7₁ knot

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.

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

Properties

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

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

its Conway polynomial is

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

and its Jones polynomial is

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

Example

See also

• Heptagram

References

1. "7_1", The Knot Atlas.