Fibered knot

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

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 $$z^2+w^3; the Hopf link (oriented correctly) is the link of the node singularity $$z^2+w^2. 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 $$S^3.

Knots that are not fibered

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 [http://mathworld.wolfram.com/TwistKnot.html twist knots] have Alexander polynomials qt−(2q+1)+qt⁻¹, where q is the number of half-twists. [http://arxiv.org/abs/dg-ga/9612014] In particular the Stevedore's knot isn't fibered.