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 7₁ knot, also known as the septoil knot, the septafoil knot, or the (7, 2)-torus knot, is one of seven

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 ...

s with crossing number seven. It is the simplest

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 ...

after the

trefoil A trefoil () is a graphic form composed of the outline of three overlapping rings, used in architecture, Pagan and Christian symbolism, among other areas. The term is also applied to other symbols with a threefold shape. A similar shape with f ...

and

cinquefoil ''Potentilla'' is a genus containing over 500 species of annual, biennial and perennial herbaceous flowering plants in the rose family, Rosaceae. Potentillas may also be called cinquefoils in English, but they have also been called five fin ...

Properties

The 7₁ knot 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^3 - t^2 + t - 1 + t^ - t^ + t^, \,

its Conway polynomial is :

\nabla(z) = z^6 + 5z^4 + 6z^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^ + q^ - q^. \,

Example

References

{{DEFAULTSORT:7 1 knot

Properties

Example

See also

References