  For a balanced sutured manifold $(M,\gamma)$, we construct a decomposition of
$SHI(M,\gamma)$ with respect to torsions in $H=H_1(M;\mathbb{Z})$, which
generalizes the decomposition of $I^\sharp(Y)$ in previous work of the authors.
This decomposition can be regarded as a candidate for the counterpart of the
torsion spin$^c$ decompositions in $SFH(M,\gamma)$. Based on this
decomposition, we define an enhanced Euler characteristic $\chi_{\rm
en}(SHI(M,\gamma))\in\mathbb{Z}[H]/\pm H$ and prove that $\chi_{\rm
en}(SHI(M,\gamma))=\chi(SFH(M,\gamma))$. This provides a better lower bound on
$\dim_\mathbb{C}SHI(M,\gamma)$ than the graded Euler characteristic $\chi_{\rm
gr}(SHI(M,\gamma))$. As applications, we prove instanton knot homology detects
the unknot in any instanton L-space and show that the conjecture $KHI(Y,K)\cong
\widehat{HFK}(Y,K)$ holds for all $(1,1)$-L-space knots and constrained knots
in lens spaces, which include all torus knots and many hyperbolic knots in lens
spaces.
