  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 $M\backslash N(K)$ 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 $L\subset
S^3$ is a fixed link and $K$ is a knot in the complement of $L$, then the
instanton link Floer homology of $L\cup K$ achieves the minimum rank if and
only if $K$ is the unknot in $S^3\backslash L$.
