Three-twist knot

Three-twist 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

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

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

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.

References

1. "5_2", The Knot Atlas.