7₁ knot

7₁ knot
Arf invariant	0
Braid length	7
Braid no.	2
Bridge no.	2
Crosscap no.	1
Crossing no.	7
Genus	3
Hyperbolic volume	0
Stick no.	9
Unknotting no.	3
Conway notation	[7]
A-B notation	71
Dowker notation	8, 10, 12, 14, 2, 4, 6
Last /Next	63 / 72
Other
alternating, torus, fibered, prime, reversible

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 knots with crossing number seven. It is the simplest torus knot after the trefoil and cinquefoil.

The 7₁ knot is invertible but not amphichiral. Its Alexander polynomial is

$\Delta(t) = t^3 - t^2 + t - 1 + t^{-1} - t^{-2} + t^{-3}, \,$

its Conway polynomial is

$\nabla(z) = z^6 + 5z^4 + 6z^2 + 1, \,$

and its Jones polynomial is

$V(q) = q^{-3} + q^{-5} - q^{-6} + q^{-7} - q^{-8} + q^{-9} - q^{-10}. \,$^[1]

See also

• Heptagram

References

1. "7_1", The Knot Atlas.