Khovanov同调型理论的研究

发布时间：2018-03-23 19:20

  本文选题：Frobenius代数　切入点：配边理论　出处：《大连理工大学》2016年博士论文

【摘要】：Khovanov同调是纽结Jones多项式不变量的范畴化。自从1999年M. Khovanov提出这个理论以来,它就一直被众多的拓扑学者所关注。近年来,Khovanov同调理论已然取得了丰硕的研究成果。目前,关于此同调理论的进一步推广以及计算问题是该领域研究的热点。本文构造一个新的链环同调理论,它是原Khovanov同调理论的推广,我们称之为"Khovanov型”同调。对于这个新型的同调理论,我们给出详细的几何解释,同时计算Kanenobu纽结的Khovanov同调及排叉结的Khovanov型同调。主要工作如下：1、对醍系数下Kanenobu纽结K(p,g)计算其Khovanov同调,并得到一个递推公式。计算结果表明：K(p,q)的Khovanov同调群的秩是关于p+q的函数。在计算过程中,我们使用纽结同调理论中基本的长正合列和一些关于Khovanov-thin纽结的结论。2、从构造一个Frobenius代数出发,通过拓扑量子场论(TQFT)作用,逐步提出Khovanov型同调理论,进而得出此同调是一个纽结不变量。通过引进亏格生成算子,给出此同调的几何解释,证明Khovanov型同调是纽结的Jones多项式的范畴化,并计算了T2,k(k∈N)环面链环的Khovanov型同调。3、计算排又纽结P(-n,-m, m)一般环R上的Khovanov型同调,并给出递推公式。计算结果表明：排叉纽结P(-n,-m, m)的Khovanov型同调是一个关于n的纽结不变量。在计算过程中,通过解开交叉点的两种方法找到同调生成元的来源,简化了Khovanov型同调计算的复杂度,从而给出计算此种纽结链环同调的一种新方法。
[Abstract]:Khovanov homology is the categorization of knots Jones polynomials invariants. Since Khovanov put forward this theory in 1999, it has been concerned by many topologists. In recent years, Khovanov homology theory has made a lot of research results. In this paper, a new chain homology theory is constructed, which is a generalization of the original Khovanov homology theory. We call it "Khovanov type" homology. For this new homology theory, we give a detailed geometric explanation. At the same time, the Khovanov homology of Kanenobu knots and the Khovanov homology of row junction are calculated. The main work is as follows: 1. The Khovanov homology of Kanenobu knots is calculated for Kanenobu knots under the Khovanov coefficient. A recursive formula is obtained. The result shows that the rank of Khovanov homology group is a function of p Q. By using the basic long positive sequence in the homology theory of knots and some conclusions about Khovanov-thin knots, starting from the construction of a Frobenius algebra, we propose the homology theory of Khovanov type step by step through the action of topological quantum field theory. By introducing genus generating operator, the geometric explanation of homology is given, and it is proved that the homology of Khovanov type is the categorization of Jones polynomials of knots. We also calculate the homology of Khovanov type. 3 and the Khovanov type homology on the general ring R of T _ 2 K ~ K _ k 鈭，

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