Knots in homology spheres which have simple knot Floer homology are trivial

TL;DR

This paper proves that in homology spheres, knots with simple knot Floer homology must be trivial, establishing a clear link between knot complexity and Floer homology ranks.

Contribution

It demonstrates that non-trivial knots in homology spheres cannot have simple knot Floer homology, providing a new criterion for knot triviality.

Findings

Non-trivial knots have higher Floer homology rank than the ambient sphere.

Simple knot Floer homology implies the knot is trivial.

The result characterizes trivial knots via Floer homology complexity.

Abstract

We show that if K is a non-trivial knot inside a homology sphere X, the rank of the knot Floer homology group associated with K is strictly bigger than the rank of the Heegaard Floer homology group associated with X.