Three-twist knot
Common name	Figure-of-nine knot
Arf invariant	0
Braid length	6
Braid no.	3
Bridge no.	2
Crosscap no.	2
Crossing no.	5
Genus	1
Hyperbolic volume	2.82812
Stick no.	8
Unknotting no.	1
Conway notation	[32]
A–B notation	52
Dowker notation	4, 8, 10, 2, 6
Last / Next	51 / 61
Other
alternating, hyperbolic, prime, reversible, twist

Inknot theory, thethree-twist knot is thetwist knot with three-half twists. It is listed as the5₂ knot^[1] in theAlexander-Briggs notation, and is one of two knots withcrossing number five, the other being thecinquefoil knot.

The three-twist knot is aprime knot, and it isinvertible but notamphichiral. ItsAlexander polynomial is

${\displaystyle \mathrm{\Delta }(t)=2t-3+2t^{-1},}$

itsConway polynomial is

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

and itsJones polynomial is

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

Because the Alexander polynomial is notmonic, the three-twist knot is notfibered.

The three-twist knot is ahyperbolic knot, with its complement having avolume of approximately 2.82812.

If the fibre of the knot in the initial image of this page were cut at the bottom right of the image, and the ends were pulled apart, it would result in a single-strandedfigure-of-nine knot (not the figure-of-nine loop).

• Knot theory
• Alternating knots and links
• Hyperbolic knots and links
• Unfibered knots and links
• Prime knots and links
• Reversible knots and links
• Non-tricolorable knots and links
• Twist knots
• Double torus knots and links

• Articles with short description
• Short description matches Wikidata