A Polynomial Invariant On Knots

Knot theory is a branch of geometric topology, which is the study of how to embedded several circles to the real three-dimensional Euclidean space. The core issue of knot theory is the classification of knots. One simplified version of this question is the following:If we have a projection of a knot,can we tell whether it is the unknot; If we have the projections of two knots,can we tell whether they are equivalence.There are many invariants which can help us to distinguish the knots,such as the component number of the links and polynomial invariants. For the polynomial invariants, the most common are Alexander polynomials, Conway polynomials,Jones polynomials and HOMFLY polynomials.The main contents of this paper is organized as follows:In chapter one, we describe the development and results of knot theory.In chapter two, we review some basic knowledge of knot theory and the important results have been achieved, we introduce knots, the projection of knots and links, prime and composite knots and related knowledge points.In chapter three, we focus on reviewing the Alexander polynomial, Conway polynomial and the Jones polynomial, and other related knowledge of knot polynomials.In chapter four, we define a knot invariant through the skein relation with two equations on the basis of the theoretical results, we give the proof of the new definined polynomial invariants through the use of induction for many times.