Fibered knot

(Redirected from Fibered link)

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 .

Figure-eight knot is fibered.

Examples

edit

Knots that are fibered

edit

For example:

Knots that are not fibered

edit
 
The stevedore 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  , where q is the number of half-twists.[1] In particular the stevedore knot is not fibered.

edit

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  .

See also

edit

References

edit
  1. ^ Fintushel, Ronald; Stern, Ronald J. (1998). "Knots, Links, and 4-Manifolds". Inventiones Mathematicae. 134 (2): 363–400. arXiv:dg-ga/9612014. doi:10.1007/s002220050268. MR 1650308.
edit