A knot

Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...

 or link

Link (knot theory)

In mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...

 
in the 3-dimensional sphere

3-sphere

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

  is called fibered or fibred if there is a 1-parameter family of Seifert surfaces

Seifert surface

In mathematics, a Seifert surface is a surface whose boundary is a given knot or link.Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface...

 for , where the parameter runs through the points of the unit circle

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

 , such that if is not equal to
then the intersection of and is exactly .

For example:

• The unknot
  Unknot
  The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...
  
  , trefoil knot
  Trefoil knot
  In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...
  
  , and figure-eight knot
  Figure-eight knot (mathematics)
  In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot...
  
   are fibered knots.

• The Hopf link
  Hopf link
  thumb|right|[[Skein relation]] for the Hopf link.In mathematical knot theory, the Hopf link, named after Heinz Hopf, is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once...
  
   is a fibered link.

Fibered knots and links arise naturally, but not exclusively, in complex algebraic geometry. For instance, each singular point

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

 of a complex plane curve can be described
topologically as the cone

Cone (topology)

In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space:CX = /\,of the product of X with the unit interval I = [0, 1]....

 on a fibered knot or link called the link of the singularity. The trefoil knot

Trefoil knot

In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...

 is the link of the cusp singularity

Cusp (singularity)

In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....

 ; the Hopf link (oriented correctly) is the link of the node singularity

Singular point of a curve

In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.-Algebraic curves in the plane:...

 . 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

Open book decomposition

In mathematics, an open book decomposition is a decomposition of a closed oriented 3-manifold M into a union of surfaces and solid tori...

 of .

Knots that are not fibered

The Alexander polynomial

Alexander polynomial

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923...

 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. http://arxiv.org/abs/dg-ga/9612014 In particular the Stevedore's knot

Stevedore knot (mathematics)

In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot. The stevedore knot is listed as the 61 knot in the Alexander–Briggs notation, and it can also be described as a twist knot with four twists, or as the pretzel...

is not fibered.