71 knot
Mathematical knot with crossing number 7
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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. This knot is used to construct the simplest counterexample to the conjecture that the unknotting number is additive under connected sum.[1][2]
| 71 knot | |
|---|---|
| |
| Arf invariant | 0 |
| Braid length | 7 |
| Braid no. | 2 |
| Bridge no. | 2 |
| Crosscap no. | 1 |
| Crossing no. | 7 |
| Genus | 3 |
| Hyperbolic volume | 0 |
| Stick no. | 9 |
| Unknotting no. | 3 |
| Conway notation | [7] |
| A–B notation | 71 |
| Dowker notation | 8, 10, 12, 14, 2, 4, 6 |
| Last / Next | 63 / 72 |
| Other | |
| alternating, torus, fibered, prime, reversible | |
Properties
The 71 knot is invertible but not amphichiral. Its Alexander polynomial is
its Conway polynomial is
and its Jones polynomial is
Example