探究 Khovanov 同调群中的挠元素

Khovanov 同调是一个非常强的纽结不变量，同时也是琼斯多项式的一个范畴化。自2000年被提出以来，Khovanov 同调已经被大量学者研究。特别的，人们关心 Khovanov 同调群中的挠元素, 因为挠元素所携带的关于纽结的信息是全新的，并不会在琼斯多项式中有任何表达。因此，尝试去理解挠元素反映着纽结的什么性质是非常有意义的工作。本项目着眼于计算纽结的Khovanov同调群，发现并总结规律，寻找含有非Z_{2}挠元素的纽结，并试图解释挠元素所反映的纽结的信息。同时着重于探索质数挠元素的来源，尝试理解质数挠元素出现在某种类型纽结的Khovanov同调群中的原因和本质。

Khovanov homology is a powerful link invariant which categorifies the Jones polynomial. Since it was introduced in 2000, Khovanov homology has been extensively studied. In particular, people are interested in the torsion elements in the homology group. (information carried by the torsion elements has no expression in the Jones polynomial, thus it is new information of knots and links). It will be exciting and meaningful to understand what torsion elements reflect on properties of knots and links. This project will start by computing Khovanov homology of certain types of knots and links, finding out patterns from the computation results, searching for certain types of knots and links that have non Z_{2} torsion and trying to dig out the information of torsion elements. In particular, it will explore the source of prime torsion Z_{p} and try to understand why prime torsion appears in the Khovanov homology of certain types of knots and links.