Little cubes and long knots

TL;DR

This paper describes the homotopy type of the space of long knots in 3D space, establishing a homotopy equivalence with a free little 2-cubes object over prime knots, revealing deep structural connections.

Contribution

It constructs a homotopy equivalence between the space of long knots and a free little 2-cubes object, linking knot decomposition with operad actions.

Findings

Homotopy equivalence between knot space and free little 2-cubes object.

Connection between knot decomposition and little cubes operad.

Extension of operad actions to embedding spaces in higher dimensions.

Abstract

This paper gives a partial description of the homotopy type of K, the space of long knots in 3-dimensional Euclidean space. The primary result is the construction of a homotopy equivalence between K and the free little 2-cubes object over the space of prime knots. In proving the freeness result, a close correspondence is discovered between the Jaco-Shalen-Johannson decomposition of knot complements and the little cubes action on K. Beyond studying long knots in 3-space, we show that for any compact manifold M the space of embeddings of R^n x M in R^n x M with support in I^n x M admits an action of the operad of little (n+1)-cubes. If M=D^k this embedding space is the space of framed long n-knots in R^{n+k}, and the action of the little cubes operad is an enrichment of the monoid structure given by the connected-sum operation.