Given a knot in the 3-sphere, the knot complement is all the points of the 3-sphere not contained in the knot.

Cameron Gordon conjectured that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot complement.

By this he was asking what non-diagrammatic properties of the knot complement would characterize alternating knots.

Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure eight knot complement.

By utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement.

Knot signatures can also be defined in terms of the Alexander module of the knot complement.

Let be the universal abelian cover of the knot complement.

Consider the Alexander module to be the first homology group of the universal abelian cover of the knot complement: .

The generators in this situation are called a longitude and a meridian of the knot complement.

The representation of the fundamental group of knot complement plays a central role in them.