Fibered_knot Knowpia

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

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

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 $qt-(2q+1)+qt^{-1}$ , 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 $z^{2}+w^{3}$ ; the Hopf link (oriented correctly) is the link of the node singularity $z^{2}+w^{2}$ . 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 $S^{3}$ .

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.

[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.

[1]