An enhanced Euler characteristic of sutured instanton homology

Abstract

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 $I^{♯}(Y)$ 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 ${\chi }_{\mathrm{en}}(SHI(M,\gamma ))\in \mathbb{Z}[H]/\pm H$ and prove that ${\chi }_{\mathrm{en}}(SHI(M,\gamma ))=\chi (SFH(M,\gamma ))$. This provides a better lower bound on ${\dim }_{\mathbb{C}}SHI(M,\gamma )$ than the graded Euler characteristic ${\chi }_{\mathrm{gr}}(SHI(M,\gamma ))$. 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.

45 pages, 12 figures; v2, published version

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Cited by in corpus (4)

• Small Dehn surgery and SU(2)
• SU(2) representations and a large surgery formula
• Instantons and Khovanov homology in ${\mathbb{RP}}^3$
• Knot surgery formulae for instanton Floer homology I: the main theorem