The Calculation Of Jones Polynomial Of 8-plat Knots By Matrix

Knot theory is a branch of geometric topology.It has many mathematical fields.In knottheory,seeking a link invariant is a very important part,that is,how to determine whether a knot can be transformed into an unknot through a series of Reideimeister moves.Using knot invariants,we can perform the classification for knots with crossings at most 16.Knot polynomial is the core of the knot invariants.Classical knot polynomials mainly include angle brackets polynomial,X polynomial,Jones polynomial,Alexander polynomial,and HOMFLY polynomial.The Jones polynomial is an important knot polynomial proposed by Vaughan Jones in1984.All knots with at most 9 crossings have different Jones polynomials.Jones polynomials also brought a lot of mathematicians in the work of the knot theory.Jones won the 1990 Fields Award.The calculation of Jones polynomial is very tme consuming.By definition,it is necessary to split each crossing,so there are 2 ^ n entries for n crossings.Obviously,the computational complexity is exponential.In this paper,a new method for calculating Jones polynomials is given for a knot with a special form.This method of calculation is obtained by calculating the product of the matrix.Each time you break a crossing,multiply a matrix,the computational complexity is linear,making the original calculation much easier.In chapter one,we introduce the research background,research objective of this paper.The research background is divided into the source and history of knots,the development of knot theory,the application of knot theory in mathematics,and the application of knot theory.In chapter two,we introduce the knowlegde for the theorems and the proofs in this paper,includes knot,link,polynomial of knot,braid,braid group,plat and tangle.In chapter three,we give the method of the calculation and its proof with the form of theorem,which is the core of the paper.