Three-twist knot

Three-twist knot
Arf invariant	0
Braid length	6
Braid no.	3
Bridge no.	2
Crosscap no.	2
Crossing no.	5
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

In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 5₂ knot in the Alexander-Briggs notation, and is one of two knots with crossing number five, the other being the cinquefoil knot.

The three-twist knot is a prime knot, and it is invertible but not amphichiral. Its Alexander polynomial is

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

its Conway polynomial is

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

and its Jones polynomial is

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

Because the Alexander polynomial is not monic, the three-twist knot is not fibered.

The three-twist knot is a hyperbolic knot, with its complement having a volume of approximately 2.82812.