Morse theory for plane algebraic curves

TL;DR

This paper applies Morse theory and link invariants to analyze the topology of algebraic curves in complex two-dimensional space, providing insights into their singularities through topological methods.

Contribution

It introduces a Morse-theoretic approach combined with link invariants to study the topology and singularities of algebraic curves in C^2.

Findings

Link changes at singular points are characterized using Morse theory.

Link invariants provide constraints on possible singularities.

Topological methods yield new insights into algebraic curve singularities.

Abstract

We use Morse theoretical arguments to study algebraic curves in C^2. We take an algebraic curve C in C^2 and intersect it with a family of spheres with fixed origin and varying radii. We explain in detail how does the resulting link change when we cross a singular point of C. Applying link invariants as Murasugi's signature and Levine--Tristram signatures we obtain some informations about possible singularities of a curve C in terms of its topology.