71 Knot
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In knot theory, 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 co ...
s with crossing number seven. It is the simplest
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 ...
after the
trefoil A trefoil () is a graphic form composed of the outline of three overlapping rings, used in architecture and Christian symbolism, among other areas. The term is also applied to other symbols with a threefold shape. A similar shape with four ring ...
and
cinquefoil ''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 perenn ...
.


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 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 :\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 polynomi ...
is :V(q) = q^ + q^ - q^ + q^ - q^ + q^ - q^. \,


Example


See also

*
Heptagram A heptagram, septagram, septegram or septogram is a seven-point star drawn with seven straight strokes. The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek suffix ''-gram''. The ''-gram'' suffix derives from ''γρ ...


References

{{DEFAULTSORT:7 1 knot