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Suppose $\Sigma$ is a compact oriented surface (possibly with boundary) thathas genus zero, and L is a link in the interior of $(-1,1)\times\Sigma$. Weprove that the Asaeda-Przytycki-Sikora (APS) homology of L has rank 2 if andonly if L is isotopic to an embedded knot in $\{0\}\times\Sigma$. As aconsequence, the APS homology detects the unknot in $(-1,1)\times\Sigma$. Thisis the first detection result for generalized Khovanov homology that is validon an infinite family of manifolds, and it partially solves a conjecture inarxiv:2005.12863. Our proof is different from the previous detection resultsobtained by instanton homology because in this case, the second page ofKronheimer-Mrowka's spectral sequence is not isomorphic to the APS homology. Wealso characterize all links in product manifolds that have minimal suturedinstanton homology, which may be of independent interest.

Instanton homology and knot detection on thickened surfaces