Operads and knot spaces

TL;DR

This paper demonstrates that the Kontsevich operad models the Taylor tower of a functor related to knot embeddings and immersions, revealing new operad structures and applications in knot space topology.

Contribution

It establishes a connection between the Kontsevich operad and the Taylor tower of knot embedding functors, introducing a novel operad structure on the two-sphere model.

Findings

Kontsevich operad models the Taylor tower of knot embedding functors

Operad structure on the simplicial two-sphere model

Application of McClure-Smith machinery to produce a little two-disk action

Abstract

We show that the Kontsevich operad, as an operad with multiplication, provides a model for the Taylor tower of the functor defined by taking the homotopy fiber of the inclusion of embeddings of an interval in a cube to the corresponding space of immersions. In developing the Kontsevich operad, we find an operad structure on the simplicial model for the two-sphere, in the opposite category to finite sets. Applications of our result include using machinery of McClure-Smith to produce a little two-disk action on our knot spaces.