Some Applications Of The Alexander Polynomial

Topology is one of the most important part of pure mathematical research,and the studyof knot is becoming a more and more significant branch of topology.Knot theory is mainly about the characters of several circles embedding in 3-dimensional Euclidean space.Very similar to other branch of topology,the central part of knot theory is the classification of knots,in other words,given two knots which look like each other,ignoring their locations and shapes in space,whether they are the same one.Currently,a lot of knot invariants can help us to distinguish them,including the most common Alexander polynomial.This article is based on the Alexander polynomial and use other mathematicians’ research conclusions as references.For these two very special knots,168 and15610,their Alexander polynomials are exactly the same,but once we change a single crossing on any of their projections,the Alexander polynomial for new knots will be different and these differences will be showed in this article.The whole article is divided into the following 4 parts:Chapter 1 is to introduce the development of knot theory and the latest results in recent studies.Chapter 2 is to introduce the preparatory knowledge of knot theory and that is the cornerstone to knot theory.Chapter 3 is mainly about some important conclusions on Seifert surgery in Rolfsen.D’s article.These conclusions are significant which is this article extended from.Chapter 4 is the research conclusions on168 and15610.When they are changed a single crossing in any of their projections,what the coefficients of new knots’ Alexander polynomial will be and all the proofs will be displayed in this part.