The Derivative Quality Of Homfly Polynomial Of Knots And Links

The main content of the knot theory is searching for the ambient isotopy invariant which is able to distinguish the knots and easy to compute. The finding of knot polynomial provides the possibility of classifying the knots.Now, some scholars have given a few of the derivative qualities of Conway polynomial. For example, the derivatives {(?)_L,d(?)_L/dt,...d~n(?)_L/dt~n}are linear independent. Because between Conway polynomial and Alexander polynomial can mutually transform, so whether the derivative of Alexander polynomial have the similar qualities? She uses the rule of derivative and matrix theory, discusses the qualities of the Alexander polynomial.In the 1980s, Freyd et al. found a invariant-Laurent polynomial which has two-variant, they called it Homfly polynomial, after appropriate replacement, the polynomial can express the same knot or the same link which is correspond to Alexander polynomial and Jones polynomial. For now, scholars have obtain the derivative qualities of Jones polynomial, especially, they have the qualities of divisibility when t = 1. Further, we think to research the derivative qualities of the general polynomial---Homfly polynomial for arbitrary knots and links. First, she give the divisibility of some knots and links at special value by the knowledge of the partial derivatives, Next, she makes use of inductive method and gives the higher order partial derivative of Homfly polynomial in arbitrary knots.By discussing these qualities, we can find the qualities of the coefficients of the polynomial of the links; with the deep discussion, we also can obtain the invariant qualities of three-manifold.