Torsion of the Khovanov homology
Torsion of the Khovanov homology
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霍瓦诺夫同调的扭转
DOI：
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发表时间：
2004
期刊：
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通讯作者：
A. Shumakovitch
中科院分区：
文献类型：
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作者：
A. Shumakovitch
Khovanov同调是最近引入的$\mathbb{R}^3 $中有向链的不变量。它归类的琼斯多项式的意义上说，（分级）欧拉特征的霍瓦诺夫同调是一个版本的琼斯多项式的联系。本文研究Khovanov同调的挠。基于我们的计算，我们制定了几个关于挠率和证明他们的前两个较弱的版本。特别地，我们证明了所有的非分裂交替链的整数Khovanov同调几乎由琼斯多项式和签名决定。唯一剩下的不确定性是，人们不能区分$\mathbb{Z}_{2^k}$因子的典型分解的Khovanov同调群为不同的值$k$。
Khovanov homology is a recently introduced invariant of oriented links in $\mathbb{R}^3$. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov homology is a version of the Jones polynomial for links. In this paper we study torsion of the Khovanov homology. Based on our calculations, we formulate several conjectures about the torsion and prove weaker versions of the first two of them. In particular, we prove that all non-split alternating links have their integer Khovanov homology almost determined by the Jones polynomial and signature. The only remaining indeterminacy is that one cannot distinguish between $\mathbb{Z}_{2^k}$ factors in the canonical decomposition of the Khovanov homology groups for different values of $k$.