| 71 knot | |
| Arf Invariant: | 0 |
| Braid Length: | 7 |
| Braid Number: | 2 |
| Bridge Number: | 2 |
| Crosscap Number: | 1 |
| Crossing Number: | 7 |
| Genus: | 3 |
| Hyperbolic Volume: | 0 |
| Stick Number: | 9 |
| Unknotting Number: | 3 |
| Conway Notation: | [7] |
| Ab Notation: | 71 |
| Dowker Notation: | 8, 10, 12, 14, 2, 4, 6 |
| Last Crossing: | 6 |
| Last Order: | 3 |
| Next Crossing: | 7 |
| Next Order: | 2 |
| Alternating: | alternating |
| Class: | torus |
| Fibered: | fibered |
| Prime: | prime |
| Symmetry: | reversible |
In knot theory, the 71 knot, also known as the septoil knot, the septafoil knot, or the (7, 2)-torus knot, is one of seven prime knots with crossing number seven. It is the simplest torus knot after the trefoil and cinquefoil.
The 71 knot is invertible but not amphichiral. Its Alexander polynomial is
\Delta(t)=t3-t2+t-1+t-1-t-2+t-3,
its Conway polynomial is
\nabla(z)=z6+5z4+6z2+1,
and its Jones polynomial is
V(q)=q-3+q-5-q-6+q-7-q-8+q-9-q-10.