Fibered knot

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 .

For example:

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

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

See also

References

  1. ↑ Fintushel, Ronald; Stern, Ronald J (1996). "[dg-ga/9612014] Knots, Links, and 4-Manifolds". arXiv:[//arxiv.org/abs/dg-ga/9612014 dg-ga/9612014].
  • 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


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