Fibered Knot

In knot theory, a branch of mathematics, a knot or link $K$
in the 3-dimensional sphere $S^3$ is called fibered or fibred (sometimes Neuwirth knot in older texts, after Lee Neuwirth) 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 $K$.

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 $qt-(2q+1)+qt^$, 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 $z^2+w^3$; 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 $S^3$.

See also

* (−2,3,7) pretzel knot

References

External links

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{{Knot theory