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# Fibered knot

Figure-eight knot is fibered.

In knot theory, a branch of mathematics, a knot or link ${\displaystyle K}$ in the 3-dimensional sphere ${\displaystyle S^{3}}$ is called fibered or fibred (sometimes Neuwirth knot in older texts, after Lee Neuwirth) if there is a 1-parameter family ${\displaystyle F_{t}}$ of Seifert surfaces for ${\displaystyle K}$, where the parameter ${\displaystyle t}$ runs through the points of the unit circle ${\displaystyle S^{1}}$, such that if ${\displaystyle s}$ is not equal to ${\displaystyle t}$ then the intersection of ${\displaystyle F_{s}}$ and ${\displaystyle F_{t}}$ is exactly ${\displaystyle K}$.

For example:

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

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## Knots that are not fibered

Stevedore's knot is 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.[1] In particular the Stevedore's knot is not fibered.