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Stevedore knot (mathematics)

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Mathematical knot with crossing number 6
Stevedore knot
Common nameStevedore knot
Arf invariant0
Braid length7
Braid no.4
Bridge no.2
Crosscap no.2
Crossing no.6
Genus1
Hyperbolic volume3.16396
Stick no.8
Unknotting no.1
Conway notation
A–B notation61
Dowker notation4, 8, 12, 10, 2, 6
Last / Next52 / 62
Other
alternating, hyperbolic, pretzel, prime, slice, reversible, twist
The common stevedore knot. If the ends were joined together, the result would be equivalent to the mathematical knot.

In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot. The stevedore knot is listed as the 61 knot in the Alexander–Briggs notation, and it can also be described as a twist knot with four half twists, or as the (5,−1,−1) pretzel knot.

The mathematical stevedore knot is named after the common stevedore knot, which is often used as a stopper at the end of a rope. The mathematical version of the knot can be obtained from the common version by joining together the two loose ends of the rope, forming a knotted loop.

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

Δ ( t ) = − 2 t + 5 − 2 t − 1 , {\displaystyle \Delta (t)=-2t+5-2t^{-1},\,}

its Conway polynomial is

∇ ( z ) = 1 − 2 z 2 , {\displaystyle \nabla (z)=1-2z^{2},\,}

and its Jones polynomial is

V ( q ) = q 2 − q + 2 − 2 q − 1 + q − 2 − q − 3 + q − 4 . {\displaystyle V(q)=q^{2}-q+2-2q^{-1}+q^{-2}-q^{-3}+q^{-4}.\,}

The Alexander polynomial and Conway polynomial are the same as those for the knot 946, but the Jones polynomials for these two knots are different. Because the Alexander polynomial is not monic, the stevedore knot is not fibered.

The stevedore knot is a ribbon knot, and is therefore also a slice knot.

The stevedore knot is a hyperbolic knot, with its complement having a volume of approximately 3.16396.

See also

References

  1. "6_1", The Knot Atlas.
  2. Weisstein, Eric W. "Stevedore's Knot". MathWorld.
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