The Differential Properties Of Alexander Polynomial Of Knots

Knot theory is a branch of topology in mathematics,and the equivalent classification of knots is a significant problem in the field of knots.We are looking for invariants that obtain the ability to distinguish different knots and are easy to calculate to classify knots.Knot invariant is a very important method to study knot classification.Polynomial,in the meanwhile,is one of the basic research objects of algebra,and plays as a tool for studying many branches of mathematics.In this paper,each differential of Alexander polynomial of knots is calculated by the skein relation and the general derivation rule to study the divisive property of results when putting the variable t a particular value.Based on the principle of promotion from special to general,we start with the trivial knot.By calculating Alexander polynomial of the Twist knot,the knot of composition by n Twist knots,(3,m)torus knot and the link of composition by n trivial knots in the method of Hopf,we figure out the divisive property of results when putting the variable t a particular value.Eventually we obtain the formula of each differential of Alexander polynomial of knots when t=4.So we are curious about the relationship between the value of t and the result.After calculating differential of higher order of Alexander polynomial relationship,we obtain the formula of each differential of Alexander polynomial of knots.