Knot Floer homology of some even 3-stranded pretzel knots

TL;DR

This paper computes the Knot Floer Homology of certain 3-stranded pretzel knots using immersed curve techniques, correcting previous results and revealing larger homology ranks in specific cases.

Contribution

It applies the theory of peculiar modules and immersed curves to accurately compute $ ext{HFK}$ for specific pretzel knots, correcting earlier inaccuracies.

Findings

Corrected previous computations of $ ext{HFK}$ for these knots.

Discovered larger homology ranks than previously predicted.

Provided explicit calculations for a family of pretzel knots.

Abstract

We apply the theory of "peculiar modules" for the Floer homology of 4-ended tangles developed by Zibrowius (specifically, the immersed curve interpretation of the tangle invariants) to compute the Knot Floer Homology ($\widehat{HFK}$) of 3-stranded pretzel knots of the form ${P(2a,-2b-1,\pm(2c+1))}$ for positive integers $a,b,c$. This corrects a previous computation by Eftekhary; in particular, for the case of ${P(2a,-2b-1,2c+1)}$ where $b<c$ and $b<a-1$, it turns out the rank of $\widehat{HFK}$ is larger than that predicted by that work.