  This paper establishes a new technique that enables us to access some
fundamental structural properties of instanton Floer homology. As an
application, we establish, for the first time, a relation between the instanton
Floer homology of a $3$-manifold or a null-homologous knot inside a
$3$-manifold and the Heegaard diagram of that $3$-manifold or knot. We further
use this relation to compute the instanton knot homology of some families of
$(1,1)$-knots, including all torus knots in $S^3$, which were mostly unknown
before. As a second application, we also study the relation between the
instanton knot homology $KHI(Y,K)$ and the framed instanton Floer homology
$I^\sharp(Y)$. In particular, we prove 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 we constructed a new decomposition of the framed
instanton Floer homology of Dehn surgeries along $K$ that corresponds to the
decomposition along torsion spin$^c$ decompositions in monopole and Heegaard
Floer theory.
