Long knots and maps between operads

TL;DR

This paper establishes a deep connection between the space of long knots in high-dimensional space and derived operad maps, confirming a conjecture linking knot theory and operad theory.

Contribution

It identifies the space of long knots with double loops on derived operad maps, verifying a conjecture of Kontsevich, Lambrechts, and Turchin.

Findings

Confirmed the conjecture relating long knots to operad maps

Connected knot spaces with double loop spaces of operad maps

Provided a new perspective on the topology of high-dimensional knots

Abstract

We identify the space of tangentially straightened long knots in R^m, for m greater than or equal to 4, as the double loops on the space of derived operad maps from the associative operad into a version of the little m-disk operad. This verifies a conjecture of Kontsevich, Lambrechts, and Turchin.