  Suppose $\Sigma$ is a compact oriented surface (possibly with boundary) that
has genus zero, and L is a link in the interior of $(-1,1)\times\Sigma$. We
prove that the Asaeda-Przytycki-Sikora (APS) homology of L has rank 2 if and
only if L is isotopic to an embedded knot in $\{0\}\times\Sigma$. As a
consequence, the APS homology detects the unknot in $(-1,1)\times\Sigma$. This
is the first detection result for generalized Khovanov homology that is valid
on an infinite family of manifolds, and it partially solves a conjecture in
arxiv:2005.12863. Our proof is different from the previous detection results
obtained by instanton homology because in this case, the second page of
Kronheimer-Mrowka's spectral sequence is not isomorphic to the APS homology. We
also characterize all links in product manifolds that have minimal sutured
instanton homology, which may be of independent interest.
