The Fundamental Group of Torus Knots

TL;DR

This paper calculates the fundamental group of torus knots using algebraic topology and group theory, and introduces an algorithm for arbitrary knots based on Wirtinger presentation.

Contribution

It provides a detailed computation of torus knot groups and presents an algorithm for fundamental group calculation of any knot.

Findings

Fundamental group of torus knots explicitly computed.

Algorithm for fundamental groups of arbitrary knots introduced.

Application of algebraic topology and group theory methods.

Abstract

This work is concerned with the calculation of the fundamental group of torus knots. Torus knots are special types of knots which wind around a torus a number of times in the longitudinal and meridional directions. We compute and describe the fundamental group of torus knots by using some concepts in algebraic topology and group theory. We also calculate the fundamental group of an arbitrary knot by using an algorithm called the Wirtinger presentation.