Fibered knot

Fibered knot

A knot or link Kin 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 the unit circle S^1, such that if s is not equal to tthen 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 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.

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 the Stevedore's knot isn't fibered.


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