Fibered knot
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![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Blue_Figure-Eight_Knot.png/170px-Blue_Figure-Eight_Knot.png)
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:
- The unknot, trefoil knot, and figure-eight knot are fibered knots.
- The Hopf link is a fibered link.
Knots that are not fibered
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Blue_Stevedore_Knot.png/170px-Blue_Stevedore_Knot.png)
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]- ^ 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]- Harer, John (1982). "How to construct all fibered knots and links". Topology. 21 (3): 263–280. doi:10.1016/0040-9383(82)90009-X. MR 0649758.
- Gompf, Robert E.; Scharlemann, Martin; Thompson, Abigail (2010). "Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures". Geometry & Topology. 14 (4): 2305–2347. arXiv:1103.1601. doi:10.2140/gt.2010.14.2305. MR 2740649.