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

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