Knot Exteriors

Topologists often study knots by looking at what remains when the knot is removed from S^3.  I suppose they’re looking (somewhat ironically) at everything but the knot.   Hmm.  In any case, it works, so why don’t we let K be a knot in S^3 and define the knot exterior to be E(K)=\overline{S^3\setminus N(K)}, the closure of S^3 minus a regular neighborhood of the knot.

Two knot exteriors, with knotOne way to visualize knot exteriors is to take the projection of S^3 to \mathbb{R}^3 where the point at infinity is chosen to lie inside N(K).  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 E(K) is homeomorphic to a solid torus then K is the unknot, is also true, because a compressing disk for the solid torus gives a Seifert surface for the knot.


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