Fibered knot
In knot theory, a branch of mathematics, a knot or link 
in the 3-dimensional sphere  is called fibered or fibred (sometimes Neuwirth knot in older texts, after Lee Neuwirth) 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 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 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 ; the Hopf link (oriented correctly) 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 . See also
(−2,3,7) pretzel knot

References

External links