Fibered knot
In knot theory, a branch of mathematics, a knot or link
in the 3-dimensional sphere is called fibered or fibred if there is a 1-parameter family of Seifert surfaces for, where the parameter runs through the points of the unit circle, such that if is not equal to
then the intersection of and is exactly.
Examples
Knots that are fibered
For example:- The unknot, trefoil knot, and [figure-eight knot (mathematics)|knot (mathematics)|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, 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 describedtopologically 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 ; the Hopf link is the link of the node singularity. 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.