Una Mirada Inicial a la Teor\'ia de Nudos y a la Homolog\'ia de Khovanov

TL;DR

This paper provides an accessible introduction to Khovanov homology, a modern knot theory tool, including its historical background, mathematical construction, and significance, aiming to promote understanding in Latin America.

Contribution

It offers an expository overview of Khovanov homology and its long exact sequence, emphasizing its historical development and aiming to bridge language barriers in the field.

Findings

Introduces Khovanov homology from the Kauffman bracket polynomial

Summarizes the historical evolution of knot theory

Aims to popularize knot theory in Latin America

Abstract

The mathematical theory of knots studies the embeddings of circles into the space $\mathbb{R}^3$, being the classification one of the fundamental problems. The introduction of homology theories results in complex mathematical structures that generate new research opportunities. On occasion of the Encuentro Internacional de Matem\'aticas (EIMAT) (International Meeting of Mathematics) to be celebrated at the Universidad del Atl\'antico in Barranquilla, Colombia in November 2023, in this article, in an expository way, we offer a first look into Khovanov homology and the long exact sequence of Khovanov homology. Moreover, we present a summary of the historical origins of the theory which can take us as early as the year 2600 BCE, passing through Italy of the XV century, Scotland of the XIX century, and we give references for further and more detailed discussions. Additionally to showing the construction of Khovanov homology from the Kauffman bracket polynomial, the main objective in publishing this article is to attack the language barrier and popularize knot theory and Khovanov homology in Colombia and Latin-America in general.