In knot theory, a branch of mathematics, a knot or link in the 3-dimensional sphere is called **fibered** or **fibred** (sometimes **Neuwirth knot** in older texts, after Lee Neuwirth) if there is a 1-parameter family of Seifert surfaces for , where the parameter runs through the points of the unit circle , such that if is not equal to then the intersection of and is exactly .

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 ; the Hopf link (oriented correctly) is the link of the node singularity . 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 .

## 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* − (2*q* + 1) + *qt*^{−1}, where *q* is the number of half-twists.^{[1]} In particular the Stevedore's knot is not fibered.

## See also

## References

**^**Fintushel, Ronald; Stern, Ronald J (1996). "[dg-ga/9612014] Knots, Links, and 4-Manifolds". arXiv:dg-ga/9612014 .

## External links

- Harer, John (1982). "How to construct all fibered knots and links".
*Topology*.**21**(3): 263–280. doi:10.1016/0040-9383(82)90009-X. - http://www.msp.warwick.ac.uk/gt/2010/14-04/p050.xhtml