Knot family: pq
(pq
= 0 (mod 2); p ³ q; pq
= qp)
Notation:
22
41
32
52
42
61
52
72
43
73
44 83
62
81
54 94
72
92
63
93
64 103
42
101
Dowker codes:
22
4 6 8 2
32
4 8 10 2 6
42
4 8 12 10 2 6
52
4 10 14 12 2 8 6
62
4 10 16 14 12 2 8
6
72
4 12 18 16 14 2 10
8 6
82
4 12 20 18 16 14 2 10
8 6
92
4 14 22 20 18 16 2 12
10 8 6
102
4 14 24 22 20
18 16 2 12 10 8 6
112
4 16 26 24 22 20 18 2
14 12 10 8 6
122
4 16 28 26 24 22 20 18
2 14 12 10 8 6
132
4 18 30 28 26 24 22 20
2 16 14 12 10 8 6
142
4 18 32 30 28 26 24 22 20
2 16 14 12 10 8 6
43
6 10 12 14 2 4
8
63
8 12 14 16 18 2 4
6 10
83
10 14 16 18 20 22 2
4 6 8 12
103
12 16 18 20 22 24 26 2
4 6 8 10 14
123
14 18 20 22 24 26 28 30
2 4 6 8 10 12 16
44
6 12 10 16 14 4 2
8
54
6 12 14 18 16 2 4
10 8
64
6 14 12 20 18 16 4
2 10 8
74
6 14 16 22 20 18 2
4 12 10 8
84
6 16 14 24 22 20 18 4
2 12 10 8
94
6 16 18 26 24 22 20 2
4 14 12 10 8
104
6 18 16 28 26 24 22 20
4 2 14 12 10 8
114
6 18 20 30 28 26 24 22
20 2 4 14 12 10 8
65
8 14 16 18 22 20 2
4 6 12 10
85
10 16 18 20 22 26 24 2
4 6 8 14 12
105
12 18 20 22 24 26 30 28
2 4 6 8 10 16 14
66
8 18 16 14 24 22 20 6
4 2 12 10
76
8 16 18 20 26 24 22 2
4 6 14 12 10
86
8 20 18 16 28 26 24 22
6 4 2 14 12 10
96
8 18 20 22 30 28 26 24
2 4 6 16 14 12 10
106
8 22 20 18 32 30 28 26
24 6 4 2 16 14 12
10
Alexander polynomials:
22
[3 -1
32
[3 -2
42
[5 -2
52
[5 -3
62
[7 -3
72
[7 -4
82
[9 -4
92
[9 -5
102
[11 -5
112
[11 -6
122
[13 -6
132
[13 -7
142
[14 -7
43
[3 -3 2
63
[3 -3 3 -2
83
[3 -3 3 -3 2
103
[3 -3 3 -3 3 -2
123
[3 -3 3 -3 3 -3 2
44
[9 -4
54
[5 -5 3
64
[13 -6
74
[7 -7 4
84
[17 -8
94
[9 -9 5
104
[21 -10
114
[11 -11 6
124
[25 -12
65
[5 -5 5 -3
85
[5 -5 5 -5 3
105
[5 -5 5 -5 5 -3
66
[19 -9
76
[7 -7 7 -4
86
[25 -12
96
[9 -9 9 -5
106
[31 -15
| D((2m)(2n))
= mn - (2mn + 1)t + mnt2 |
|
| D((2m+1)(2n))
= (m + 1) + (2m + 1) |
2n-1
å
i = 1 |
(-1)i ti + (m + 1)t2n |
|
| D((2m)(2n+1))
= (n + 1) + (2n + 1) |
2m-1
å
i= 1 |
(-1)i ti + (n + 1)t2m |
|
Jones polynomials:
22
-2 2
1 -1 1 -1 1
32
1 6
1 -1 2 -1 1 -1
42
-4 2
1 -1 1 -2 2 -1 1
52
1 8
1 -1 2 -2 2 -1 1 -1
62
-6 2
1 -1 1 -2 2 -2 2 -1 1
72
1 10 1 -1
2 -2 2 -2 2 -1 1 -1
82
-8 2
1 -1 1 -2 2 -2 2 -2 2 -1 1
92
1 12 1 -1
2 -2 2 -2 2 -2 2 -1 1 -1
102
-10 2
1 -1 1 -2 2 -2 2 -2 2 -2 2 -1 1
112
1 14
1 -1 2 -2 2 -2 2 -2 2 -2 2 -1 1
-1
122
-12 2
1 -1 1 -2 2 -2 2 -2 2 -2 2 -2 2
-1 1
132
1 16 1 -1
2 -2 2 -2 2 -2 2 -2 2 -2 2 -1
1 -1
142
-14 2
1 -1 1 -2 2 -2 2 -2 2 -2 2 -2 2
-2 2 -1 1
43
2 9
1 -1 2 -2 3 -2 1 -1
63
3 12 1 -1
2 -2 3 -3 3 -2 1 -1
83
4 15 1 -1
2 -2 3 -3 3 -3 3 -2 1 -1
103
5 18 1 -1
2 -2 3 -3 3 -3 3 -3 3 -2 1 -1
123
6 21 1 -1
2 -2 3 -3 3 -3 3 -3 3 -3 3 -2 1 -1
44
-4 4
1 -1 2 -3 3 -3 2 -1 1
54
2 11 1 -1
2 -3 4 -3 3 -2 1 -1
64
-6 4
1 -1 2 -3 3 -4 4 -3 2 -1 1
74
2 13 1 -1
2 -3 4 -4 4 -3 3 -2 1 -1
84
-8 4
1 -1 2 -3 3 -4 4 -4 4 -3 2 -1
1
94
2 15 1 -1
2 -3 4 -4 4 -4 4 -3 3 -2 1 -1
104
-10 4
1 -1 2 -3 3 -4 4 -4 4 -4 4 -3
2 -1 1
114
2 17
1 -1 2 -3 4 -4 4 -4 4 -4 4 -3
3 -2 1 -1
124
-12 4
1 -1 2 -3 3 -4 4 -4 4 -4 4 -4
4 -3 2 -1 1
65
3 14 1 -1
2 -3 4 -4 5 -4 3 -2 1 -1
85
4 17 1 -1
2 -3 4 -4 5 -5 5 -4 3 -2 1 -1
105
5 20 1 -1
2 -3 4 -4 5 -5 5 -5 5 -4 3 -2 1 -1
66
-6 6
1 -1 2 -3 4 -5 5 -5 4 -3 2 -1 1
76
3 16 1 -1
2 -3 4 -5 6 -5 5 -4 3 -2 1 -1
86
-8 6
1 -1 2 -3 4 -5 5 -6 6 -5 4 -3 2 -1
1
96
3 18 1 -1
2 -3 4 -5 6 -6 6 -5 5 -4 3 -2 1 -1
106
-10 6
1 -1 2 -3 4 -5 5 -6 6 -6 6 -5 4 -3
2 -1 1
Symmetry groups: D2
if p = q; otherwise D1.
Symmetry type: fully amphicheiral for p
= q; otherwise cheiral and reversible.
Signatures:
0 if
p
= q = 0 (mod 2);
p if p
= 0 (mod 2) and q = 1 (mod 2);
q if
q
= 0 (mod 2) and p = 1 (mod 2).
Unknotting numbers:
u(p2)
= 1;
u(p3) = p/2; u(pq) = min (u(p-2,q),
u(p,q-2))
+ 1
hence:
u((2m)(2n))
= n;
u((2m+1)(2n)) = n; u((2m)(2n+1))
= m.
|