Links, 3-polytopes And Some Applications

It is well known that there is a one-to-one correspondence between link diagrams and signed plane graphs.Basing on this fact,Kauffman established a relation between the Kauffman brackets of links and the Tutte polynomials of the corresponding signed plane graphs.In this dissertation,we define universal graphs to simplify the calculations of the Jones polynomials of links and the Homfly polynomials of a type of oriented links.Then we establish a relation between links and cubic 3-polytopes,and reduce the set of universal graphs to the graphs of cubic 3-polytopes and two special graphs {R₁¹,R₂¹}.Finally we study new configurations of benzenoid hydrocarbons,called benzenoid links,and determine the structures of the smallest and the second smallest nontrivial benzenoid links.This dissertation includes four chapters.In the first chapter,we introduce some backgrounds and recall some definitions in knot theory.Then the main results of this dissertation are presented.In the second chapter,we introduce the concept of universal graphs.The Kauffman bracket of any prime link can be easily obtained by the chain polynomial of the corresponding universal graph by a special parametrization.Using this concept,we can simplify the computations of the Kauffman brackets of links so that the computing can be done in a unified way for many infinite families of links. Thus we also simplify the calculations of the Jones polynomials of links.We reduce the universal graphs to be the cubic 3-connected graphs,that is the graphs of cubic 3-polytopes,and two special graphs {R₁¹,R₂¹}(see Fig.2.12) step by step.Then we establish a relation between the prime links and cubic 3-polytopes.Let S be the set of links such that each L∈S has a diagram whose corresponding reduced signed plane graph is the graph of a cubic 3-polytope.We show that all nontrivial prime links,except(2,n)-torns links and(p,q,r)-pretzel links,can be obtained from S by using some operations of untwining.Thus we find an unexpected relation between links and a classical research object,that is the cubic 3-polytopes.Furthermore we extend this relation to general links and generalized cubic 3-polytope chains(cf. Definition 2.3.34).Finally we establish a relation between the crossing number of the link diagrams and the cyclomatic numbers of the corresponding reduced signed plane graphs,and exhaust all the chain polynomials of the 11 universal graphs with cyclomatic number less than 7.Then we can obtain the Kauffman brackets of all the links with crossing numbers no more than 13,and many infinite families of links with crossing numbers more than 13 from the chain polynomials of these 11 graphs.In the third chapter,we simplify the calculations of the Homfly polynomials of a type of oriented links by using the results in the precious chapter and the relation between the Homfly polynomials of this type of oriented links and the chain polynomials of their corresponding plane graphs.Similarly,we establish a relation between the crossing numbers of this type of oriented link diagrams and the cyclomatic numbers of the corresponding plane graphs.Then we can obtain the Homfly polynomials of all of this type of oriented links with crossing numbers no more than 13,and many infinite families of this type of oriented links with crossing numbers more than 13.In the final chapter,we study new configurations of benzenoid hydrocarbons, called benzenoid links,we determine the structures of the smallest and the second smallest nontrivial benzenoid links of three different types.Their numbers of Kekule structures are also provided.Finally we list all the benzenoid Hopf links of typeâ…¢with 22 - 25 hexagons by their canonical codes.We hope that the chemist can synthesize these benzenoid links in the future.