Knot Equivalence Through Braids and Rational Tangles

A major goal in the study of knot theory is to discover more practical and universal methods that determine knot equivalence. In this paper, we will explore two methods of doing so through the use of rational tangles and braids. We begin by determining a method of converting from knots to rational tangles, then find equivalence between the tangles by means of continued fractions and Prime Fractional Sets, and then convert the tangles back into knots. This leads to the conclusion that two knots have corresponding rational tangles which are equivalent if and only if the original two knots were themselves equivalent. Through a similar course of events, we then do the same with braids through the use of Markovs Moves and come to the conclusion of Markovs Theorem, where two knots are shown to be equivalent if and only if their corresponding braids are determined to be Markov equivalent.