Links, 3-polytopes and some applications

众所周知，链环的投影图与符号平图一一对应，Kauffman以此建立了前者的Kauffman括号多项式与后者的Tutte多项式之间的关系。为了简化链环的Jones多项式的计算，本论文引入了universal图的概念，并建立了链环与三正则多面体之间的关系。我们也可以利用universal图来简化一部分有向链环的Homfly多项式的计算。本论文还研究了一类新的碳氢化合物－－苯型链环，确定了它的最小和第二小的结构。论文共分四章。在第一章，我们介绍了纽结理论的研究背景及我们的主要工作。在第二章，我们引入了universal图的概念，任一个链环的Kauffman括号多项式可由与其对应的universal图...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 Homflypolynomi...学位：理学博士院系专业：数学科学学院数学与应用数学系_应用数学学号：B20042301

Links and cubic 3-polytopes

It is well known that a prime link diagram corresponds to a signed plane graph without cut vertices (Kauffman, 1989). In this paper, we present a new relation between prime links and cubic 3-polytopes. Let S be the set of links such that each L. S has a diagram whose corresponding signed plane graph is the graph of a cubic 3-polytope. We show that all nontrivial prime links, except (2, n)-torus links and (p, q, r)-pretzel links, can be obtained from S by using some operation of untwining. Furthermore, we de. ne the generalized cubic 3-polytope chains and then show that any nontrivial link can be obtained from S by some untwining operations, where S is the set of links corresponding to generalized cubic 3-polytope chains. These results are used to simplify the computation of the Kauffman brackets of links so that the computing can be done in a unified way for many in finite families of links