An enhanced Euler characteristic of sutured instanton homology

Abstract

For a balanced sutured manifold $(M,{\hat{I}}^3)$, we construct a decomposition of $SHI(M,{\hat{I}}^3)$ with respect to torsions in $H=H_1(M;\mathbb{Z})$, which generalizes the decomposition of $I^{♯}(Y)$ in previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin${}^c$ decompositions in $SFH(M,{\hat{I}}^3)$. Based on this decomposition, we define an enhanced Euler characteristic $\ddot{I}{‡}_{\mathrm{en}}(SHI(M,{\hat{I}}^3))\in \mathbb{Z}[H]/\pm H$ and prove that $\ddot{I}{‡}_{\mathrm{en}}(SHI(M,{\hat{I}}^3))=\ddot{I}‡(SFH(M,{\hat{I}}^3))$. This provides a better lower bound on ${\dim }_{\mathbb{C}}SHI(M,{\hat{I}}^3)$ than the graded Euler characteristic $\ddot{I}{‡}_{\mathrm{gr}}(SHI(M,{\hat{I}}^3))$. 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.

45 pages, 12 figures; v2, published version