A note on a Geography problem in knot Floer homology

TL;DR

This paper proves that for a specific class of knots, their knot Floer homology is non-trivial in a certain Alexander grading, partially answering a broader question about all non-trivial knots.

Contribution

It establishes non-triviality of knot Floer homology in a specific Alexander grading for a class of knots, advancing understanding in knot theory.

Findings

Knot Floer homology is non-trivial in next-to-top Alexander grading for certain knots.

Partial affirmative answer to Baldwin and Vela-Vick's question.

Provides new insights into the structure of knot Floer homology.

Abstract

We prove that knot Floer homology of a certain class of knots is non-trivial in next-to-top Alexander grading. This gives a partial affirmative answer to a question posed by Baldwin and Vela-Vick which asks if the same is true for all non-trivial knots in $S^3$.