Properties And Calculations Of Knot Invariants

A fundamental question of knot theory is how to distinguish different knots.It is important to find knot invariants which is easy to calculate and has srtong ability to distinguish knots.Knot invariants can help us to classify knots.Common knot invariants include tricolorablity of knot diagrams,the linking number of a two-components oriented link,knot groups and polynomials of knots,among which knot polynomials is an important part of knot invariants.The goal of this paper is to introduce the properties and calculations of three knot invariants.These three invariants are: regional knot invariants,knot invariants with multiple skein relations and non-homogeneous knot invariants.Like the Wirtinger presentation of a knot group,a link diagram divides the plane into many regions.Then for each region one can introduce a generator,and each crossing one can introduce a relator.Then one can get an algebraic structure called a Tridle.Tridle also has a linear version.We call it a linear Tridle.One can get a presentation matrix for a linear Tridle,and then a polynomial invariant is obtained.We then define knot invariants using skein relations.There are several differences between knot invariants with multiple skein relations and the traditional polynomial invariants.First,several new ways to smooth crossings are introduced into skein relations.Second,for any crossings,different skein relation is selected according to whether the two strands of the crossing come from the same component or not.In this paper,a simplified version of knot invariants with multiple skein relations is given.The simplified version is easy to calculate and is an extension of Kauffman 2-variable polynomial.Finally,a non-homogeneous knot invariant is introduced.It has four skein relations,and some of them contain non-zero constant terms.This kind of non-homogeneous invariants has never appeared before.This paper will discuss the relation between the coefficients in the skein relations and the relation between the non-zero constant and the coefficients.