An operad for splicing

Abstract

A new topological operad is introduced, called the splicing operad. This operad acts on a broad class of spaces of self-embeddings N --> N where N is a manifold. The action of this operad on EC(j,M) (self embeddings R^j x M --> R^j x M with support in I^j x M) is an extension of the action of the operad of (j+1)-cubes on this space. Moreover the action of the splicing operad encodes Larry Siebenmann's splicing construction for knots in S^3 in the j=1, M=D^2 case. The space of long knots in R^3 (denoted K_{3,1}) was shown to be a free 2-cubes object with free generating subspace P, the subspace of long knots that are prime with respect to the connect-sum operation. One of the main results of this paper is that K_{3,1} is free with respect to the splicing operad action, but the free generating space is the much `smaller' space of torus and hyperbolic knots TH \subset K_{3,1}. Moreover, the splicing operad for K_{3,1} has a `simple' homotopy-type as an operad.