Two Algorithms Distinguishing Distinct Link Diagrams Up To Planar Isotopy

Topology,which is developed in modern times,is an important and basic branch of mathematics,focusing on the phenomenon of continuity.Knot theory is an attractive field of topology,and its central topic is the classification of knots.In present,mathematicians have been able to determine whether two knots are equivalent.In theory,we can create a machine, which can determine whether two input of knot projections are equivalent.This can be determined theoretically,but not practically.In actual calculation,mathematicians have invented a number of Polynomial Invariants,and those invariants include more information than just the Alexander Polynomial.This article makes the following expansion and innovation,it introduces two algorithms distinguishing distinct link diagrams up to planar isotopy.This article is structured as follows:For the problems of classification we discussed is about knot theory,so,first of all,we introduce topology,knot theory and their history of development,and its application in various fields in the first chapter.In the second chapter,we introduce the basic concepts of knot theory,including regular projection,DT code,as well as the details on how to get a DT code from a knot projection. Finally,we have a little discussion on the number of DT codes in some special knots projections.ChapterⅢis a central part of this article,focusing on two methods to distinguish distinct Link diagrams up to Planar Isotopy.One method is an expansion of the literature[46],the other is finding a map from a knot diagram to another diagram which satisfies some restrictions.If we could find it,we can say that they are planar isotopic.Examples show that both of these methods are feasible.At last,we compare the two methods.