Spaces of knots in the solid torus, knots in the thickened torus, and links in the 3-sphere

TL;DR

This paper determines the homotopy types of spaces of knots and links in various 3-manifolds, generalizing previous results and answering a question posed by Arnold about knots in the solid torus.

Contribution

It provides a recursive method to compute the homotopy type of spaces of irreducible framed links and extends these results to knots in the solid torus and links in the 3-sphere.

Findings

Homotopy type of space of irreducible framed links in S^3 determined

Homotopy type of space of knots in the solid torus characterized

Generators of fundamental groups explicitly described

Abstract

We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of any framed link except the unknot. Its generators can be viewed as generalizations of the Gramain loop in the space of long knots. Taking the quotient by certain such rotations relates the spaces we study. All of our results generalize previous work of Hatcher and Budney. We provide many examples and explicitly describe generators of fundamental groups.