A note on the rank of Heegaard Floer homology

TL;DR

This paper proves that for a non-trivial knot in a homology sphere, the knot Floer homology rank exceeds the Heegaard Floer homology rank of the ambient space, highlighting a fundamental inequality in knot theory.

Contribution

It establishes a strict inequality between the ranks of knot Floer homology and Heegaard Floer homology for non-trivial knots in homology spheres, providing new insight into their relationship.

Findings

Rank of knot Floer homology is strictly greater than that of Heegaard Floer homology for non-trivial knots.

The result applies to knots inside homology spheres, emphasizing their topological complexity.

The inequality offers a new tool for distinguishing non-trivial knots from trivial ones.

Abstract

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