Fibered knot

A knot or link <math>K<math> in the 3-dimensional sphere <math>S^3<math> is called fibered (sometimes spelled fibred) in case there is a 1-parameter family <math>F_t<math> of Seifert surfaces for <math>K<math>, where the parameter <math>t<math> runs through the points of the unit circle <math>S^1<math>, such that if <math>s<math> is not equal to <math>t<math> then the intersection of <math>F_s<math> and <math>F_t<math> is exactly <math>K<math>.

For example:

• The unknot, trefoil knot, and figure-eight knot are fibered knots.

• The Hopf link is a fibered link.

Fibered knots and links arise naturally, but not exclusively, in complex algebraic geometry. For instance, each singular point of a complex plane curve can be described topologically as the cone on a fibered knot or link called the link of the singularity. The trefoil knot is the link of the cusp singularity <math>z^2+w^3<math>; the Hopf link (oriented correctly) is the link of the node singularity <math>z^2+w^2<math>. In these cases, the family of Seifert surfaces is an aspect of the Milnor fibration of the singularity.