# Genome of Descartes Folium via Normalization

## TL;DR

This paper explores the algebraic and topological structures of the Descartes Folium, revealing natural group laws and exotic structures through normalization and diagram manipulation techniques.

## Contribution

It introduces a normalization process for the Descartes Folium and uncovers its hidden algebraic and topological group structures using novel diagram-based methods.

## Key findings

- Descartes Folium has natural group structures
- Normalization reveals exotic algebraic structures
- Diagram techniques effectively analyze algebraic curves

## Abstract

The Folium of Descartes in $\mathbb{K}\times\mathbb{K}$ carries group laws, defined entirely in terms of algebraic operations over the field $\mathbb{K}$. The problems discussed in this paper include: normalization of Descartes Folium, group laws and morphisms, exotic structures, exotic structures, second exotic structure, some topologies on Descartes Folium, differential structure on Descartes Folium, first isomorphism of algebraic Lie groups over $\mathbb{K}$, second isomorphism of algebraic Lie groups over $\mathbb{K}$, derived structures of algebraic Lie groups, a differential/complex analytic structure on Descartes Folium, Descartes Folium as a topological field, etc. For predicting these terms, we focus on methods that exploit diagram manipulation techniques (as alternatives to algebraic method of proofs). All our results confirm that the Descartes Folium stores natural group structures, unsuspected till now.