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discussion of unknown values
statistics of invariants
introduction and help
Download KNOTORIOUS (32 bit version)
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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
You may freely contact the authors in case of any questions.

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 -1 -1 -1 0 -1 -1 -1
0 -1 -1 -1 0 0 -1 -1
0 0 -1 -1 0 0 0 -1
0 0 0 -1 0 0 0 0
-1 -1 -1 -1 -1 -1 -1 -1
0 -1 -1 -1 0 -1 -1 -1
0 0 -1 -1 0 0 -1 -1
0 0 0 -1 0 0 0 -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 -1
0 -4
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 -1 -1 -1 -1 -1
0 -1 -1 0 -1 -1
0 0 -1 0 0 -1
0 -1 -1 -1 -1 -1
0 0 -1 0 -1 -1
0 0 0 0 0 -2
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 -1 1 -1
0 -1 1 0
0 0 -3 0
0 -1 1 -1
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
-3 1
0 -2
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 0 0 -1 0 0
0 -1 -1 -1 0 -1
0 0 -1 -1 0 0
0 0 0 -2 0 0
0 -1 -1 -1 -1 -1
0 0 -1 -1 0 -1
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 0 1 0
0 -1 1 0
0 0 -3 0
0 -1 1 -1
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 0 0 0
-1 -1 0 0
0 0 -1 0
1 1 1 2
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 -1 0 -1 -1 0
0 -1 0 0 -1 0
0 0 -1 0 -1 0
0 -1 0 -1 -1 0
0 0 0 0 -2 0
0 0 -1 0 -1 -1
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 1 -1 -1
0 -2 1 1
0 0 -1 -1
0 0 0 -2
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 0 0 0 1 0
0 -1 -1 0 0 0
0 0 1 0 0 0
1 -1 0 1 1 1
0 0 0 0 1 0
1 0 0 0 1 1
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 0 0 0
0 -2 0 0
-1 0 -1 0
-1 1 -1 1
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 0 -1 0
0 -2 0 0
0 -1 -2 0
-1 0 -1 -1
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 0 0 0
-1 -1 -1 0
-1 0 -1 1
0 0 0 2
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 -1 0 0
0 1 0 0
0 0 1 0
1 0 -1 2
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 0 -1 -1 -1 0
0 -1 -1 -1 -1 0
0 0 -2 -1 -2 0
0 0 -2 -2 -2 0
0 0 -1 -1 -2 0
0 -1 -1 -1 -1 -1
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 0 0 0 0 0
-1 -1 -1 0 -1 -1
-1 0 -1 0 0 -1
0 0 0 1 0 0
-1 0 -1 0 -1 -1
-1 0 0 -1 0 -1
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 0 0 -1
0 -2 1 1
0 0 -1 -1
0 0 0 -2
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 0 0 0
-1 -1 0 1
0 0 1 0
1 0 1 -2
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 0 0 0 0 0
0 -1 0 0 -1 0
-1 0 -1 0 0 0
-1 0 -1 1 -1 0
0 0 0 0 -1 0
0 -1 0 0 -1 -1
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
2 0 0 0
0 -1 -1 0
0 0 1 0
-1 -1 0 1
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 0 0 0 0 0
-1 -1 0 -1 -1 -1
0 0 1 0 0 0
-1 0 -1 -1 -1 -1
-1 0 -1 0 -1 0
-1 0 -1 0 -1 -1
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 0 -1 0
0 -1 0 -1
0 0 -2 0
0 0 1 -2
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 0 0 0 0 0
-1 -1 0 0 0 0
-1 -1 1 0 -1 0
-1 -1 1 1 -1 1
0 0 0 0 -1 0
-1 -1 1 0 -1 1
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 -1 0 -1
0 -2 0 -1
0 0 1 -1
0 -1 0 -2
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 0 0 0 0 0
-1 -1 0 0 -1 0
0 0 1 0 0 0
0 0 1 1 0 1
-1 0 -1 -1 -1 -1
0 0 1 0 0 1
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 0 0 0 0 0
0 -1 0 0 -1 0
-1 0 -1 0 0 0
-1 0 -1 1 1 1
0 0 0 0 1 0
0 0 0 0 1 1
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 0 0 0 0 0
-1 -1 0 0 0 0
-1 -1 1 1 1 0
0 0 0 1 0 0
0 0 0 0 -1 0
0 0 0 0 -1 -1
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 0 0 1 1 1
-1 -1 0 0 -1 0
0 0 -1 0 0 0
0 0 -1 1 0 0
0 0 -1 1 1 1
0 0 -1 1 0 1
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 -1 -1 -1 0 0
0 -1 -1 0 0 0
0 0 1 0 0 0
0 -1 -1 -1 0 0
0 -1 0 -1 1 1
0 0 0 0 0 1
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 0 0 0 0 0
0 -1 0 0 -1 0
-1 0 -1 0 0 0
-1 0 -1 1 0 1
0 0 0 0 -1 0
0 1 0 0 1 1
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 0 0 0 0 0
-1 -1 -1 0 -1 -1
0 0 1 1 0 1
0 0 0 1 0 0
-1 0 -1 0 -1 0
0 0 0 1 0 1
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 -1 0 0 0 0
0 1 0 0 0 0
-1 -1 -1 0 0 0
-1 0 -1 1 -1 1
-1 -1 -1 0 -1 0
0 0 -1 0 -1 1
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 0 -1 -1 0 0
-1 -1 -1 -1 0 0
0 0 -1 -1 -1 0
0 0 0 1 0 0
0 0 0 1 1 0
-1 0 -1 0 0 1
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
-3 -1
-2 -3
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 -1 0 0 0 0
0 1 0 0 0 0
-1 0 1 0 1 0
-1 0 1 1 1 1
0 0 0 0 1 0
-1 0 1 0 1 1
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
1 0 0 0
0 1 0 0
1 0 -2 1
-1 -1 0 -1
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
-2 1 0 1
0 -2 -1 -2
-1 -1 -2 -1
0 -1 -1 -2
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 0 -1 1
0 2 1 1
0 0 1 0
0 1 0 2
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
1 0 0 0 -1 1
-1 -1 0 -1 0 -1
0 0 -1 0 -1 0
-1 0 0 -1 0 0
0 0 0 0 -1 0
0 0 -1 0 -1 1
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
-1 1 -1 1
0 2 0 1
0 0 -1 1
0 1 0 2
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 0 0 0
-1 0 1 -1
1 1 -2 1
-1 0 1 0
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 -1 -1 -1 0 0
0 -1 -1 0 0 0
0 0 1 0 0 0
0 -1 -1 -1 0 0
0 -1 0 -1 -1 0
0 0 0 0 -1 -1
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 -1 0 -1
0 0 0 -1
0 0 1 0
0 -1 1 -2
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
0 2 -1 -1
1 1 1 0
0 0 1 0
0 1 -1 0
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
3 2
1 0
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
-1 0 -1 -1 0 0
-1 -1 -1 -1 0 0
0 0 -1 -1 -1 0
0 0 0 1 0 0
0 0 0 1 1 0
-1 0 -1 0 0 -1
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
1 0 0 0
0 1 0 0
1 0 1 1
-1 -1 0 -1
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
-1 1 -1 1
0 -2 0 -1
0 0 -1 1
0 -1 0 -2
2
25
First homology
of the double branched cover of 9_49
Z/5+Z/5
-4 4