Zeros Of The Jones Polynomial For Knots And Links And Their Distributions

The main subject of the knot theory in mathematics is searching for an ambient isotopy invariant which is able to distinguish the knots strongly, and can be calculated conveniently. Polynomials which is one of the research object in algebra, is a tool in researching many branches of mathematics. Particularly, polynomial opens up the road for the knot theory. The finding of Alexander polynomial is a milestone in the knot theory, but it can not distinguish the knot from its mirror image. In the 1984s, New Zealand mathematician Jones found a new invariant-Jones polynomial, it was an ambient isotopy invariant, and the calculation was convenient. His finding made the knot theory known as one of the focus at the mathematical field in the world.We have known a Laurent polynomial with integer coefficientsΔ(t) is a Alexander polynomial of some knot iff (1)Δ(1)=1:(2)Δ(t)=Δ(t^-1). For seeking when a Laurent polynomial with integer coefficients can be a Jones polynomial of some knot. Some scholars started from the relation between the Jones polynomial and the polynomial with integer coefficients whose degree was less than 5 and equaled to 5 for now. In this paper , she discusses the relations between the polynomials of degree 6 and 7 with integer coefficients and their constant coefficients are 0 by the theory of polynomials and matrix transformation, and gets some of the main results; We have known the zeros of the Jones polynomial of the torus knots T_p,q(where(p,q)=1) were distributed on the united circle, but if a unity was a zero of the Jones polynomial? She proves that(?)和(?)(m was a positive integer) can not be the zeros of the Jones polynomial for torus knots T_p,q by the knowledge of the trigonometric function; For the 2-bridge knot, nobody discussed the zeros distribution. She elicites the normal form of the Jones polynomials of the 2-bridge knot C(-2,2,···(-1)^r2) (r≥3) by the recursive form, and discussed the distribution of their zeros; For now, some scholars have discussed the zeros distribution of the Jones polynomials for the pretzel links P(k,k,k),(?),(?) and (?)(k>0). She discusses the Jones polynomials of the pretzel links (?) and (?)(k>0,l>0) and the distribution of their zeros by the means of special to general and thought of discussing classifyly when k ,l .and n take values classifyly.