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 $K$ .

Examples

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 stevedore 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 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.