Khovanov同調群的研究

Abstract

Khovanov發現了一個神祕的環的不變量。大致的想法是造一個chain complex，使它的尤拉特徵數和它的Jones多項式一樣，然後我們就計算這個chain complex的同調群。這是一個環的不變量，但它每一項所代表的意義尚未被發現。有一些專家們發現了許多它的特性。Dror Bar-Natan的論文裡記載一個特別令人驚奇的性質，它已經被Eun Soo Lee證明了。這個性質是，對prime alternative的環來說，如果我們將它的同調群畫在方格紙上，那麼它將只出現在兩條平行線上。

在這篇文章裡，我想要找出結的connected sum和disjoint union之間的關係。在Knovanov的文章裡已經有了一個方法，是將兩者做成一個exact sequence，而我的方法就是引用這個。Dror Bar-Natan寫的程式也幫了我很大的忙。

A new link invariant found by Khovanov is a mysterious invariant. The brief idea is to build a chain complex for a knot so that its Euler characteristic is its Jones polynomial, and we can compute the Khovanov homology for this chain complex. It is a link invariant, but the meaning of the terms in it is not yet varified. Instead some masters drill out many properties inside this invariant. One amazing property of the Khovanov homology of prime alternative knots is stated in Dror Bar-Natan's paper and is proved by Eun Soo Lee. It says that the Khovanov homology of prime alternative knots appears only in two skew parallel lines if we draw them in a table.

In this article I want to find some relationship between connected sum and disjoint union of two knots. In Knovanov's paper he introduce a nice relation between connected sum and disjoint union of two knots. It is long exact sequences, and my computation is relied on it. The program released by Dror Bar-Natan really does great help to me.