---
title: Framed Instanton Homology
url: https://www.emergentmind.com/topics/framed-instanton-homology
type: topic
---

# Framed Instanton Homology

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 [1401.2093; 1912.08982].

## 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 \(T^3\to T^2\). This is a finite-dimensional complex vector space with a relative \(\mathbb Z/4\)-grading and a canonical absolute \(\mathbb Z/2\)-grading, and its isomorphism type depends only on \(Y\) and the mod 2 homology class of \(\lambda\) [2010.03800]. In the sutured formulation, one removes a ball from \(Y\), writes \(Y(1)=Y\setminus B^3\), chooses a simple closed curve \(\delta\subset \partial Y(1)\), and identifies \(I^\#(Y)\) with \(SHI(Y(1),\delta)\) [2010.07836].

A second, older description views \(I^\#(Y,\omega)\) as a \(\mathbb Z_2\)-graded abelian group defined from the framed configuration space of an \(SO(3)\)-bundle \(E\to Y\) with \(w_2(E)\) Poincaré dual to \([\omega]\), and as four consecutive gradings of Floer’s relatively \(\mathbb Z_8\)-graded instanton homology for a non-trivial admissible bundle over \(Y\#T^3\) restricting to \(E\) over \(Y\) and a non-trivial bundle over \(T^3\) [1605.05394]. Scaduto’s formulation emphasizes counting ASD instantons on \(\mathbb R\times (Y\#T^3)\) modulo a framed gauge group, and equips \(I^\#(Y)\) with an absolute \(\mathbb Z/4\)-grading [1401.2093].

For knots, Daemi–Scaduto’s equivariant singular framework introduces a different object: the framed singular instanton complex \(\widetilde C_*(Y,K)\), whose homology is \(\widetilde I_*(Y,K)\). This theory is built from singular \(SU(2)\) connections with prescribed meridional holonomy and a basepoint framing on the knot, and it is naturally chain homotopy equivalent to Kronheimer–Mrowka’s \(I^\natural(Y,K)\) [1912.08982].

| Variant | Input | Defining feature |
|---|---|---|
| \(I^\#(Y,\lambda)\) | closed 3-manifold and multicurve \(\lambda\) | \(\mathrm{Fix}(\tfrac12\mu(\mathrm{pt}))\subset I_*(Y\#T^3,\lambda\cup\gamma)\) |
| \(I^\#(Y)\cong SHI(Y(1),\delta)\) | punctured 3-manifold with a suture \(\delta\) | sutured instanton realization |
| \(\widetilde I_*(Y,K)\) | based knot in an integer homology 3-sphere | homology of the \(S\)-complex \(\widetilde C_*(Y,K)\) |
| \(I^\natural(Y,K)\) | based knot | naturally chain homotopy equivalent to \(\widetilde I_*(Y,K)\) |

These constructions are closely related but not identical. A useful organizing principle is that the phrase “framed instanton homology” covers both the \(T^3\)-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 \(Y\#T^3\) with a non-trivial bundle on the torus factor. Scaduto’s formulation uses a framed gauge group \(\mathcal G^\#\subset \mathcal G(Y^\#)\), where \(Y^\#\) is the relevant \(SO(3)\)-bundle over \(Y\#T^3\), and defines the chain complex by counting signed rigid ASD trajectories on \(\mathbb R\times (Y\#T^3)\). The resulting homology is functorial under cobordisms, carries an absolute \(\mathbb Z/4\)-grading, and satisfies a cobordism degree formula involving \(-\frac32(\chi+\sigma)\), a \(b_1\)-correction, and, for non-trivial bundles, a characteristic term \(2P(X)\) [1401.2093].

In the singular knot setting, the analytic starting point is more explicit. For a knot \(K\subset Y\) in an integer homology 3-sphere, one fixes a rank-2 Hermitian bundle \(E\to Y\), a reduction \(E|_K=L\oplus L^*\), and a model singular connection
\[
B_0=\lambda_0\oplus \lambda_0^*,
\]
where \(\lambda_0=b(r)(1/4)i\,d\theta\) in polar coordinates near \(K\). The resulting singular connections have meridional holonomy of order 4 in \(SU(2)\). After fixing a basepoint on \(K\) and a trivialization of \(E\) there, one obtains the framed configuration space \(\widetilde B(Y,K)\). Changing the framing at the basepoint defines an \(S^1\)-action on \(\widetilde B(Y,K)\); generic framed connections have stabilizer \(\{\pm1\}\), while framed connections whose underlying singular connection is \(S^1\)-reducible have stabilizer \(U(1)\). The Chern–Simons functional has critical points given by singular flat connections, including a distinguished isolated non-degenerate reducible \(\theta\) [1912.08982].

The equivariant character of the singular theory is central. The formal \(L^2\)-gradient is
\[
(\mathrm{grad\,CS})_B=\frac{1}{4\pi^2}*F_B,
\]
and on \(\mathbb R\times Y\) the downward gradient equation becomes the perturbed ASD equation
\[
F_A^+ + \widehat{V}_\pi(A)=0.
\]
For irreducible critical points \(\alpha_1,\alpha_2\), the moduli space \(M_z(\alpha_1,\alpha_2)\) has virtual dimension \(\mathrm{gr}_z(\alpha_1,\alpha_2)=\mathrm{ind}(D_A)\), and passing modulo 4 yields a well-defined relative grading; fixing \(\mathrm{CS}(\theta)=0\) gives an absolute \(\mathbb Z/4\)-grading [1912.08982].

The two framings appearing in the literature are technically different. In the closed theory, “framed” usually refers to the \(T^3\)-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 \(S^1\)-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 \(S\)-complex
\[
\widetilde{C}_*(Y,K)=C_*(Y,K)\oplus C_{*-1}(Y,K)\oplus \mathbb Z
\]
with differential
\[
\widetilde{d}=
\begin{bmatrix}
d & 0 & 0 \\
v & -d & \delta_2 \\
\delta_1 & 0 & 0
\end{bmatrix}.
\]
Here \(C_*(Y,K)\) is generated by irreducible critical points, \(d\) is the usual Floer differential counting 0-dimensional cylindrical instantons, \(v\) is a holonomy cut-down map, and \(\delta_1,\delta_2\) count trajectories to and from the reducible \(\theta\). From this one constructs three equivariant \(\mathbb Z}[x]\)-complexes, with \(\deg x=-2\), whose homologies \(\widehat I_*(Y,K)\), \(\check I_*(Y,K)\), and \(\overline I_*(Y,K)\) fit into exact triangles analogous to Borel and Tate packages in equivariant Floer theory [1912.08982].

The closed-manifold theory is governed by surgery triangles. For a knot \(K\subset Y\), one has an exact triangle
\[
I^\#(Y_n(K)) \to I^\#(Y_{n+1}(K)) \to I^\#(Y)\to \cdots,
\]
with maps induced by the corresponding 2-handle cobordisms. In the absolute \(\mathbb Z/4\)-graded refinement, the three cobordism degrees are constrained so that their sum is \(-1\pmod 4\), and exactly one of the three cobordisms is non-spin [2003.03329]. Scaduto’s metric-stretching argument gives a related link-surgeries spectral sequence, and in the branched-cover context the \(E^1\)-page is a direct sum of framed instanton homologies of surgery manifolds [1401.2093].

A major structural development is the knot surgery formula in sutured instanton homology. For a rationally null-homologous knot \(K\subset Y\) and \(m\neq 0\), Li and Ye construct bent complexes \(A(K,s)\), \(B^\pm(K,s)\), together with maps \(\pi^\pm\) and an isomorphism \(\Xi_m\), and prove the mapping-cone formula
\[
I^\sharp(-Y_{-m}(K)) \cong H\!\left(\mathrm{Cone}\!\left(\pi^- + \Xi_m\circ \pi^+ :
\bigoplus_s H(A(K,s)) \to \bigoplus_s H(B^-(K,s))\right)\right).
\]
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 [2206.10077].

Another algebraic package appears for \(Y=\Sigma\times S^1\) with non-trivial bundle. There, framed instanton homology fits into the twisted Gysin exact sequence
\[
\cdots \to I^\#(Y)_w \to I(Y)'_w \xrightarrow{u^2-64} I(Y)'_w \to I^\#(Y)_w \to \cdots,
\]
so \(I^\#(Y)_w\) is the mapping cone of \(u^2-64\) on \(I(Y)'_w\). In Muñoz’s ring model for \(I(\Sigma\times S^1)'_w\), the \(u\)-map is denoted \(\beta\), and the nilpotency degree of \(u^2-64\) is
\[
\min\{n\ge 1:(u^2-64)^n=0\}=2\left\lceil g/2\right\rceil-1
\]
for genus \(g\) [1612.08690].

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 \(Y_\Gamma\) | \(I^\#(Y_\Gamma)\cong \widehat{HF}(Y_\Gamma;\mathbb C)\) as \(\mathbb Z/2\)-graded complex vector spaces | [2010.03800] |
| integral surgeries on a knot with an instanton L-space surgery | for \(g=g(K)\), \(I^\#(S^3_n(K))\) is given by a piecewise \(\mathbb Z/2\)-graded formula, and for \(n\ge 2g-1\) one has \(I^\#(S^3_n(K))\cong \mathbb C_{(0)}^n\) | [2003.03329] |
| nontrivial circle bundles \(Y_m^g\) over \(\Sigma_g\) | if \(|m|\ge 2g-1\), then \(\dim I^\sharp(Y_m^g)=2^{2g}|m|\); for smaller \(|m|\) there are explicit binomial-sum corrections | [2209.11018] |
| \(\Sigma\times S^1\) with non-trivial bundle \(w\) | \(\dim I^\#(\Sigma\times S^1)_w = 2(g+1)\binom{2g}{g}-2^g(1+2^g)\) | [1612.08690] |
| two-fold quasi-alternating branched covers | \(I^\#(\Sigma(L),\omega)\) is free abelian of rank \(\det(L)\) and supported in gradings \(\{0,2\}\subset \mathbb Z_4\) | [1605.05394] |

For almost-rational plumbings, the key mechanism is an identification of the even-graded part of \(I^\#\) with lattice homology \(H(\Gamma)\), followed by Némethi’s description of \(\widehat{HF}\) as \(\ker U\subset HF^+\). This yields the isomorphism \(I^\#(Y_\Gamma)\cong \widehat{HF}(Y_\Gamma;\mathbb C)\) and, in particular, establishes the Kronheimer–Mrowka conjectural correspondence on a large class containing all Seifert fibered rational homology spheres [2010.03800].

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 [2003.03329; 2209.11018].

The singular framed knot theory also admits concrete examples. For spherical knots, the ADHM description of instantons on \(S^4\) 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 \(h=0\), while the right-handed trefoil and the \((3,4)\) and \((3,5)\) torus knots have \(h=1\) in the notation of that paper [1912.08982].

A recurrent pattern is that explicit computations often determine only dimensions or parity gradings, while finer \(\mathbb Z/4\)-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 \(I^\#(Y)\cong \widehat{HF}(Y;\mathbb C)\) 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 \(\mathbb{RP}^2\) [2010.03800]. In a different direction, Wang proves the inequality
\[
\dim_{\mathbb C} I^\sharp(Y)\le \dim_{\mathbb C}KHI(Y,K)
\]
for all rationally null-homologous knots \(K\subset Y\), and constructs a decomposition of \(I^\sharp\) for Dehn surgeries that parallels torsion \(\mathrm{spin}^c\) splittings in monopole and Heegaard Floer theories [2010.07836].

The link to Khovanov-type theories begins with Scaduto’s spectral sequence
\[
E^2\cong \widetilde{Kh}^{\mathrm{odd}}(L)\Longrightarrow I^\#(\overline{\Sigma(L)}),
\]
where the \(E^2\)-page carries a compatible \(\mathbb Z/4\)-grading
\[
\frac{3}{2}q-t+\frac{1}{2}(\sigma+\nu)\pmod 4.
\]
For quasi-alternating links this spectral sequence collapses, implying that \(I^\#(\Sigma(L))\) is free of rank \(\det(L)\) and supported in even gradings [1401.2093]. Two-fold marked refinements extend this picture to non-trivial \(SO(3)\)-bundles over branched double covers: for two-fold quasi-alternating links, the twisted Khovanov theory is thin and the corresponding \(I^\#(\Sigma(L),\omega)\) is completely determined, including its \(\mathbb Z_4\)-grading distribution [1605.05394].

A further deformation replaces ordinary or odd Khovanov homology by Bar–Natan and \(F_5\) theories in characteristic 2. Kronheimer and Mrowka’s deformed framed instanton homology \(I^\sharp(K;\Gamma)\), built on the singular bifold \((Y,K^\sharp)\) with a local system over
\[
\mathcal R=\mathbb F_2[T_0^{\pm1},T_1^{\pm1},T_2^{\pm1},T_3^{\pm1}],
\]
fits into a spectral sequence whose \(E_2\)-page is \(F_5\)-homology, and after specialization yields a spectral sequence from Bar–Natan homology to \(I^\sharp(K;\Gamma_{\mathrm{BN}})\). The deformation is controlled by explicit elements
\[
P=T_1T_2T_3+T_1T_2^{-1}T_3^{-1}+T_2T_3^{-1}T_1^{-1}+T_3T_1^{-1}T_2^{-1},
\qquad
Q=\sum_{j=0}^3(T_j^2+T_j^{-2}),
\]
which determine the Frobenius algebra on the \(E_1\)-page [1910.11128].

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 \(I^\#(Y)\cong SHI(Y(1),\delta)\), so it mediates between closed instanton Floer groups, knot instanton homology, and surgery mapping cones [2010.07836]. 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 \(\nu^\sharp(K)\) and \(\tau^\sharp(K)\), where \(\nu^\sharp\) is extracted from the vanishing pattern of integer surgery cobordism maps and \(\tau^\sharp\) is the homogenization of \(\nu^\sharp\). They prove that \(\nu^\sharp\) is a smooth concordance invariant, that \(\tau^\sharp\) is a concordance homomorphism, and that
\[
\dim_{\mathbb C} I^\sharp(S^3_{p/q}(K))=q\cdot r_0(K)+|p-q\nu^\sharp(K)|
\]
for coprime \(p,q\) with \(q\ge 1\), subject to the special behavior at zero in the \(W\)-shaped case. They also show that \(2\tau^\sharp\) is a slice-torus invariant, that
\[
\mathrm{sl}(K)\le 2\tau^\sharp(K)-1\le \nu^\sharp(K),
\]
and that for instanton L-space knots one has \(\nu^\sharp(K)=r_0(K)=2g(K)-1\) [2004.08699].

The sequel develops a conjugation symmetry for decomposed cobordism maps, proves that \(\nu^\sharp(K)\) is either zero or odd, and defines
\[
\epsilon^\sharp(K):=2\tau^\sharp(K)-\nu^\sharp(K)\in\{-1,0,1\}.
\]
It also establishes a denominator bound for surgery descriptions: if \(Y=S^3_{p/q}(K)\) for a nontrivial knot \(K\) that is not a trefoil or the figure-eight, then
\[
q\le \tfrac13 \dim_{\mathbb Q} I^\#(Y).
\]
The same paper proves that for nonzero slopes the dimension of \(I^\#(S^3_{p/q}(K),\lambda)\) is independent of \([\lambda]\in H_1(S^3_{p/q}(K);\mathbb Z/2)\) [2206.11531].

A different concordance-oriented development uses the minus version \(KHI^{-}(Y,K)\). For specially decorated knot cobordisms \((W,\Sigma_g,\mathcal D)\), one has \(\mathbb C[U]\)-equivariant cobordism maps
\[
F_{W,\Sigma_g,\mathcal D}:KHI^{-}(Y_0,K_0)\to KHI^{-}(Y_1,K_1),
\]
and the tube attachment lemma states that attaching a tube to \(\Sigma_g\) multiplies the map by \(U\). From this, the torsion-order inequality
\[
\mathrm{ord}_U(K)\le \mathrm{br}(K)-1
\]
is recovered, and for alternating knots of bridge index at most \(3\) one obtains
\[
\dim_{\mathbb C} I^\sharp(S^3_r(K))=\mathrm{rk}_{\mathbb Z_2}\widehat{HF}(S^3_r(K))
\]
for every nonzero rational slope \(r\) [2312.15417].

Integral coefficients reveal substantial torsion phenomena. Li and Ye show that if \(I^\sharp(S^3_n(K);\mathbb Z)\) has no 2-torsion for some nonzero integral surgery, then \(K\) must be fibered, and that for every nontrivial knot the surgeries of slopes \(1\), \(1/2\), and \(1/4\) 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 [2405.16252]. Over \(\mathbb F=\mathbb Z/2\), an additional pair of invariants \(M(K),r_2(K)\in 4\mathbb Z\) governs surgery dimensions:
\[
\dim_{\mathbb F} I^\#(S^3_{p/q}(K),w;\mathbb F)=
\begin{cases}
r_2(K)+2,& p/q=M(K)\text{ and }[w]_2=0,\\
q\,r_2(K)+|p-qM(K)|,& \text{otherwise}.
\end{cases}
\]
The same work relates these dimensions to a Frøyshov-type invariant \(q_3\) and deduces that \(r\)-surgery on a nontrivial knot cannot be nondegenerate \(SU(2)\)-abelian for any
\[
|r|\le 4\left\lceil \frac{g(K)}{2}\right\rceil
\]
[2511.18885].

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 \(J_i^S\), \(\Gamma^R\), and Kronheimer–Mrowka concordance invariants [1912.08982]. The sutured surgery program isolates the \(m=0\) 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\(^c\)-like or equivariant refinement [2206.10077].

Source: https://www.emergentmind.com/topics/framed-instanton-homology