Fibered knot

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Figure-eight knot is fibered.

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 .

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.

Related constructions[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.

External links[edit]