On the geography and botany of knot Floer homology

TL;DR

This paper investigates the limitations and distinguishing capabilities of knot Floer homology, revealing new constraints, detection results, and the existence of infinite families of knots sharing the same Floer homology.

Contribution

It demonstrates that not all bigraded groups satisfying known constraints are realizable as knot Floer homology and shows Floer homology can distinguish certain knots and detect the trefoil.

Findings

Existence of bigraded groups not arising from knots.

Rank of knot Floer homology detects the trefoil.

Infinite families of knots share the same Floer homology.

Abstract

This note explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.