(−2,3,7) pretzel knot | |
---|---|

| |

Arf invariant | 0 |

Crosscap no. | 2 |

Crossing no. | 12 |

Hyperbolic volume | 2.828122088 |

Unknotting no. | 5 |

Conway notation | [7;-2 1;2] |

Dowker notation | 4, 8, -16, 2, -18, -20, -22, -24, -6, -10, -12, -14 |

D-T name | 12n242 |

Last /Next | 12n241<sub> </sub> / 12n243<sub> </sub> |

Other | |

hyperbolic, fibered, pretzel, reversible |

In geometric topology, a branch of mathematics, the **(−2, 3, 7) pretzel knot**, sometimes called the **Fintushel–Stern knot** (after Ron Fintushel and Ronald J. Stern), is an important example of a pretzel knot which exhibits various interesting phenomena under three-dimensional and four-dimensional surgery constructions.

## Mathematical properties

The (−2, 3, 7) pretzel knot has 7 *exceptional* slopes, Dehn surgery slopes which give non-hyperbolic 3-manifolds. Among the enumerated knots, the only other hyperbolic knot with 7 or more is the figure-eight knot, which has 10. All other hyperbolic knots are conjectured to have at most 6 exceptional slopes.

## Further reading

- Kirby, R., (1978). "Problems in low dimensional topology",
*Proceedings of Symposia in Pure Math.*, volume 32, 272-312. (see problem 1.77, due to Gordon, for exceptional slopes)