The Jones Polynomial And Properties Of T(4,n)

In this paper, we studied the Jones polynomial of a class of torus knot and its properties. Torus is a class of interlinkage that got the earliest system research in the history. Torus is to occupy the important position of knot, and explore the nature of the torus knot, will help us to deep the understanding of the nature of the interlinkage. This paper mainly studied the kauffman polynomial of torus knot, which launched the Jones polynomial of a torus knot. and explored the chiral and cross index properties torus knot.The first part we mainly derived the Jones polynomial of the torus knot. In Abdullah KOPUZLU, Abdulgani S, AHIN and Tamer UGUR’s paper "On Polyno-mials of K(2,n) Torus Knot", they found the recursive relation and Jones polynomials of T2,n torus knot. From T2,n to T4,n,the standard projection and polynomials of the torus knot changes a lot. referencing the method of the two authors, we deduce the Kaufman Polynomial and the Jones Polynomial of T4,nThe second part we mainly explored the important properties of torus knot, including the chiral,reversibility and isotopic property etc. Among them, Exploring the same mark invariance enabled us to get the complete classification of torus knot. Using the Jones polynomial, we calculated cross indicator of the torus knot. For connected sum of torus knot, we also got a very good properties of torus knot that the cross index of two torus knots connected sum is equal to two turus knots connected added together. common link do not have this property.