2)$7

These constructions are closely related but not identical. A useful organizing principle is that the phrase “framed instanton homology” covers both the $SU(2)$8-stabilized closed-manifold theory and the basepoint-framed singular knot theory. This suggests a common theme: framing is used to rigidify the gauge problem sufficiently to obtain finite-dimensional Floer groups with good functoriality and exact-triangle structures.

2. Gauge-theoretic foundations

In the closed-manifold theory, the geometric input is the Chern–Simons functional on a configuration space over $SU(2)$9 with a non-trivial bundle on the torus factor. Scaduto’s formulation uses a framed gauge group $SO(3)$0, where $SO(3)$1 is the relevant $SO(3)$2-bundle over $SO(3)$3, and defines the chain complex by counting signed rigid ASD trajectories on $SO(3)$4. The resulting homology is functorial under cobordisms, carries an absolute $SO(3)$5-grading, and satisfies a cobordism degree formula involving $SO(3)$6, a $SO(3)$7-correction, and, for non-trivial bundles, a characteristic term $SO(3)$8 (Scaduto, 2014).

In the singular knot setting, the analytic starting point is more explicit. For a knot $SO(3)$9 in an integer homology 3-sphere, one fixes a rank-2 Hermitian bundle $Y$0, a reduction $Y$1, and a model singular connection

$Y$2

where $Y$3 in polar coordinates near $Y$4. The resulting singular connections have meridional holonomy of order 4 in $Y$5. After fixing a basepoint on $Y$6 and a trivialization of $Y$7 there, one obtains the framed configuration space $Y$8. Changing the framing at the basepoint defines an $Y$9-action on $\lambda\subset Y$0; generic framed connections have stabilizer $\lambda\subset Y$1, while framed connections whose underlying singular connection is $\lambda\subset Y$2-reducible have stabilizer $\lambda\subset Y$3. The Chern–Simons functional has critical points given by singular flat connections, including a distinguished isolated non-degenerate reducible $\lambda\subset Y$4 (Daemi et al., 2019).

The equivariant character of the singular theory is central. The formal $\lambda\subset Y$5-gradient is

$\lambda\subset Y$6

and on $\lambda\subset Y$7 the downward gradient equation becomes the perturbed ASD equation

$\lambda\subset Y$8

For irreducible critical points $\lambda\subset Y$9, the moduli space $$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$0 has virtual dimension $$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$1, and passing modulo 4 yields a well-defined relative grading; fixing $$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$2 gives an absolute $$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$3-grading (Daemi et al., 2019).

The two framings appearing in the literature are technically different. In the closed theory, “framed” usually refers to the $$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$4-stabilized gauge setup that avoids reducibles. In the singular knot theory, it refers to fixing the meridional holonomy at a chosen basepoint and then exploiting the residual $$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$5-symmetry of framed singular connections. A plausible implication is that the terminology reflects not a single construction but a common strategy for turning reducible phenomena into algebraically tractable structure.

3. Complexes, equivariance, and exact triangles

The framed singular knot theory packages its data into an $$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$6-complex

$$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$7

with differential

$$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$8

Here $$I^\#(Y,\lambda):=\mathrm{Fix}\Big(\tfrac{1}{2}\mu(\mathrm{pt})\Big)\subset I_*(Y\#T^3,\lambda\cup\gamma),$$9 is generated by irreducible critical points, $\gamma$0 is the usual Floer differential counting 0-dimensional cylindrical instantons, $\gamma$1 is a holonomy cut-down map, and $\gamma$2 count trajectories to and from the reducible $\gamma$3. From this one constructs three equivariant $\gamma$4-complexes, with $\gamma$5, whose homologies $\gamma$6, $\gamma$7, and $\gamma$8 fit into exact triangles analogous to Borel and Tate packages in equivariant Floer theory (Daemi et al., 2019).

The closed-manifold theory is governed by surgery triangles. For a knot $\gamma$9, one has an exact triangle

$I^\#(Y)$00

with maps induced by the corresponding 2-handle cobordisms. In the absolute $I^\#(Y)$01-graded refinement, the three cobordism degrees are constrained so that their sum is $I^\#(Y)$02, and exactly one of the three cobordisms is non-spin (Lidman et al., 2020). Scaduto’s metric-stretching argument gives a related link-surgeries spectral sequence, and in the branched-cover context the $I^\#(Y)$03-page is a direct sum of framed instanton homologies of surgery manifolds (Scaduto, 2014).

A major structural development is the knot surgery formula in sutured instanton homology. For a rationally null-homologous knot $I^\#(Y)$04 and $I^\#(Y)$05, Li and Ye construct bent complexes $I^\#(Y)$06, $I^\#(Y)$07, together with maps $I^\#(Y)$08 and an isomorphism $I^\#(Y)$09, and prove the mapping-cone formula

$I^\#(Y)$10

The proof is based on sutured instanton homology, bypass exact triangles, and the octahedral lemma in the derived category rather than on the Heegaard Floer surgery formalism (Li et al., 2022).

Another algebraic package appears for $I^\#(Y)$11 with non-trivial bundle. There, framed instanton homology fits into the twisted Gysin exact sequence

$I^\#(Y)$12

so $I^\#(Y)$13 is the mapping cone of $I^\#(Y)$14 on $I^\#(Y)$15. In Muñoz’s ring model for $I^\#(Y)$16, the $I^\#(Y)$17-map is denoted $I^\#(Y)$18, and the nilpotency degree of $I^\#(Y)$19 is

$I^\#(Y)$20

for genus $I^\#(Y)$21 (Chen et al., 2016).

These algebraic formalisms do more than organize computations. They identify the precise places where reducibles enter the theory: as extra summands, as equivariant variables, as distinguished critical points, or as nilpotent endomorphisms. This is one of the defining differences between framed instanton homology and more classical irreducible-only instanton packages.

4. Computations and explicit families

Framed instanton homology is unusually rich computationally because several independent calculi coexist: surgery exact triangles, bent-complex mapping cones, equivariant singular complexes, lattice homology for plumbings, and branched-cover spectral sequences. The resulting calculations range from Seifert fibered spaces to torus knots, twist knots, branched double covers, and surface bundles.

Family	Result	Source
almost-rational plumbings $I^\#(Y)$22	$I^\#(Y)$23 as $I^\#(Y)$24-graded complex vector spaces	(Alfieri et al., 2020)
integral surgeries on a knot with an instanton L-space surgery	for $I^\#(Y)$25, $I^\#(Y)$26 is given by a piecewise $I^\#(Y)$27-graded formula, and for $I^\#(Y)$28 one has $I^\#(Y)$29	(Lidman et al., 2020)
nontrivial circle bundles $I^\#(Y)$30 over $I^\#(Y)$31	if $I^\#(Y)$32, then $I^\#(Y)$33; for smaller $I^\#(Y)$34 there are explicit binomial-sum corrections	(Li et al., 2022)
$I^\#(Y)$35 with non-trivial bundle $I^\#(Y)$36	$I^\#(Y)$37	(Chen et al., 2016)
two-fold quasi-alternating branched covers	$I^\#(Y)$38 is free abelian of rank $I^\#(Y)$39 and supported in gradings $I^\#(Y)$40	(Scaduto et al., 2016)

For almost-rational plumbings, the key mechanism is an identification of the even-graded part of $I^\#(Y)$41 with lattice homology $I^\#(Y)$42, followed by Némethi’s description of $I^\#(Y)$43 as $I^\#(Y)$44. This yields the isomorphism $I^\#(Y)$45 and, in particular, establishes the Kronheimer–Mrowka conjectural correspondence on a large class containing all Seifert fibered rational homology spheres (Alfieri et al., 2020).

For knots, several surgery calculi are available. Baldwin and Sivek compute all integral surgeries on a knot with an instanton L-space surgery, Li and Ye derive integral, rational, and partially zero-surgery formulas for general rationally null-homologous knots, and the companion applications paper computes framed instanton homology for nontrivial circle bundles, many Seifert fibered spaces with nonzero orbifold degree, surgeries on a family of alternating knots, twisted Whitehead doubles, and splicings with twist knots (Lidman et al., 2020, Li et al., 2022).

The singular framed knot theory also admits concrete examples. For spherical knots, the ADHM description of instantons on $I^\#(Y)$46 gives a concrete characterization of the relevant moduli spaces, and for two-bridge knots the branched-cover analysis relates singular instanton moduli spaces to instantons on lens spaces. In particular, two-bridge knots satisfy $I^\#(Y)$47, while the right-handed trefoil and the $I^\#(Y)$48 and $I^\#(Y)$49 torus knots have $I^\#(Y)$50 in the notation of that paper (Daemi et al., 2019).

A recurrent pattern is that explicit computations often determine only dimensions or parity gradings, while finer $I^\#(Y)$51-graded or module-theoretic structure remains subtler. This suggests that framed instanton homology is computationally accessible at the rank level over broad classes, but that its richer structure is still unevenly understood.

5. Relations with Heegaard Floer, Khovanov-type theories, and sutured instanton homology

A central theme in the subject is comparison with Heegaard Floer homology. Kronheimer and Mrowka conjectured that $I^\#(Y)$52 for all closed 3-manifolds, and this has been verified for boundaries of almost-rational plumbings and, by a direct argument, for the remaining Seifert fibered rational homology spheres with base $I^\#(Y)$53 (Alfieri et al., 2020). In a different direction, Wang proves the inequality

$I^\#(Y)$54

for all rationally null-homologous knots $I^\#(Y)$55, and constructs a decomposition of $I^\#(Y)$56 for Dehn surgeries that parallels torsion $I^\#(Y)$57 splittings in monopole and Heegaard Floer theories (Li et al., 2020).

The link to Khovanov-type theories begins with Scaduto’s spectral sequence

$I^\#(Y)$58

where the $I^\#(Y)$59-page carries a compatible $I^\#(Y)$60-grading

$I^\#(Y)$61

For quasi-alternating links this spectral sequence collapses, implying that $I^\#(Y)$62 is free of rank $I^\#(Y)$63 and supported in even gradings (Scaduto, 2014). Two-fold marked refinements extend this picture to non-trivial $I^\#(Y)$64-bundles over branched double covers: for two-fold quasi-alternating links, the twisted Khovanov theory is thin and the corresponding $I^\#(Y)$65 is completely determined, including its $I^\#(Y)$66-grading distribution (Scaduto et al., 2016).

A further deformation replaces ordinary or odd Khovanov homology by Bar–Natan and $I^\#(Y)$67 theories in characteristic 2. Kronheimer and Mrowka’s deformed framed instanton homology $I^\#(Y)$68, built on the singular bifold $I^\#(Y)$69 with a local system over

$I^\#(Y)$70

fits into a spectral sequence whose $I^\#(Y)$71-page is $I^\#(Y)$72-homology, and after specialization yields a spectral sequence from Bar–Natan homology to $I^\#(Y)$73. The deformation is controlled by explicit elements

$I^\#(Y)$74

which determine the Frobenius algebra on the $I^\#(Y)$75-page (Kronheimer et al., 2019).

Sutured instanton homology is the mechanism behind many of these comparisons. It supplies bypass triangles, Alexander-type gradings from Seifert surfaces, contact handle maps, and the identification $I^\#(Y)$76, so it mediates between closed instanton Floer groups, knot instanton homology, and surgery mapping cones (Li et al., 2020). In practice, framed instanton homology often appears as the closed endpoint of a much larger sutured package.

6. Concordance, torsion, and surgery obstructions

Framed instanton homology supports several knot concordance invariants. Baldwin and Sivek define $I^\#(Y)$77 and $I^\#(Y)$78, where $I^\#(Y)$79 is extracted from the vanishing pattern of integer surgery cobordism maps and $I^\#(Y)$80 is the homogenization of $I^\#(Y)$81. They prove that $I^\#(Y)$82 is a smooth concordance invariant, that $I^\#(Y)$83 is a concordance homomorphism, and that

$I^\#(Y)$84

for coprime $I^\#(Y)$85 with $I^\#(Y)$86, subject to the special behavior at zero in the $I^\#(Y)$87-shaped case. They also show that $I^\#(Y)$88 is a slice-torus invariant, that

$I^\#(Y)$89

and that for instanton L-space knots one has $I^\#(Y)$90 (Baldwin et al., 2020).

The sequel develops a conjugation symmetry for decomposed cobordism maps, proves that $I^\#(Y)$91 is either zero or odd, and defines

$I^\#(Y)$92

It also establishes a denominator bound for surgery descriptions: if $I^\#(Y)$93 for a nontrivial knot $I^\#(Y)$94 that is not a trefoil or the figure-eight, then

$I^\#(Y)$95

The same paper proves that for nonzero slopes the dimension of $I^\#(Y)$96 is independent of $I^\#(Y)$97 (Baldwin et al., 2022).

A different concordance-oriented development uses the minus version $I^\#(Y)$98. For specially decorated knot cobordisms $I^\#(Y)$99, one has $\widetilde I_*(Y,K)$00-equivariant cobordism maps

$\widetilde I_*(Y,K)$01

and the tube attachment lemma states that attaching a tube to $\widetilde I_*(Y,K)$02 multiplies the map by $\widetilde I_*(Y,K)$03. From this, the torsion-order inequality

$\widetilde I_*(Y,K)$04

is recovered, and for alternating knots of bridge index at most $\widetilde I_*(Y,K)$05 one obtains

$\widetilde I_*(Y,K)$06

for every nonzero rational slope $\widetilde I_*(Y,K)$07 (Ghosh et al., 2023).

Integral coefficients reveal substantial torsion phenomena. Li and Ye show that if $\widetilde I_*(Y,K)$08 has no 2-torsion for some nonzero integral surgery, then $\widetilde I_*(Y,K)$09 must be fibered, and that for every nontrivial knot the surgeries of slopes $\widetilde I_*(Y,K)$10, $\widetilde I_*(Y,K)$11, and $\widetilde I_*(Y,K)$12 always have 2-torsion. They also prove 2-torsion statements for unreduced singular instanton knot homology of genus-one knots with nontrivial Alexander polynomial and for unknotting-number-one knots (Li et al., 2024). Over $\widetilde I_*(Y,K)$13, an additional pair of invariants $\widetilde I_*(Y,K)$14 governs surgery dimensions: $\widetilde I_*(Y,K)$15 The same work relates these dimensions to a Frøyshov-type invariant $\widetilde I_*(Y,K)$16 and deduces that $\widetilde I_*(Y,K)$17-surgery on a nontrivial knot cannot be nondegenerate $\widetilde I_*(Y,K)$18-abelian for any

$\widetilde I_*(Y,K)$19

(Ghosh et al., 24 Nov 2025).

Several directions remain open in the existing literature. The singular equivariant theory raises questions about extending functoriality beyond negative definite pairs, constructing an Alexander grading on the framed and equivariant groups, and clarifying the relation between $\widetilde I_*(Y,K)$20, $\widetilde I_*(Y,K)$21, and Kronheimer–Mrowka concordance invariants (Daemi et al., 2019). The sutured surgery program isolates the $\widetilde I_*(Y,K)$22 case as exceptional because scalar ambiguities can affect the mapping cone, suggesting that a fully satisfactory zero-surgery formalism may require a more intrinsic spin$\widetilde I_*(Y,K)$23-like or equivariant refinement (Li et al., 2022).