71 knot

Mathematical knot with crossing number 7

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.

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

Assembling of 7₁ knot.

See also

• Heptagram

References

1. "7_1", The Knot Atlas.