Sur l'homologie des espaces de noeuds non-compacts

TL;DR

This paper describes the first term of Vassiliev's spectral sequence for non-compact knots in ${f R}^d$, linking it to Hochschild homology of algebra operads and simplifying its computation.

Contribution

It provides a new description of the spectral sequence's first term using Hochschild homology of Poisson and Gerstenhaber algebra operads, and relates it to chord diagrams.

Findings

First term expressed via Hochschild homology of algebra operads

Chord diagrams form a subspace of this homology

Simplified method for calculating the spectral sequence

Abstract

The spectral sequence constructed by V.A.Vassiliev computes the homology of the spaces of non-compact knots in ${\bf R}^d$, $d\ge 3$. In this work the first term of this spectral sequence is described in terms of the homology of the Hochschild complex for the Poisson algebras operad, if d is odd (resp. for the Gerstenhaber algebras operads, if d is even). In particular the bialgebra of chord diagrams arises as some subspace of this homology (in this case d=3). Also a simplification for the calculation of the Vassiliev spectral sequence in the first term is provided.