Hodge-type decomposition in the homology of long knots

Abstract

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.