Stevedore knot (mathematics)

Stevedore knot
Common name	Stevedore knot
Arf invariant	0
Braid length	7
Braid no.	4
Bridge no.	2
Crosscap no.	2
Crossing no.	6
Genus	1
Hyperbolic volume	3.16396
Stick no.	8
Unknotting no.	1
Conway notation	[42]
A–B notation	61
Dowker notation	4, 8, 12, 10, 2, 6
Last / Next	52 / 62
Other
alternating, hyperbolic, pretzel, prime, slice, reversible, twist

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

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

its Conway polynomial is

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

and its Jones polynomial is

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

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.^[2] 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.

• Figure-eight knot (mathematics)