Stevedore knot (mathematics)

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.

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

The common stevedore knot. If the ends were joined together, the result would be equivalent to the mathematical 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.