The space of short ropes and the classifying space of the space of long knots

TL;DR

This paper proves that the space of short ropes is weakly homotopy equivalent to the classifying space of long knots in 3D, using advanced manifold and homotopy techniques.

Contribution

It confirms a conjecture by Mostovoy and constructs an explicit geometric map linking short ropes to the classifying space of long knots.

Findings

Space of short ropes is weakly homotopy equivalent to classifying space of long knots

Constructs a manifold space model for the classifying space

Provides an explicit geometric map from short ropes to the model

Abstract

We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in $\mathbb{R}^3$. We make use of techniques developed by S. Galatius and O. Randal-Williams to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way.