Hodge‐type decomposition in the homology of long knots
Hodge‐type decomposition in the homology of long knots
复制标题
长结同调中的 Hodge 型分解
DOI：
10.1112/jtopol/jtq015
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发表时间：
2010
影响因子：
1.1
通讯作者：
V. Turchin
中科院分区：
文献类型：
--
作者：
V. Turchin
本文描述了长纽结空间的有理同调和同伦的自然分裂。这种分解可能是由布线映射产生的，就像循环空间的同调中的自然分解是由幂映射产生的一样。给出了分裂项的欧拉特征线的母函数。基于这个生成函数，它表明，无论是同调和同伦的问题空间的排名增长至少指数。使用自然图复形，我们可以证明弦图双代数水平上的这种分裂正是Bar-Natan博士先前定义的分裂。附录中列出了欧拉特征线的计算机计算表。这些计算给出了一定的乐观，即阶数大于20的Vassiliev不变量可以区分结和它们的逆。
The paper describes a natural splitting in the rational homology and homotopy of the spaces of long knots. This decomposition presumably arises from the cabling maps in the same way as a natural decomposition in the homology of loop spaces arises from power maps. The generating function for the Euler characteristics of the terms of this splitting is presented. Based on this generating function, it is shown that both the homology and homotopy ranks of the spaces in question grow at least exponentially. Using natural graph‐complexes, one shows that this splitting on the level of the bialgebra of chord diagrams is exactly the splitting defined earlier by Dr. Bar‐Natan. The appendix presents tables of computer calculations of the Euler characteristics. These computations give a certain optimism that the Vassiliev invariants of order greater than 20 can distinguish knots from their inverses.