In
knot theory
In 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 joined so it cannot be und ...
, the three-twist knot is 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 fam ...
with three-half twists. It is listed as the 5
2 knot in the
Alexander-Briggs notation, and is one of two knots with
crossing number five, the other being the
cinquefoil knot
In knot theory, the cinquefoil knot, also known as Solomon's seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. It is listed as the 51 knot in the Alexander-Briggs notation, an ...
.
Properties
The three-twist knot is a
prime knot
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non- trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be ...
, and it is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
but not
amphichiral. Its
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 ...
is
:
since
is a possible
Seifert matrix
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 exampl ...
, or because of its
Conway polynomial, which is
:
and its
Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polyno ...
is
:
Because the Alexander polynomial is not
monic, the three-twist knot is not
fibered.
The three-twist knot is a
hyperbolic knot
Hyperbolic may refer to:
* of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics
** Hyperbolic geometry, a non-Euclidean geometry
** Hyperbolic functions, analogues of ordinary trigonometric functions, defined us ...
, with its complement having a
volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
of approximately 2.82812.
If the fibre of the knot in the initial image of this page were cut at the bottom right of the image, and the ends were pulled apart, it would result in a single-stranded
figure-of-nine knot (not the figure-of-nine loop).
Example
References
{{Knot theory, state=collapsed
Double torus knots and links