In
knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
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 surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable 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 ...
s for
, where the parameter
runs through the points of the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
, such that if
is not equal to
then the intersection of
and
is exactly
.
Examples
Knots that are fibered
For example:
* The
unknot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...
,
trefoil knot
In knot theory, 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. As the simplest kno ...
, and
figure-eight knot
The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...
are fibered knots.
* The
Hopf link
In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf.
Geometric realization
A concrete model consists of ...
is a fibered link.
Knots that are not fibered
![Blue Stevedore Knot](https://upload.wikimedia.org/wikipedia/commons/8/8b/Blue_Stevedore_Knot.png)
The
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. In 1969, John Conway showed a ve ...
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 knot
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite f ...
s have Alexander polynomials
, where ''q'' is the number of half-twists.
In particular the
stevedore knot
The stevedore knot is a stopper knot, often tied near the end of a rope. It is more bulky and less prone to jamming than the closely related figure-eight knot.
Naming
There is a lack of consensus among knot experts regarding the origin of t ...
is not fibered.
Related constructions
Fibered knots and links arise naturally, but not exclusively, in
complex algebraic geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
. For instance, each
singular point of a
complex plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic p ...
can be described
topologically as the
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
on a fibered knot or link called the link of the singularity. The
trefoil knot
In knot theory, 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. As the simplest kno ...
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 surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable 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 ...
s is an aspect of the
Milnor fibration In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book ''Singular Points of Complex Hypersurfaces'' (Princeton University Press, 1968) and earlier lectures. The most studied ...
of the singularity.
A knot is fibered if and only if it is the binding of some
open book decomposition In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3-manifold ''M'' into a union of surfaces (necessarily with boundary) and solid tori. Open books have relevance to contact geometry, with a ...
of
.
See also
*
(−2,3,7) pretzel knot
In geometric topology, a branch of mathematics, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot (after Ron Fintushel and Ronald J. Stern), is an important example of a pretzel knot which exhibits various in ...
References
External links
*
*
{{Knot theory