Fibered knot
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In knot theory, a branch of mathematics, a knot or link <math>K</math> in the 3-dimensional sphere <math>S^3</math> is called fibered or fibred (sometimes Neuwirth knot in older texts, after Lee Neuwirth) if there is a 1-parameter family <math>F_t</math> of Seifert surfaces for <math>K</math>, where the parameter <math>t</math> runs through the points of the unit circle <math>S^1</math>, such that if <math>s</math> is not equal to <math>t</math> then the intersection of <math>F_s</math> and <math>F_t</math> is exactly <math>K</math>.
ExamplesEdit
Knots that are fiberedEdit
For example:
- The unknot, trefoil knot, and figure-eight knot are fibered knots.
- The Hopf link is a fibered link.
Knots that are not fiberedEdit
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 <math>qt-(2q+1)+qt^{-1}</math>, where q is the number of half-twists.<ref>Template:Cite journal</ref> In particular the stevedore knot is not fibered.
Related constructionsEdit
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 <math>z^2+w^3</math>; the Hopf link (oriented correctly) is the link of the node singularity <math>z^2+w^2</math>. 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 <math>S^3</math>.