Non-trivalent graph cocycle and cohomology of the long knot space

TL;DR

This paper demonstrates that a non-trivalent graph cocycle can produce a non-zero cohomology class in the space of long knots, revealing new insights into knot space topology and operad actions.

Contribution

It introduces a novel use of non-trivalent graph cocycles to generate non-zero cohomology classes in higher codimensional long knot spaces.

Findings

Non-trivalent graph cocycle yields non-zero cohomology class

Browder operation induced by operad action is non-trivial

Advances understanding of knot space cohomology

Abstract

In this paper we show that via the configuration space integral construction a non-trivalent graph cocycle can also yield a non-zero cohomology class of the space of higher (and even) codimensional long knots. This simultaneously proves that the Browder operation induced by the operad action defined by R. Budney is not trivial.