71 knot

Template:Short description Template:Infobox knot theory 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. This knot is used to construct the simplest known counterexample to the conjecture that the unknotting number is additive under connected sum.

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

A 3D-printed model of the 7₁ knot. The tube follows the (2,7) torus knot and encloses a channel in which loose steel balls roll.

See also

• Heptagram