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.

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

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

its Conway polynomial is

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

and its Jones polynomial is

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

• Heptagram

• Knot theory
• Alternating knots and links
• Torus knots and links
• Fibered knots and links
• Prime knots and links
• Reversible knots and links
• Non-tricolorable knots and links

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• Short description matches Wikidata