Jones Polynomial And The Distribution Of Their Zeros Of Links

In this dissertation, we mainly focus on the computation of Jones polynomial and some its related polynomials of links in knot theory and analyze the distribution of their zeros. The methods we used are basically combinatorial. In order to calculate the Jones polynomials of some familes of links, we develop some results of several kinds of graph polynomials. This dissertation includes four chapters altogether.In the first chapter, we make praperations for the subsequent chapters. We present some defintions on links firstly, then the one-to-one correspondence between link diagrams and signed plane graphs, and finally the Jones polynomials of links and some related polynomials. All results in this chapter are known.In the second chapter, we study graph polynomials and obtain the following three main results: the first is the expansion of the flow polynomial for the general graph using the concept of broken cuts. This is the dual results of Whitney's expansion of the chromatic polynomial using broken cycles; the second is to introduce a new graph polynomial, W-polynomial, which includes many graphs polynomials as its special cases; the third is the splitting formula of the Negami polynomials for doubly weighted graphs.In the third chapter, we calculate the Jones polynomial of a family of links based on wheels firstly, then we establish the relations between the Kauffman brackets of link diagrams and the chain, the sheaf, and the W-polynomials of graphs. By means of this relations we obtain the Kauffman brackets for some equivalence classes of links and some familes of rational links. Finally we associate each plane graph with two links and obtain the relatons of the Homfly polynomials of the two links with the Whitney-Tutte polynomial of the graph.In the final chapter, stimulated by the close connection between the Jones polynomial in knot theory and the Potts model in statistical mechanics, we study the zeros distribution of the Jones polynomials for some familes of links, and determine the accumulation sets of their zeros successfully.