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7 2 knot

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Pentatwist knot, five-twist knot
7 2 knot
Arf invariant 1
Braid length 9
Braid no. 4
Bridge no. 2
Crosscap no. 2
Crossing no. 7
Genus 1
Hyperbolic volume 2.82812
Stick no. 9
Unknotting no. 1
Conway notation [52]
A–B notation 72
Dowker notation 4, 8, 12, 14, 2, 6, 10
Last / Next 71 / 73
Other
alternating, prime, reversible

In knot theory, the Pentatwist knot, also known as the five-twist knot, or the 72, is one of seven prime knots with crossing number seven. It is the fifth twist knot.

Invariants

Its Alexander polynomial is

Δ ( t ) = 3 t − 5 + 3 t − 1 , {\displaystyle \Delta (t)=3t-5+3t^{-1},\,} {\displaystyle \Delta (t)=3t-5+3t^{-1},\,}

its Conway polynomial is

∇ ( z ) = 3 z 2 + 1 , {\displaystyle \nabla (z)=3z^{2}+1,\,} {\displaystyle \nabla (z)=3z^{2}+1,\,}

and its Jones polynomial is

V ( q ) = − q − 8 + q − 7 − q − 6 + 2 q − 5 − 2 q − 4 + 2 q − 3 − q − 2 + q − 1 . {\displaystyle V(q)=-q^{-8}+q^{-7}-q^{-6}+2q^{-5}-2q^{-4}+2q^{-3}-q^{-2}+q^{-1}.\,} {\displaystyle V(q)=-q^{-8}+q^{-7}-q^{-6}+2q^{-5}-2q^{-4}+2q^{-3}-q^{-2}+q^{-1}.\,}[1]

Example

References

This article is adapted from “7 2 knot” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

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