Fibered knot
A knot or link in the 3-dimensional sphere is called fibered (sometimes spelled fibred) in case there is a 1-parameter family of Seifert surfaces for , where the parameter runs through the points of the unit circle , such that if is not equal to then the intersection of and is exactly .
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 ; the Hopf link (oriented correctly) is the link of the node singularity . In these cases, the family of Seifert surfaces is an aspect of the Milnor fibration of the singularity.
A knot is fibered if and only if it is the binding of some open book decomposition of .
Knots that are not fibered
![](http://upload.wikimedia.org/wikipedia/commons/f/f4/Knot-stevedore-sm.png)
The Alexander polynomial of a fibered knot is monic, i.e. the coefficients of the highest and lowest powers of t are plus or minus 1. Examples of knots with nonmonic Alexander polynomials abound, for example the twist knots have Alexander polynomials qt−(2q+1)+qt−1, where q is the number of half-twists. [1] In particular the Stevedore's knot isn't fibered.