Nudos y Superficies

TL;DR

This paper introduces knot theory concepts focusing on surfaces, including isotopies, Reidemeister moves, invariants like Alexander polynomial, and the group structure of knots via concordance and connected sum.

Contribution

It provides an accessible introduction to knot theory from a surface perspective, covering fundamental invariants and the algebraic structure of knots.

Findings

Introduction of knot invariants via Seifert surfaces

Explanation of the group structure on knots through concordance

Discussion of fundamental concepts like isotopies and Reidemeister moves

Abstract

These notes are an introduction to knot theory from the perspective of surfaces. The notes cover fundamental concepts such as isotopies, Reidemeister moves, torus knots, and (orientable, connected) surfaces with one boundary component. They also present knot invariants defined through Seifert surfaces and their associated matrices, including the 3-genus, the Alexander polynomial, and the signature. Finally, starting from the notion of concordance and the connected sum operation, a group structure on the set of knots is introduced.