| Three-twist knot | |
| Practical Name: | Figure-of-nine knot |
| Arf Invariant: | 0 |
| Braid Length: | 6 |
| Braid Number: | 3 |
| Bridge Number: | 2 |
| Crosscap Number: | 2 |
| Crossing Number: | 5 |
| Genus: | 1 |
| Hyperbolic Volume: | 2.82812 |
| Stick Number: | 8 |
| Unknotting Number: | 1 |
| Conway Notation: | [32] |
| Ab Notation: | 52 |
| Dowker Notation: | 4, 8, 10, 2, 6 |
| Last Crossing: | 5 |
| Last Order: | 1 |
| Next Crossing: | 6 |
| Next Order: | 1 |
| Alternating: | alternating |
| Class: | hyperbolic |
| Prime: | prime |
| Symmetry: | reversible |
| Twist: | twist |
In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 52 knot[1] 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)=2z2+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.
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-stranded figure-of-nine knot (not the figure-of-nine loop).