Fibered knot
From Wikipedia, the free encyclopedia
A knot or link K in the 3-dimensional sphere S3 is called fibered or fibred in case there is a 1-parameter family Ft of Seifert surfaces for K, where the parameter t runs through the points of the unit circle S1, such that if s is not equal to t then the intersection of Fs and Ft 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 z2 + w3; the Hopf link (oriented correctly) is the link of the node singularity z2 + w2. 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 S3.
[edit] 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 qt−(2q+1)+qt−1, where q is the number of half-twists. [1] In particular the Stevedore's knot isn't fibered.
