- Fibered knot
A knot or link $K$in 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 theunit circle $S^1$, such that if $s$ is not equal to $t$then 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 acomplex plane curve can be described topologically as the cone on a fibered knot or link called the**link of the singularity**. Thetrefoil 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 theMilnor 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 theStevedore's knot isn't fibered.

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