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71 knot

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(Redirected from 7₁ knot) Mathematical knot with crossing number 7
71 knot
Arf invariant0
Braid length7
Braid no.2
Bridge no.2
Crosscap no.1
Crossing no.7
Genus3
Hyperbolic volume0
Stick no.9
Unknotting no.3
Conway notation
A–B notation71
Dowker notation8, 10, 12, 14, 2, 4, 6
Last / Next63 / 72
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.

Properties

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

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

its Conway polynomial is

∇ ( z ) = z 6 + 5 z 4 + 6 z 2 + 1 , {\displaystyle \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 . {\displaystyle V(q)=q^{-3}+q^{-5}-q^{-6}+q^{-7}-q^{-8}+q^{-9}-q^{-10}.\,}

Example

Assembling of 71 knot.


See also

References

  1. "7_1", The Knot Atlas.
Knot theory (knots and links)
Hyperbolic
Satellite
Torus
Invariants
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