# Perko pair

Perko pair | |
---|---|

Arf invariant | 1 |

Braid length | 10 |

Braid no. | 3 |

Bridge no. | 3 |

Crosscap no. | 2 |

Crossing no. | 10 |

Genus | 3 |

Hyperbolic volume | 5.63877 |

Unknotting no. | 3 |

Conway notation | [3:-20:-20] |

A-B notation | 10_{161}/10_{162} |

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

Last /Next | 10_{160} / 10_{162} |

Other | |

hyperbolic, fibered, prime, reversible |

In the mathematical theory of knots, the **Perko pair**, named after Kenneth Perko, is a pair of entries in classical knot tables that actually represent the same knot; in Dale Rolfsen's knot table, this supposed pair of distinct knots is labeled 10_{161} and 10_{162}. In 1973, while working to complete the Tait–Little knot tables of knots up to 10 crossings (dating from the late 19th century),^{[1]} Perko found the duplication in Charles Newton Little's table,^{[2]} this duplication had been missed by John Horton Conway several years before in his knot table and subsequently found its way into Rolfsen's table.^{[3]} The Perko pair gives a counterexample to a "theorem" claimed by Little in 1900 that the writhe of a reduced diagram of a knot is an invariant (see Tait conjectures), as the two diagrams for the pair have different writhes.

In some later knot tables, the knots have been renumbered slightly (knots 10_{163} to 10_{166} are renumbered as 10_{162} to 10_{165}) so that knots 10_{161} and 10_{162} are different. Some authors have mistaken these two renumbered knots for the Perko pair and claimed incorrectly that they are the same.^{[4]}

## References[edit]

**^**Charles Newton Little, Non-alternating +/- knots, Trans. Roy. Soc. Edinburgh 39 (1900), page 774.**^**Kenneth A. Perko Jr.(b.1943),*On the classification of knots.*Proc. Amer. Math. Soc. 45 (1974), 262—266.**^**Dale Rolfsen,*Knots and Links*(see Appendix C for the knot table), 1976, ISBN 0-914098-16-0.**^**"The Revenge of the Perko Pair",*RichardElwes.co.uk*. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.

## External links[edit]

- "10_161",
*The Knot Atlas*. - Pictures of the equivalence between the two knots: "Perko pair knots",
*KnotPlot*. Accessed February 2016.

- 1 Arf invariant knots and links
- 10 braid length knots and links
- 3 braid number knots and links
- 3 bridge number knots and links
- 2 crosscap number knots and links
- 10 crossing number knots and links
- 3 genus knots and links
- 3 unknotting number knots and links
- Non-alternating knots and links
- Hyperbolic knots and links
- Fibered knots and links
- Prime knots and links
- Reversible knots and links
- Non-tricolorable knots and links