An enhanced Euler characteristic of sutured instanton homology

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.