Little cubes and long knots

Abstract

This paper gives a partial description of the homotopy type of $K$, the space of long knots in $R^3$. The primary result is the construction of a homotopy equivalence $K\simeq C_2(P\bigsqcup \{\ast \})$ where $C_2(P\bigsqcup \{\ast \})$ is the free little 2-cubes object on the pointed space $P\bigsqcup \{\ast \}$, where $P\subset K$ is the subspace of prime knots, and $\ast$ is a disjoint base-point. 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 $R^3$ we show that for any compact manifold $M$ the space of embeddings of $R^n\times M$ in $R^n\times M$ with support in $I^n\times M$ 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.