Fibered knot

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Figure-eight knot is fibered.

In knot theory, a branch of mathematics, a knot or link K in the 3-dimensional sphere S^{3} is called fibered or fibred 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.

For example:

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

Stevedore's 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. In particular the Stevedore's knot is not fibered.

See also

References

http://www.sciencedirect.com/science/article/pii/004093838290009X

http://www.msp.warwick.ac.uk/gt/2010/14-04/p050.xhtml


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