關於結的同構Khovanov chain complex的性質

Abstract

在計算結的不變量時，Khovanov發展出了一套理論將結與鏈復形結合起來，稱作Khovanov homology。他所得到的結果是Jones Polynomial可以用Khovanov同調群表示出來。本論文要證明的是當兩個連通結的Khovanov鏈復形為同構時，這兩個結是相等的。首先我們用Bar-Natan論文裡的符號把結轉換成另一種圖形，叫做圖表示法(Graph Presentation)。之後再利用鏈復形cobordism的特性去說明如果它們為同構，那麼這兩個結的每個局部結構一定得保持相同，進一步說明它們是同樣的結。

When calculating link invariants, Khovanov has developed a theory which turns the link diagram into a chain complex of direct sum of graded vector spaces defined from the topological quantum field theory (TQFT), then assigns a suitable differential mapping to compute the homology of the chain complex. The result turns out to be related to the Jones polynomial of the link. First we assign each link a new diagram called the graph presentation by replacing the overcrossings of the link by edges, the circles in the Khovaonv chain complexes as vertices. It will help us study the structure of the chain complexes of the link. From Bar Natan’s papers, we use the mapping cobordisms of the chain complex to show that any two link diagrams which are isomorphic by Khovanov chain complexes, then the link diagrams are identical.