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 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, and can also be described as the (5,2)-

torus knot In knot theory, a torus knot is a special kind of knot (mathematics), knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link (knot theory), link which lies on the surface of a torus in the same way. Each t ...

. 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 ...

. Its writhe 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 ...

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 :

\Delta(t) = t^2 - t + 1 - t^ + t^

, since

\begin1 & -1 & 0&0\\ 0 & 1 &-1 &0 \\ 0& 0& 1&-1 \\ 0& 0& 0&1\end

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 :

\nabla(z) = z^4 + 3z^2 + 1

, 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 :

V(q) = q^ + q^ - q^ + q^ - q^.

These are the same as the Alexander, Conway, and Jones polynomials of the knot 10₁₃₂. However, the Kauffman polynomial 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 500 species of Annual plant, annual, Biennial plant, biennial and Perennial plant, perennial herbaceous plant, herbaceous flowering plants in the rose family (biology), family, Rosaceae. Potentillas m ...

''.

Properties

History

See also

References

Further reading