Today I discovered this nice video of a lecture by Thurston:

https://youtu.be/daplYX6Oshc

in which he explains how a knot can be turned into a “fabric for universes”. For example, the unknot can be thought as a portal to Narnia, and when you pass again you switch back to Narnia. This forms in a sense a $mathbb{Z}/2mathbb{Z}$. Then he proceeds to explore what fabric one gets with the treefoil and you get an $S_3$. I am sure there is some real geometry behind but I can’t grasp how to translate a portal into something homotopic.

A way to formalize this would be the following. Take a knot $K$ in $mathbb{R}^3$. Fix a tubular neighborhood $N$ of $K$. For each point $x$ in the knot take the loop $L_x$ obtained as the sphere bundle of $N$ at $x$ (a small circle around $x$ that jumps into the portal). Then there exist a connected 3-manifold $M$ with a (finite?) cover $M to mathbb{R}^3 setminus K$ such that the “monodromy in small circles” around $L_x$ has order two for all $x$. Then we set the “group of universes” as the group of cover automorphisms.

I think this captures the previous idea in the following sense: to each locus of $mathbb{R}^3 setminus K$ we have $n$ counterimages that represent the different worlds. Some branch should be chosen to make distinguishing between worlds possible. The constraint on monodromy ensures that if you jump twice through the same portal (at least for the portals very close to the boundary) you get back.

Does such a manifold exist for all knots? Is this construction just some simplification of the fundamental group of the complement?