Heegaard Floer homology and plane curves with non-cuspidal singularities

TL;DR

This paper explores the application of Floer theory to classify singular points on algebraic curves in complex projective planes, providing a new formula for the $H_1$-action on knot Floer complexes.

Contribution

It introduces a general formula for the $H_1$-action on the knot Floer complex of linkifications, linking Floer theory with algebraic curve singularity configurations.

Findings

Derived a formula for the $H_1$-action on knot Floer complexes

Connected Floer theory with algebraic curve singularity analysis

Potentially new insights into non-cuspidal singularities

Abstract

We study possible configurations of singular points occuring on general algebraic curves in $\mathbb{C}P^2$ via Floer theory. To achieve this, we describe a general formula for the $H_{1}$-action on the knot Floer complex of the knotification of a link in $S^3$, in terms of natural actions on the link Floer complex of the original link. This result may be interest on its own.