Fibered knot



In knot theory, a branch of mathematics, a knot or link $$K$$ in the 3-dimensional sphere $$S^3$$ is called fibered or fibred (sometimes Neuwirth knot in older texts, after Lee Neuwirth) if 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 $$t$$ then the intersection of $$F_s$$ and $$F_t$$ is exactly $$K$$.

Knots that are fibered
For example:


 * The unknot, trefoil knot, and figure-eight knot are fibered knots.
 * The Hopf link is a fibered link.

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 twist knots have Alexander polynomials $$qt-(2q+1)+qt^{-1}$$, where q is the number of half-twists. In particular the stevedore knot is not fibered.

Related constructions
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 $$S^3$$.