Polynomials With Integer Coefficients Of Knots And The Jones Polynomial

In this paper, we mainly deal with some relations between the polynomials with integer coefficients of knots and the Jones polynomial by using their basic properties and some special values of Jones polynomial.In section one, we introduce briefly some basic concepts and conclusions to be involved in this paper about knots and links. Such as equivalence of knots、the invariant of knot groups and the method of judging knots. In section two, we introduce some basic properties of links. Let VK(t) be the Jones polynomial of knot And the relations between the oriented knot and its mirror image of the Jones polynomial. In section three, we give necessary and sufficient conditions that a Laurent polynomial with integer coefficients, whose breadth is less than six, is the Jones polynomial of a certain knot. We give necessary conditions that a Laurent polynomial, with degree ten or eleven, is the Jones polynomial of a certain knot. When breadth is three, the Jones polynomial of trefoil knot is t+t3-t4; When breadth is four, the Jones polynomial of a knot is When breadth is five, the Jones polynomial of knots are and t3+t5-t8. In section four, we introduce when a polynomial with ten degree, we give necessary conditions that the Jones polynomial of a certain knot and when a polynomial with eleven degree, we give necessary conditions that the Jones polynomial of a certain knot.The Jones polynomial is one of important invariant of knots, which play an important role in the classification of knots. The basic properties of Jones polynomials can also be applied to the breadth is greater than six of the polynomial, the corresponding conclusions about the knot.