Topologists often study knots by looking at what remains when the knot is *removed* from . I suppose they’re looking (somewhat ironically) at everything *but* the knot. Hmm. In any case, it works, so why don’t we let be a knot in and define the** knot exterior** to be , the closure of minus a regular neighborhood of the knot.

One way to visualize knot exteriors is to take the projection of to where the point at infinity is chosen to lie inside . In this case, the knot exterior looks like a 3–ball minus a knotted arc inside. Two such exteriors are drawn at the right. I’ve added the knot (a neighborhood of which we removed, remember) in blue. The first is the exterior of the unknot, and the second is the exterior of a trefoil knot. It’s pretty easy to see in the first case that the exterior is a solid torus.

The converse of this fact, that if is homeomorphic to a solid torus then is the unknot, is also true, because a compressing disk for the solid torus gives a Seifert surface for the knot.

## 0 Responses to “Knot Exteriors”