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 |

Knot | u_a | Alexander polynomial | Nak. index | det. | sign | max LT. | ||||||||||||||||||||||||||||||||||||

8_1 | 1 detected by an unknotting move | -3+7t-3t^2 Seifert matrix of 8_1
| 1 Generator of the Alexander module (0,1) the Blanchfield form on it t^-1-2+t | 13 First homology of the double branched cover of 8_1 Z/13 | 0 | 0 | ||||||||||||||||||||||||||||||||||||

8_2 | 2 detected by the signature | -1+3t-3t^2+3t^3-3t^4+3t^5-t^6 Seifert matrix of 8_2
| 1 Generator of the Alexander module (0,0,0,1,0,0) the Blanchfield form on it -t^-2+3t^-1-3+3t-t^2 | 17 First homology of the double branched cover of 8_2 Z/17 | -4 | 4 | ||||||||||||||||||||||||||||||||||||

8_3 | 1 detected by an unknotting move | -4+9t-4t^2 Seifert matrix of 8_3
| 1 Generator of the Alexander module (0,1) the Blanchfield form on it 2t^-1-4+2t | 17 First homology of the double branched cover of 8_3 Z/17 | 0 | 0 | ||||||||||||||||||||||||||||||||||||

8_4 | 1 detected by an unknotting move | -2+5t-5t^2+5t^3-2t^4 Seifert matrix of 8_4
| 1 Generator of the Alexander module (0,0,0,1) the Blanchfield form on it t^-2-2t^-1+2-2t+t^2 | 19 First homology of the double branched cover of 8_4 Z/19 | 2 | 2 | ||||||||||||||||||||||||||||||||||||

8_5 | 2 detected by the signature | -1+3t-4t^2+5t^3-4t^4+3t^5-t^6 Seifert matrix of 8_5
| 1 Generator of the Alexander module (0,0,1,0,0,0) the Blanchfield form on it -t^-2+3t^-1-3+3t-t^2 | 21 First homology of the double branched cover of 8_5 Z/21 | -4 | 4 | ||||||||||||||||||||||||||||||||||||

8_6 | 1 detected by an unknotting move | -2+6t-7t^2+6t^3-2t^4 Seifert matrix of 8_6
| 1 Generator of the Alexander module (0,0,1,0) the Blanchfield form on it t^-2-3t^-1+4-3t+t^2 | 23 First homology of the double branched cover of 8_6 Z/23 | -2 | 2 | ||||||||||||||||||||||||||||||||||||

8_7 | 1 detected by an unknotting move | 1-3t+5t^2-5t^3+5t^4-3t^5+t^6 Seifert matrix of 8_7
| 1 Generator of the Alexander module (0,0,0,1,0,0) the Blanchfield form on it -1 | 23 First homology of the double branched cover of 8_7 Z/23 | 2 | 2 | ||||||||||||||||||||||||||||||||||||

8_8 | 2 detected by the Lickorish test | 2-6t+9t^2-6t^3+2t^4 Seifert matrix of 8_8
| 1 Generator of the Alexander module (0,0,0,1) the Blanchfield form on it t^-2-3t^-1+4-3t+t^2 | 25 First homology of the double branched cover of 8_8 Z/25 | 0 | 0 | ||||||||||||||||||||||||||||||||||||

8_9 | 1 detected by an unknotting move | -1+3t-5t^2+7t^3-5t^4+3t^5-t^6 Seifert matrix of 8_9
| 1 Generator of the Alexander module (0,0,0,0,1,-1) the Blanchfield form on it -t^-1+2-t | 25 First homology of the double branched cover of 8_9 Z/25 | 0 | 0 | ||||||||||||||||||||||||||||||||||||

8_10 | 1 detected by an unknotting move | 1-3t+6t^2-7t^3+6t^4-3t^5+t^6 Seifert matrix of 8_10
| 1 Generator of the Alexander module (0,0,1,0,0,0) the Blanchfield form on it -t^-2+3t^-1-5+3t-t^2 | 27 First homology of the double branched cover of 8_10 Z/27 | 2 | 2 | ||||||||||||||||||||||||||||||||||||

8_11 | 1 detected by an unknotting move | -2+7t-9t^2+7t^3-2t^4 Seifert matrix of 8_11
| 1 Generator of the Alexander module (0,0,1,0) the Blanchfield form on it t^-2-4t^-1+6-4t+t^2 | 27 First homology of the double branched cover of 8_11 Z/27 | -2 | 2 | ||||||||||||||||||||||||||||||||||||

8_12 | 1 detected by an unknotting move | 1-7t+13t^2-7t^3+t^4 Seifert matrix of 8_12
| 1 Generator of the Alexander module (0,0,1,1) the Blanchfield form on it -t^-1+3-t | 29 First homology of the double branched cover of 8_12 Z/29 | 0 | 0 | ||||||||||||||||||||||||||||||||||||

8_13 | 1 detected by an unknotting move | 2-7t+11t^2-7t^3+2t^4 Seifert matrix of 8_13
| 1 Generator of the Alexander module (0,0,1,1) the Blanchfield form on it 1 | 29 First homology of the double branched cover of 8_13 Z/29 | 0 | 0 | ||||||||||||||||||||||||||||||||||||

8_14 | 1 detected by an unknotting move | -2+8t-11t^2+8t^3-2t^4 Seifert matrix of 8_14
| 1 Generator of the Alexander module (0,1,0,0) the Blanchfield form on it 1 | 31 First homology of the double branched cover of 8_14 Z/31 | -2 | 2 | ||||||||||||||||||||||||||||||||||||

8_15 | 2 detected by the signature | 3-8t+11t^2-8t^3+3t^4 Seifert matrix of 8_15
| 1 Generator of the Alexander module (1,0,0,0) the Blanchfield form on it 2t^-1-3+2t | 33 First homology of the double branched cover of 8_15 Z/33 | -4 | 4 | ||||||||||||||||||||||||||||||||||||

8_16 | 2 detected by the Lickorish test | 1-4t+8t^2-9t^3+8t^4-4t^5+t^6 Seifert matrix of 8_16
| 1 Generator of the Alexander module (0,1,-t,0,0,0) the Blanchfield form on it -t^-2+3t^-1-5+3t-t^2 | 35 First homology of the double branched cover of 8_16 Z/35 | 2 | 2 | ||||||||||||||||||||||||||||||||||||

8_17 | 1 detected by an unknotting move | -1+4t-8t^2+11t^3-8t^4+4t^5-t^6 Seifert matrix of 8_17
| 1 Generator of the Alexander module (0,0,1,0,0,0) the Blanchfield form on it 1 | 37 First homology of the double branched cover of 8_17 Z/37 | 0 | 0 | ||||||||||||||||||||||||||||||||||||

8_18 | 2 detected by the Nakanishi index | -1+5t-10t^2+13t^3-10t^4+5t^5-t^6 Seifert matrix of 8_18
| 2 | 45 First homology of the double branched cover of 8_18 Z/15+Z/3 | 0 | 0 | ||||||||||||||||||||||||||||||||||||

8_19 | 3 detected by the signature | 1-t+t^3-t^5+t^6 Seifert matrix of 8_19
| 1 Generator of the Alexander module (0,1,0,0,0,0) the Blanchfield form on it t^-2-t^-1+1-t+t^2 | 3 First homology of the double branched cover of 8_19 Z/3 | -6 | 6 | ||||||||||||||||||||||||||||||||||||

8_20 | 1 detected by an unknotting move | 1-2t+3t^2-2t^3+t^4 Seifert matrix of 8_20
| 1 Generator of the Alexander module (1,0,0,0) the Blanchfield form on it 2t^-1-3+2t | 9 First homology of the double branched cover of 8_20 Z/9 | 0 | 0 | ||||||||||||||||||||||||||||||||||||

8_21 | 1 detected by an unknotting move | -1+4t-5t^2+4t^3-t^4 Seifert matrix of 8_21
| 1 Generator of the Alexander module (1,0,0,0) the Blanchfield form on it 1 | 15 First homology of the double branched cover of 8_21 Z/15 | -2 | 2 |