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.

Properties

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

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

its Conway polynomial is

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

and its Jones polynomial is

$V(q)=q^{-3}+q^{-5}-q^{-6}+q^{-7}-q^{-8}+q^{-9}-q^{-10}.$

Example

File:7₁ knot.webm

See also

• Heptagram