An operad for splicing

TL;DR

This paper introduces the splicing operad, a new topological operad acting on spaces of self-embeddings and knots, revealing new algebraic structures and homotopy properties of knot spaces, especially for long knots in R^3.

Contribution

It defines the splicing operad and demonstrates its action on knot spaces, showing that the space of long knots is free with respect to this operad, generated by torus and hyperbolic knots.

Findings

The space of long knots in R^3 is a free algebra over the splicing operad.

The free generating subspace consists of torus and hyperbolic knots.

The splicing operad has a simple homotopy-type.

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.