A Geometric Homology Representative in the Space of Long Knots

TL;DR

This paper constructs explicit geometric representatives of nontrivial homology classes in the space of long knots in even-dimensional Euclidean spaces, extending previous work and employing configuration space integrals.

Contribution

It introduces new geometric cycles in the knot space that are nontrivial in homology, generalizing prior results and utilizing spectral sequences and configuration space integrals.

Findings

Explicit geometric representatives are constructed for certain homology classes.

The classes pair nontrivially with known cohomology classes, confirming their significance.

The work extends previous results to a broader class of knot spaces.

Abstract

We produce explicit geometric representatives of nontrivial homology classes in the space of long knots in R^d, when d is even. We generalize results of Cattaneo, Cotta-Ramusino and Longoni to define cycles which live off of the vanishing line of a homology spectral sequence due to Sinha. We use configuration space integrals to show our classes pair nontrivially with cohomology classes due to Longoni.