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 5
1 knot in the
Alexander-Briggs notation, and can also be described as the (5,2)-
torus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of cop ...
. The cinquefoil is the closed version of the
double overhand knot
The double overhand knot or barrel knot is simply an extension of the regular overhand knot, made with one additional pass. The result is slightly larger and more difficult to untie. It forms the first part of the surgeon's knot and both sides ...
.
Properties
The cinquefoil 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 co ...
. Its
writhe
In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amou ...
is 5, 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 is ...
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 ve ...
is
:
,
its
Conway polynomial 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 polynomi ...
is
:
These are the same as the Alexander, Conway, and Jones polynomials of the knot 10
132. However, the
Kauffman polynomial
In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as
:F(K)(a,z)=a^L(K)\,,
where w(K) is the writhe of the link diagram and L(K) is a polynomial in ''a'' and ' ...
can be used to distinguish between these two knots.
History
The name “cinquefoil” comes from the five-petaled flowers of plants in the genus ''
Potentilla
''Potentilla'' is a genus containing over 300Guillén, A., et al. (2005)Reproductive biology of the Iberian species of ''Potentilla'' L. (Rosaceae).''Anales del Jardín Botánico de Madrid'' 1(62) 9–21. species of annual, biennial and perenni ...
''.
See also
*
Pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
*
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 ...
*
7₁ knot
*
Skein relation
Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invar ...
References
Further reading
*
{{Knot theory, state=collapsed