BV-structures on the homology of the framed long knot space

TL;DR

This paper introduces two BV-algebra structures on the homology of the space of framed long knots, connecting string topology and Hochschild homology, and explores their potential equivalence.

Contribution

It presents two novel BV-algebra structures on framed long knot homology and relates them through spectral sequences and operad theory.

Findings

Two BV-structures are defined on the homology of framed long knots.

The structures are constructed via string topology and Hochschild homology.

Conjecture that the two BV-structures coincide.

Abstract

We introduce BV-algebra structures on the homology of the space of framed long knots in $\mathbb{R}^n$ in two ways. The first one is given in a similar fashion to Chas-Sullivan's string topology. The second one is defined on the Hochschild homology associated with a cyclic, multiplicative operad of graded modules. The latter can be applied to Bousfield-Salvatore spectral sequence converging to the homology of the space of framed long knots. Conjecturally these two structures coincide with each other.