An Enhanced Euler Characteristic of Sutured Instanton Homology

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