Knowing a knot from The Abelianization of its Fundamental Group on \(S_3\)

Abstract

The main goal of this article is to determine the nature of a knot set named twotwist \((m,n)\), where \(m\), \(n\) are whole positive numbers, using the abelianization of its fundamental group on \(S_{3}\). First, a basic theory to define the concept of knots as topological objects was established. Then, a detailed description of a process named "Mechanical Wheel," which was created and developed in this article to identify such abelianization, was provided. Lastly, a description of each two-twist for all \(m\) and \(n\) was attained.