On Floer minimal knots in sutured manifolds

Abstract

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