本文選題：Frobenius代數(shù)　切入點(diǎn)：配邊理論　出處：《大連理工大學(xué)》2016年博士論文

【摘要】：Khovanov同調(diào)是紐結(jié)Jones多項(xiàng)式不變量的范疇化。自從1999年M. Khovanov提出這個(gè)理論以來,它就一直被眾多的拓?fù)鋵W(xué)者所關(guān)注。近年來,Khovanov同調(diào)理論已然取得了豐碩的研究成果。目前,關(guān)于此同調(diào)理論的進(jìn)一步推廣以及計(jì)算問題是該領(lǐng)域研究的熱點(diǎn)。本文構(gòu)造一個(gè)新的鏈環(huán)同調(diào)理論,它是原Khovanov同調(diào)理論的推廣,我們稱之為"Khovanov型”同調(diào)。對于這個(gè)新型的同調(diào)理論,我們給出詳細(xì)的幾何解釋,同時(shí)計(jì)算Kanenobu紐結(jié)的Khovanov同調(diào)及排叉結(jié)的Khovanov型同調(diào)。主要工作如下：1、對醍系數(shù)下Kanenobu紐結(jié)K(p,g)計(jì)算其Khovanov同調(diào),并得到一個(gè)遞推公式。計(jì)算結(jié)果表明：K(p,q)的Khovanov同調(diào)群的秩是關(guān)于p+q的函數(shù)。在計(jì)算過程中,我們使用紐結(jié)同調(diào)理論中基本的長正合列和一些關(guān)于Khovanov-thin紐結(jié)的結(jié)論。2、從構(gòu)造一個(gè)Frobenius代數(shù)出發(fā),通過拓?fù)淞孔訄稣?TQFT)作用,逐步提出Khovanov型同調(diào)理論,進(jìn)而得出此同調(diào)是一個(gè)紐結(jié)不變量。通過引進(jìn)虧格生成算子,給出此同調(diào)的幾何解釋,證明Khovanov型同調(diào)是紐結(jié)的Jones多項(xiàng)式的范疇化,并計(jì)算了T2,k(k∈N)環(huán)面鏈環(huán)的Khovanov型同調(diào)。3、計(jì)算排又紐結(jié)P(-n,-m, m)一般環(huán)R上的Khovanov型同調(diào),并給出遞推公式。計(jì)算結(jié)果表明：排叉紐結(jié)P(-n,-m, m)的Khovanov型同調(diào)是一個(gè)關(guān)于n的紐結(jié)不變量。在計(jì)算過程中,通過解開交叉點(diǎn)的兩種方法找到同調(diào)生成元的來源,簡化了Khovanov型同調(diào)計(jì)算的復(fù)雜度,從而給出計(jì)算此種紐結(jié)鏈環(huán)同調(diào)的一種新方法。
[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，

本文編號：1654824