Cinquefoil | |
---|---|

Common name | Double overhand knot |

Arf invariant | 1 |

Braid length | 5 |

Braid no. | 2 |

Bridge no. | 2 |

Crosscap no. | 1 |

Crossing no. | 5 |

Genus | 2 |

Hyperbolic volume | 0 |

Stick no. | 8 |

Unknotting no. | 2 |

Conway notation | [5] |

A–B notation | 5_{1} |

Dowker notation | 6, 8, 10, 2, 4 |

Last /Next | 4_{1} / 5_{2} |

Other | |

alternating, torus, fibered, prime, reversible |

In knot theory, the **cinquefoil knot**, also known as **Solomon's seal knot** or the **pentafoil knot**, is one of two knots with crossing number five, the other being the three-twist knot. It is listed as the **5 _{1} knot** in the Alexander-Briggs notation, and can also be described as the (5,2)-torus knot. The cinquefoil is the closed version of the double overhand knot.

The cinquefoil is a prime knot. Its writhe is 5, and it is invertible but not amphichiral.^{[1]} Its Alexander polynomial is

- ,

its Conway polynomial is

- ,

and its Jones polynomial is

These are the same as the Alexander, Conway, and Jones polynomials of the knot 10_{132}. However, the Kauffman polynomial can be used to distinguish between these two knots.

The name “cinquefoil” comes from the five-petaled flowers of plants in the genus *Potentilla*.

- A Pentafoil Knot at the Wayback Machine (archived June 4, 2004)