nLab
space of knots

Contents

Context

Knot theory

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

Given a manifold Σ\Sigma, such as the 3-sphere, its space of knots is a topological space parametrizing knots in Σ\Sigma, hence something like the topological subspace

Emb(S 1,Σ)Maps(S 1,Σ) Emb(S^1, \Sigma) \;\subset\; Maps(S^1,\Sigma)

of the space of maps S 1ΣS^1 \to \Sigma on those maps which are embeddings. This space is often denoted just 𝒦\mathcal{K}.

A knot invariant is then equivalently a locally constant function on the space of knots.

References

General

Using graph complexes

Discussion of the rational homotopy type of knot spaces in terms of graph complexes:

Last revised on October 5, 2019 at 12:22:46. See the history of this page for a list of all contributions to it.