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.