Knot family: p1q
(p = 1 (mod
2) or q = 1 (mod 2);
p³
q; p1q = q1p)
Notation:
312
62
313
74
512
82
413
84
513
95
712
102
613
104
514
108
Dowker codes:
312
4 8 10 12 2 6
512
4 10 12 14 16 2 6
8
712
4 12 14 16 18 20 2
6 8 10
912
4 14 16 18 20 22 24 2
6 8 10 12
1112
4 16 18 20 22 24 26 28
2 6 8 10 12 14
1312
4 18 20 22 24 26 28 30
32 2 6 8 10 12 14 16
313
6 10 12 14 4 2 8
413
6 10 12 16 14 4 2
8
513
6 12 14 18 16 4 2
10 8
613
6 12 14 20 18 16 4
2 10 8
713
6 14 16 22 20
18 4 2 12 10 8
813
6 14 16 24 22 20 18 4
2 12 10 8
913
6 16 18 26 24
22 20 4 2 14 12 10
8
1013
6 16 18 28 26 24 22 20
4 2 14 12 10 8
1113
6 18 20 30 28 26 24 22
4 2 16 14 12 10 8
1213
6 18 20 32 30 28 26 24
22 4 2 16 14 12 10
8
514
6 14 12 16 18 20 4
2 8 10
714
6 16 14 18 20 22 24 4
2 8 10 12
914
6 18 16 20 22 24 26 28
4 2 8 10 12 14
1114
6 20 18 22 24 26 28 30
32 4 2 8 10
12 14 16
515
8 16 14 18 22 20 6
4 2 12 10
615
8 14 16 18 24 22 20 4
6 2 12 10
715
8 18 16 20 26 24 22 6
4 2 14 12 10
815
8 16 18 20 28 26 24 22
4 6 2 14 12 10
915
8 20 18 22 30
28 26 24 6 4 2 16
14 12 10
1015
8 18 20 22 32 30 28 26
24 4 6 2 16 14 12
10
716
8 20 18 16 22 24 26 28
6 4 2 10 12 14
916
8 22 20 18 24 26 28 30
32 6 4 2 10 12 14
16
717
10 22 20 18 24 30 28 26
8 6 4 2 16 14 12
817
10 18 20 22 24 32 30 28
26 4 6 8 2 16
14 12
Alexander polynomials:
312
[3 -3 1
512
[3 -3 3 -1
712
[3 -3 3 -3 1
912
[3 -3 3 -3 3 -1
1112
[3 -3 3 -3 3 -3 1
1312
[3 -3 3 -3 3 -3 3 -1
313
[7 -4
413
[5 -5 2
513
[11 -6
613
[7 -7 3
713
[15 -8
813
[9 -9 4
913
[19 -10
1013
[11 -11 5
1113
[23 -12
1213
[13 -13 6
514
[5 -5 5 -2
714
[5 -5 5 -5 2
914
[5 -5 5 -5 5 -2
1114
[5 -5 5 -5 5 -5 2
515
[17 -9
615
[7 -7 7 -3
715
[23 -12
815
[9 -9 9 -4
915
[29 -15
1015
[11 -11 11 -5
716
[7 -7 7 -7 3
916
[7 -7 7 -7 7 -3
717
[31 -16
817
[9 -9 9 -9 4
| D((2m + 1)1(2n
+
1)) = (m + 1)(n + 1) - (2mn + 2m + 2n
+
1)t +(m + 1)(n + 1)t2 |
|
| D((2m)1(2n
+ 1)) = m + (2m + 1) |
2n+1
å
i = 1 |
(-1)i ti + mt2n+2 |
|
| D((2m+1)1(2n))
= n + (2n + 1) |
2m+1
å
i = 1 |
(-1)i ti + nt2m+2 |
|
Jones polynomials:
312
-5 1
1 -2 2 -2 2 -1 1
512
-8 0
1 -2 2 -3 3 -2 2 -1 1
712
-11 -1 1 -2
2 -3 3 -3 3 -2 2 -1 1
912
-14 -2 1 -2
2 -3 3 -3 3 -3 3 -2 2 -1 1
1112
-17 -3 1 -2
2 -3 3 -3 3 -3 3 -3 3 -2 2 -1 1
1312
-19 -4 1 -2
2 -3 3 -3 3 -3 3 -3 3 -3 3 -2 2 -1
1
313
1 8 1
-2 3 -2 3 -2 1 -1
413
-3 5
1 -1 2 -3 3 -3 3 -2 1
513
1 10
1 -2 3 -3 4 -3 3 -2 1 -1
613
-5 5
1 -1 2 -3 3 -4 4 -3 3 -2 1
713
1 12
1 -2 3 -3 4 -4 4 -3 3 -2 1 -1
813
-7 5
1 -1 2 -3 3 -4 4 -4 4 -3 3 -2 1
913
1 14
1 -2 3 -3 4 -4 4 -4 4 -3 3 -2 1 -1
1013
-9 5
1 -1 2 -3 3 -4 4 -4 4 -4 4 -3 3 -2
1
1113
1 16
1 -2 3 -3 4 -4 4 -4 4 -4 4 -3 3 -2
1 -1
1213
-11 5 1 -1
2 -3 3 -4 4 -4 4 -4 4 -4 4 -3 3 -2
1
514
-8 2
1 -2 3 -4 4 -4 4 -3 2 -1 1
714
-11 1 1 -2
3 -4 4 -5 5 -4 4 -3 2 -1 1
914
1114
515
1 12 1 -2 3
-4 5 -4 5 -4 3 -2 1 -1
615
-4 8
1 -1 2 -3 4 -5 5 -5 5 -4 3 -2 1
715
1 14
1 -2 3 -4 5 -5 6 -5 5 -4 3 -2 1 -1
815
-6 8
1 -1 2 -3 4 -5 5 -6 6 -5 5 -4 3 -2
1
915
16 1 -2 3 -4
5 -5 6 -6 6 -5 5 -4 3 -2 1 -1
1015
-8 8
1 -1 2 -3 4 -5 5 -6 6 -6 6 -5 5 -4
3 -2 1
716
-11 3 1 -2
3 -4 5 -6 6 -6 6 -5 4 -3 2 -1 1
916
-14 2 1 -2
3 -4 5 -6 6 -7 7 -6 6 -5 4 -3 2 -1
1
717
1 16 1 -2 3
-4 5 -6 7 -6 7 -6 5 -4 3 -2 1 -1
817
-5 11 1 -1
2 -3 4 -5 6 -7 7 -7 7 -6 5 -4 3 -2
1
Symmetry groups: D4
if p = q; otherwise D2.
Symmetry type: chiral and reversible.
Signatures:
2 if
p
= q = 1 (mod 2);
p-1 if
p
= 1 (mod 2) and q = 0 (mod 2);
q-1 if
q
= 1 (mod 2) and p = 0 (mod 2);
Unknotting numbers:
u(p2)
= 1;
u(p3) = 2;
u(p1q)
= min (u((p-2)1q),
u(p1(q-2),
u((p-2)1(q-2)))
+ 1
except for
the subfamily (2m+1)1(2m), where
u((2m+1)1(2m))
= min (u((2m-1)1(2m)), u((2m+1)1(2m-2)),
u((2m)1(2m-1)) + 1
hence:
u((2m+1)1(2n+1))
= n+1; u((2m+1)(2n)) = m; u((2m)1(2n+1))
= n+1.
|