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. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_1 | 4 detected by the signature | 1-t+t^2-t^3+t^4-t^5+t^6-t^7+t^8 Seifert matrix of 9_1
| 1 Generator of the Alexander module (0,0,0,0,1,0,0,0) the Blanchfield form on it t^-3-t^-2+t^-1-1+t-t^2+t^3 | 9 First homology of the double branched cover of 9_1 Z/9 | -8 | 8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_2 | 1 detected by an unknotting move | 4-7t+4t^2 Seifert matrix of 9_2
| 1 Generator of the Alexander module (1,0) the Blanchfield form on it 1 | 15 First homology of the double branched cover of 9_2 Z/15 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_3 | 3 detected by the signature | 2-3t+3t^2-3t^3+3t^4-3t^5+2t^6 Seifert matrix of 9_3
| 1 Generator of the Alexander module (0,0,1,0,0,0) the Blanchfield form on it 2t^-2-3t^-1+3-3t+2t^2 | 19 First homology of the double branched cover of 9_3 Z/19 | -6 | 6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_4 | 2 detected by the signature | 3-5t+5t^2-5t^3+3t^4 Seifert matrix of 9_4
| 1 Generator of the Alexander module (t+t^3-t^4,0,1+t-t^3,0) the Blanchfield form on it 3t^-1-5+3t | 21 First homology of the double branched cover of 9_4 Z/21 | -4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_5 | 1 detected by an unknotting move | 6-11t+6t^2 Seifert matrix of 9_5
| 1 Generator of the Alexander module (1,0) the Blanchfield form on it -2t^-1+4-2t | 23 First homology of the double branched cover of 9_5 Z/23 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_6 | 3 detected by the signature | 2-4t+5t^2-5t^3+5t^4-4t^5+2t^6 Seifert matrix of 9_6
| 1 Generator of the Alexander module (t+t^2-t^3,0,-t^2,-1-t,-1+t^3,-t) the Blanchfield form on it 2t^-2-2t^-1+3-2t+2t^2 | 27 First homology of the double branched cover of 9_6 Z/27 | -6 | 6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_7 | 2 detected by the signature | 3-7t+9t^2-7t^3+3t^4 Seifert matrix of 9_7
| 1 Generator of the Alexander module (-1,0,0,1) the Blanchfield form on it 3t^-1-4+3t | 29 First homology of the double branched cover of 9_7 Z/29 | -4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_8 | 1 detected by an unknotting move | -2+8t-11t^2+8t^3-2t^4 Seifert matrix of 9_8
| 1 Generator of the Alexander module (0,0,0,1) the Blanchfield form on it -t^-2+3t^-1-4+3t-t^2 | 31 First homology of the double branched cover of 9_8 Z/31 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_9 | 3 detected by the signature | 2-4t+6t^2-7t^3+6t^4-4t^5+2t^6 Seifert matrix of 9_9
| 1 Generator of the Alexander module (0,0,1,0,0,0) the Blanchfield form on it 2t^-2-3t^-1+3-3t+2t^2 | 31 First homology of the double branched cover of 9_9 Z/31 | -6 | 6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_10 | 2 detected by the signature | 4-8t+9t^2-8t^3+4t^4 Seifert matrix of 9_10
| 1 or 2 | 33 First homology of the double branched cover of 9_10 Z/33 | -4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_11 | 2 detected by the signature | -1+5t-7t^2+7t^3-7t^4+5t^5-t^6 Seifert matrix of 9_11
| 1 Generator of the Alexander module (0,1,0,0,0,0) the Blanchfield form on it 2t^-2-3t^-1+3-3t+2t^2 | 33 First homology of the double branched cover of 9_11 Z/33 | 4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_12 | 1 detected by an unknotting move | -2+9t-13t^2+9t^3-2t^4 Seifert matrix of 9_12
| 1 Generator of the Alexander module (0,0,1,1) the Blanchfield form on it 1 | 35 First homology of the double branched cover of 9_12 Z/35 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_13 | 2 detected by the signature | 4-9t+11t^2-9t^3+4t^4 Seifert matrix of 9_13
| 1 Generator of the Alexander module (0,0,1,1-t) the Blanchfield form on it 4t^-1-7+4t | 37 First homology of the double branched cover of 9_13 Z/37 | -4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_14 | 1 detected by an unknotting move | 2-9t+15t^2-9t^3+2t^4 Seifert matrix of 9_14
| 1 Generator of the Alexander module (0,1,0,0) the Blanchfield form on it 2t^-1-3+2t | 37 First homology of the double branched cover of 9_14 Z/37 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_15 | 2 detected by the Lickorish test | -2+10t-15t^2+10t^3-2t^4 Seifert matrix of 9_15
| 1 Generator of the Alexander module (0,0,0,1) the Blanchfield form on it -t^-2+5t^-1-8+5t-t^2 | 39 First homology of the double branched cover of 9_15 Z/39 | 2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_16 | 3 detected by the signature | 2-5t+8t^2-9t^3+8t^4-5t^5+2t^6 Seifert matrix of 9_16
| 1 Generator of the Alexander module (-1,0,0,0,0,1) the Blanchfield form on it 2t^-2-3t^-1+4-3t+2t^2 | 39 First homology of the double branched cover of 9_16 Z/39 | -6 | 6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_17 | 2 detected by the Lickorish test | 1-5t+9t^2-9t^3+9t^4-5t^5+t^6 Seifert matrix of 9_17
| 1 Generator of the Alexander module (0,0,0,1,0,-1) the Blanchfield form on it t^-2-3t^-1+3-3t+t^2 | 39 First homology of the double branched cover of 9_17 Z/39 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_18 | 2 detected by the signature | 4-10t+13t^2-10t^3+4t^4 Seifert matrix of 9_18
| 1 or 2 | 41 First homology of the double branched cover of 9_18 Z/41 | -4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_19 | 1 detected by an unknotting move | 2-10t+17t^2-10t^3+2t^4 Seifert matrix of 9_19
| 1 Generator of the Alexander module (0,0,1,1) the Blanchfield form on it t^-1-3+t | 41 First homology of the double branched cover of 9_19 Z/41 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_20 | 2 detected by the signature | -1+5t-9t^2+11t^3-9t^4+5t^5-t^6 Seifert matrix of 9_20
| 1 Generator of the Alexander module (0,0,0,1,0,0) the Blanchfield form on it -2t^-2+4t^-1-5+4t-2t^2 | 41 First homology of the double branched cover of 9_20 Z/41 | -4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_21 | 1 detected by an unknotting move | -2+11t-17t^2+11t^3-2t^4 Seifert matrix of 9_21
| 1 or 2 | 43 First homology of the double branched cover of 9_21 Z/43 | 2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_22 | 1 detected by an unknotting move | 1-5t+10t^2-11t^3+10t^4-5t^5+t^6 Seifert matrix of 9_22
| 1 Generator of the Alexander module (0,0,0,0,1,0) the Blanchfield form on it t^-2-5t^-1+9-5t+t^2 | 43 First homology of the double branched cover of 9_22 Z/43 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_23 | 2 detected by the signature | 4-11t+15t^2-11t^3+4t^4 Seifert matrix of 9_23
| 1 Generator of the Alexander module (0,1,0,0) the Blanchfield form on it 2t^-1-3+2t | 45 First homology of the double branched cover of 9_23 Z/45 | -4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_24 | 1 detected by an unknotting move | -1+5t-10t^2+13t^3-10t^4+5t^5-t^6 Seifert matrix of 9_24
| 1 Generator of the Alexander module (0,0,1,0,0,0) the Blanchfield form on it t^-2-5t^-1+7-5t+t^2 | 45 First homology of the double branched cover of 9_24 Z/45 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_25 | 1 detected by an unknotting move | -3+12t-17t^2+12t^3-3t^4 Seifert matrix of 9_25
| 1 Generator of the Alexander module (1,0,0,0) the Blanchfield form on it -2t^-1+5-2t | 47 First homology of the double branched cover of 9_25 Z/47 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_26 | 1 detected by an unknotting move | 1-5t+11t^2-13t^3+11t^4-5t^5+t^6 Seifert matrix of 9_26
| 1 Generator of the Alexander module (0,0,0,-1,1,0) the Blanchfield form on it -1 | 47 First homology of the double branched cover of 9_26 Z/47 | 2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_27 | 1 detected by an unknotting move | -1+5t-11t^2+15t^3-11t^4+5t^5-t^6 Seifert matrix of 9_27
| 1 Generator of the Alexander module (0,1,0,0,0,0) the Blanchfield form on it t^-2-3t^-1+5-3t+t^2 | 49 First homology of the double branched cover of 9_27 Z/49 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_28 | 1 detected by an unknotting move | 1-5t+12t^2-15t^3+12t^4-5t^5+t^6 Seifert matrix of 9_28
| 1 Generator of the Alexander module (0,0,0,0,0,1) the Blanchfield form on it t^-2-3t^-1+5-3t+t^2 | 51 First homology of the double branched cover of 9_28 Z/51 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_29 | 1 detected by an unknotting move | 1-5t+12t^2-15t^3+12t^4-5t^5+t^6 Seifert matrix of 9_29
| 1 Generator of the Alexander module (0,-1,-1,0,-t,0) the Blanchfield form on it -t^-2+3t^-1-5+3t-t^2 | 51 First homology of the double branched cover of 9_29 Z/51 | 2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_30 | 1 detected by an unknotting move | -1+5t-12t^2+17t^3-12t^4+5t^5-t^6 Seifert matrix of 9_30
| 1 Generator of the Alexander module (0,0,0,0,1,0) the Blanchfield form on it 2t^-1-5+2t | 53 First homology of the double branched cover of 9_30 Z/53 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_31 | 2 detected by the Lickorish test | 1-5t+13t^2-17t^3+13t^4-5t^5+t^6 Seifert matrix of 9_31
| 1 Generator of the Alexander module (0,0,t,1+t,0,0) the Blanchfield form on it t^-2-3t^-1+5-3t+t^2 | 55 First homology of the double branched cover of 9_31 Z/55 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_32 | 1 detected by an unknotting move | 1-6t+14t^2-17t^3+14t^4-6t^5+t^6 Seifert matrix of 9_32
| 1 Generator of the Alexander module (0,0,0,0,1,0) the Blanchfield form on it t^-1-1+t | 59 First homology of the double branched cover of 9_32 Z/59 | 2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_33 | 1 detected by an unknotting move | -1+6t-14t^2+19t^3-14t^4+6t^5-t^6 Seifert matrix of 9_33
| 1 Generator of the Alexander module (0,0,0,1,0,0) the Blanchfield form on it -t^-1+1-t | 61 First homology of the double branched cover of 9_33 Z/61 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_34 | 1 detected by an unknotting move | -1+6t-16t^2+23t^3-16t^4+6t^5-t^6 Seifert matrix of 9_34
| 1 Generator of the Alexander module (0,1,1,0,-1,1) the Blanchfield form on it -t^-2+4t^-1-7+4t-t^2 | 69 First homology of the double branched cover of 9_34 Z/69 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_35 | 2 detected by the Nakanishi index | 7-13t+7t^2 Seifert matrix of 9_35
| 2 | 27 First homology of the double branched cover of 9_35 Z/3+Z/9 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_36 | 2 detected by the signature | -1+5t-8t^2+9t^3-8t^4+5t^5-t^6 Seifert matrix of 9_36
| 1 Generator of the Alexander module (0,0,1,0,0,0) the Blanchfield form on it t^-2-5t^-1+7-5t+t^2 | 37 First homology of the double branched cover of 9_36 Z/37 | 4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_37 | 2 detected by the Nakanishi index | 2-11t+19t^2-11t^3+2t^4 Seifert matrix of 9_37
| 2 | 45 First homology of the double branched cover of 9_37 Z/15+Z/3 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_38 | 2 detected by the signature | 5-14t+19t^2-14t^3+5t^4 Seifert matrix of 9_38
| 1 or 2 | 57 First homology of the double branched cover of 9_38 Z/57 | -4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_39 | 1 detected by an unknotting move | -3+14t-21t^2+14t^3-3t^4 Seifert matrix of 9_39
| 1 Generator of the Alexander module (1,0,0,1) the Blanchfield form on it -t^-2+4t^-1-6+4t-t^2 | 55 First homology of the double branched cover of 9_39 Z/55 | 2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_40 | 2 detected by the Nakanishi index | 1-7t+18t^2-23t^3+18t^4-7t^5+t^6 Seifert matrix of 9_40
| 2 | 75 First homology of the double branched cover of 9_40 Z/15+Z/5 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_41 | 2 detected by the Nakanishi index | 3-12t+19t^2-12t^3+3t^4 Seifert matrix of 9_41
| 2 | 49 First homology of the double branched cover of 9_41 Z/7+Z/7 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_42 | 1 detected by an unknotting move | -1+2t-t^2+2t^3-t^4 Seifert matrix of 9_42
| 1 Generator of the Alexander module (0,t,1,0) the Blanchfield form on it t^-1-2+t | 7 First homology of the double branched cover of 9_42 Z/7 | 2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_43 | 2 detected by the signature | -1+3t-2t^2+t^3-2t^4+3t^5-t^6 Seifert matrix of 9_43
| 1 Generator of the Alexander module (0,0,0,0,0,1) the Blanchfield form on it -t^-2+3t^-1-3+3t-t^2 | 13 First homology of the double branched cover of 9_43 Z/13 | -4 | 4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_44 | 1 detected by an unknotting move | 1-4t+7t^2-4t^3+t^4 Seifert matrix of 9_44
| 1 Generator of the Alexander module (1,0,0,0) the Blanchfield form on it 1 | 17 First homology of the double branched cover of 9_44 Z/17 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_45 | 1 detected by an unknotting move | -1+6t-9t^2+6t^3-t^4 Seifert matrix of 9_45
| 1 Generator of the Alexander module (-1+t,0,-1,0) the Blanchfield form on it -1 | 23 First homology of the double branched cover of 9_45 Z/23 | 2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_46 | 2 detected by the Nakanishi index | -2+5t-2t^2 Seifert matrix of 9_46
| 2 | 9 First homology of the double branched cover of 9_46 Z/3+Z/3 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_47 | 2 detected by the Nakanishi index | 1-4t+6t^2-5t^3+6t^4-4t^5+t^6 Seifert matrix of 9_47
| 2 | 27 First homology of the double branched cover of 9_47 Z/9+Z/3 | -2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_48 | 2 detected by the Nakanishi index | -1+7t-11t^2+7t^3-t^4 Seifert matrix of 9_48
| 2 | 27 First homology of the double branched cover of 9_48 Z/9+Z/3 | 2 | 2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

9_49 | 3 detected by the Stoimenow criterion | 3-6t+7t^2-6t^3+3t^4 Seifert matrix of 9_49
| 2 | 25 First homology of the double branched cover of 9_49 Z/5+Z/5 | -4 | 4 |