Hodge decomposition in the homology of long knots

TL;DR

This paper introduces a natural Hodge-type splitting in the rational homology and homotopy of long knot spaces, revealing exponential growth and connections to chord diagram algebra, with implications for knot invariants.

Contribution

It establishes a natural splitting in the homology and homotopy of long knots, relates it to cabling maps and chord diagram algebra, and provides computational evidence for knot invariants.

Findings

Homology and homotopy ranks grow at least exponentially.

The splitting matches Dr. Bar-Natan's earlier decomposition.

Computations suggest Vassiliev invariants >20 can distinguish knots from their inverses.

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 we show that both the homology and homotopy ranks of the spaces in question grow at least exponentially. Using natural graph-complexes we show that this splitting on the level of the bialgebra of chord diagrams is exactly the splitting defined earlier by Dr. Bar-Natan. In the Appendix we present tables of computer calculations of the Euler characteristics. These computations give a certain optimism that the Vassiliev invariants of order > 20 can distinguish knots from their inverses.