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Stevedore's Knot

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StevedoresKnot
StevedoresKnot3D

The stevedore's knot is the 6-crossing prime knot 6_1. It is implemented in the Wolfram Language as KnotData["Stevedore"].

It has braid word sigma_1^(-1)sigma_2sigma_1^(-1)sigma_3sigma_2^(-1)sigma_3sigma_2. It has Arf invariant 0 and is not amphichiral, although it is invertible. It is a slice knot (Rolfsen 1976, p. 225).

The Alexander polynomial Delta(x), BLM/Ho polynomial Q(x), Conway polynomial del (x), HOMFLY polynomial P(l,m), and Jones polynomial V(t) of Stevedore's knot are

Delta(x)=5-2x-2x^(-1)
(1)
Q(x)=2x^5+4x^4-4x^3-6x^2+4x+1
(2)
del (x)=1-2x^2
(3)
P(l,m)=m^2(1-l^2)+(l^4+l^2-l^(-2))
(4)
V(t)=t^2-t+2-2t^(-1)+t^(-2)-t^(-3)+t^(-4).
(5)

Surprisingly, the knot 09-046 shares the same Alexander polynomial with the stevedore's knot. However, no knots on 10 or fewer crossings share the same BLM/Ho polynomial or Jones polynomial with it.

See also

Figure Eight Knot, Knot, Miller Institute Knot, Prime Knot, Solomon's Seal Knot, Trefoil Knot

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References

Bar-Natan, D. "The Knot 6_1." http://www.math.toronto.edu/~drorbn/KAtlas/Knots/6.1.html.KnotPlot. "6_1." http://newweb.cecm.sfu.ca/cgi-bin/KnotPlot/KnotServer/kserver?ncomp=1&ncross=6&id=1.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 225, 1976.

Cite this as:

Weisstein, Eric W. "Stevedore's Knot." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StevedoresKnot.html

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