Suppose $(M, gamma)$ is a balanced sutured manifold and $K$ is a rationally null-homologous knot in $M$. It is known that the rank of the sutured Floer homology of $Mbackslash N(K)$ is at least twice the rank of the sutured Floer homology of $M$. This paper studies the properties of $K$ when the equality is achieved for instanton homology. As an application, we show that if $Lsubset S^3$ is a fixed link and $K$ is a knot in the complement of $L$, then the instanton link Floer homology of $Lcup K$ achieves the minimum rank if and only if $K$ is the unknot in $S^3backslash L$.
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