On Floer minimal knots in sutured manifolds

TL;DR

This paper investigates the properties of knots in sutured manifolds when the sutured Floer homology rank reaches a minimal value, revealing conditions under which knots are unknots in link complements.

Contribution

It characterizes knots achieving minimal sutured Floer homology rank in sutured manifolds, especially in the context of instanton homology and link complements.

Findings

Minimal rank of sutured Floer homology implies the knot is an unknot in the complement.

Achieving minimal instanton link Floer homology corresponds to the knot being trivial.

Provides criteria for identifying unknots via Floer homology ranks.

Abstract

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 $S^3\backslash L$.