7₁ knot

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7₁ knot
Blue 7 1 Knot.png
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 6372
Other
alternating, torus, fibered, prime, reversible

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

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

its Conway polynomial is

\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}. \, [1]

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