Shorthands | denote |
---|---|

u_a | algebraic unknotting number |

Nak | Nakanishi index |

det | determinant |

sign | signature |

max LT | maximum absolute value of Levine-Tristram signatures |

Hidden features | |

Click on | to see |

algebraic unknotting number | how it has been detected |

Alexander polynomial | a Seifert matrix (nondegenerate representative in the S-equivalence class) |

Nakanishi index | generator of the Alexander module, if Nakanishi index is 1 |

Determinant | H_1 of the double branched cover |

Welcome to the KNOTORIOUS world wide web page! | ||

set up by | Maciej Borodzik | mcboro'at'mimuw;edu;pl |

and | Stefan Friedl | sfriedl'at'gmail;com |

last update of the webpage | 19 Feb 2012 | |

last update of the knotorious data | 01 Dec 2011 |

Criterion | # of cases it detects u_a. |
---|---|

the signature | 883 (30%) |

the span of the Levine--Tristram signature | 888 (30%) |

the number of generators of H_1(double cover) | 170 (6%) |

higher covers | 199 (7%) |

the Lickorish obstruction | 435 (15%) |

the Stoimenow criterion | 4 (0.1%) |

the new u(K)=2 obstruction | 32 (1%) |

Total number of knots in the database: | 2977 |

Remarks:

- higher covers mean "The number of generators of H_1 of the branched cover". We checked double, triple and quadruple covers. Note, that this bounds the Nakanishi index, which is not put into the table, because in many cases we did not compute it.

We want to show, how often the given invariant is the unique criterion among others. We restrict ourselves to the knots, for which we were able to show that u_a>1. The precise meaning of each entry is discussed below.

Criterion | How often is it unique | |
---|---|---|

(1) | H_1(double cover) | 125 (8%) |

(2) | higher covers | 44 (3%) |

(3) | signature | 815 (53%) |

(4) | the span of signatures | 1 (0.1%) |

(5) | the Stoimenow criterion | 4 (0.3%) |

(6) | the Lickorish obstruction | 371 (24%) |

(7) | the new u(K)=2 obstruction | 21 (1%) |

Number of knots with unknotting 2 or more: | 1526 |

Explanation.

- The entry
*H_1 of double cover*counts all the knots for which the number of generators of H_1(double branched cover) is equal to the algebraic unknotting number, but the invariants in points (3),(4),(6) and (7) are insufficient to detect the unknotting number. - The entry
*higher covers*counts all the knots for which the number of generators of H_1(branched cover of order 3 or more) is equal to the algebraic unknotting, but the invariants in points (1),(3),(4),(5),(6),(7) are insufficient. - The entry
*signature*counts all the knots for which |signature(K)/2|=u_a(K), but all the invariants in points (1),(2),(5),(6),(7) cannot detect the unknotting number. -
The entry
*the span of signatures*shows the number of the knots for which the invariants in points (1),(2),(3),(5),(6) and (7) do not detect the algebraic unknotting number, but u_a is equal to half the difference between the maximal and minimal value of the Levine--Tristram signature. -
The entry
*the Stoimenow criterion*shows the number of knots, for which the Stoimenow's criterion shows that u_a(K)=3, but no invariants from (1),(2),(3),(4),(6) can give that result. -
The entry
*the Lickorish obstruction*shows the number of all knots, for which the Lickorish obstruction shows that u_a=2, but no invariant from (1),(2),(3),(4),(5),(7) can show that result. -
The entry
*the new u(K)=2 obstruction*shows the number of all knots, for which the new unknotting=2 obstruction detect that u_a(K)=3, but no criterion mentioned in points (1),(2),(3),(4),(5) and (6) is capable of doing that. - The number of knots with unknotting 2 or more does not count 25 knots for which we do not know the unknotting number.