The topology of spaces of knots

TL;DR

This paper introduces two models for the space of knots with fixed endpoints in manifolds with boundary, utilizing configuration space maps and cosimplicial structures, to analyze their homotopy and cohomology properties.

Contribution

It develops novel models based on isotopy calculus for studying knot spaces, connecting differential topology and spectral sequence analysis.

Findings

Models are weakly homotopy equivalent to knot spaces in high dimensions.

Spectral sequences converge to cohomology and homotopy groups of knot spaces.

Explicit vanishing lines are established in the spectral sequences.

Abstract

We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of spaces of maps between configuration spaces and another which is cosimplicial. These models build on the calculus of isotopy functors and are weakly homotopy equivalent to knot spaces when the ambient dimension is greater than three. The mapping space model, and the evaluation map on which it builds, is suitable for analysis through differential topology. The cosimplicial model gives rise to spectral sequences which converge to cohomology and homotopy groups of spaces of knots when they are connected. We explicitly identify and establish vanishing lines in these spectral sequences.