Geometry of Knot Complements

The Mathematics

Knot and link theory studies loops embedded in space up to deformation. Much of a knot’s character lives not in the loop itself but in its complement — everything around it — whose topology, and often its geometry, is a powerful invariant of the knot. These images give an inside view of that complement: a compact region of ℝ³ bounded by a toroidal surface, whose topology is the knot complement itself. They were inspired by Henry Segerman’s 3D print of the figure-eight knot.

Technique

The torus knots are parameterized directly in the three-sphere, while for the Whitehead link I took a parameterization in ℝ³ and inverse-stereographically projected it into S³. Given these curves, we use the spherical analog of the Frenet frame to build tubes of constant spherical width around them. Stereographically projecting into ℝ³ from a point on the knot then encloses a compact region — the knot complement — inside the image of the tube.