Fibered knot

Figure-eight knot is fibered.

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

Examples

Knots that are fibered

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

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 ${\displaystyle qt-(2q+1)+qt^{-1}}$ , where q is the number of half-twists.^[1] In particular the stevedore knot is not fibered.

Related constructions

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 ${\displaystyle z^{2}+w^{3}}$ ; the Hopf link (oriented correctly) is the link of the node singularity ${\displaystyle z^{2}+w^{2}}$ . 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 ${\displaystyle S^{3}}$ .

See also

• (−2,3,7) pretzel knot

References

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

• 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.