Knot family: p111q
(pq = 0 (mod 2); p³q;
p111q = q111p)
Notation:
21112
77
31112
813
41112
914
51112
1010
41113
1019
Dowker codes:
21112
4 8 10 12 2 14 6
31112
4 10 12 14 2 16 8
6
41112
4 10 12 16 14 2 18
8 6
51112
4 12 14 18 16 2 20 10
8 6
61112
4 12 14 20 18 16 2 22
10 8 6
71112
4 14 16 22 20 18 2 24
12 10 8 6
81112
4 14 16 24 22
20 18 2 26 12 10 8
6
91112
4 16 18 26 24
22 20 2 28 14 12 10
8 6
101112
4 16 18 28 26
24 22 20 2 30 14 12 10
8 6
111112
4 18 20 30 28 26 24 22
2 32 16 14 12 10 8
6
41113
6 12 14 16 18 2 4
20 10 8
61113
6 14 16 18 20 22 4
2 24 8 10 12
81113
6 16 18 20 22
24 26 4 2 28 8 10
12 14
101113
6 18 20 22 24 26 28 30
4 2 32 8 10 12 14
16
41114
6 14 12 16 20
18 4 2 22 10 8
51114
6 14 16 18 22 20 2
4 24 12 10 8
61114
6 16 14 18 24 22 20 4
2 26 12 10 8
71114
6 16 18 20 26 24 22 2
4 28 14 12 10 8
81114
6 18 16 20 28 26 24 22
4 2 30 14 12 10 8
91114
6 18 20 22 30 28 26 24
2 4 32 16 14 12 10
8
61115
8 16 18 20 22 26 24 2
4 6 28 14 12 10
81115
8 20 18 22 24 26 28 30
6 4 2 32 10 12 14
16
61116
8 20 18 16 22 28 26 24
6 4 2 30 14 12 10
71116
8 18 20 22 24 30 28 26
2 4 6 32 16 14 12
10
Alexander polynomials:
21112
[9 -5 1
31112
[11 -7 2
41112
[15 -9 2
51112
[17 -11 3
61112
[21 -13 3
71112
[23 -15 4
81112
[27 -17 4
91112
[29 -19 5
101112
[33 -21 5
111112
[35 -23 6
41113
[11 -11 7 -2
61113
[11 -11 11 -7 2
81113
[11 -11 11 -11 7 -2
101113
[11 -11 11 -11 11 -7 2
41114
[25 -16 4
51114
[17 -17 11 -3
61114
[35 -23 6
71114
[23 -23 15 -4
81114
[45 -30 8
91114
[29 -29 19 -5
61115
[17 -17 17 -11 3
81115
[17 -17 17 -17 11 -3
61116
[49 -33 9
71116
[23 -23 23 -15 4
| D((2m)111(2n))
= mn - (3mn+m+n)t + (2m+1)(2n+1)t2
- (3mn+m+n)t3 + mnt4 |
|
| D((2m+1)111(2n))
= m+1 - (4m+3)t + (6m+5) |
2n
å
i = 2 |
(-1)iti - (4m-3)t2n+1
+ (m+1)t2n+2 |
|
| D((2m)111(2n+1))
= n+1 - (4n+3)t + (6n+5) |
2m
å
i = 2 |
(-1)iti - (4n-3)t2m+1
+ (n+1)t2m+2 |
|
Jones polynomials:
21112
-4 3
1 -2 3 -4 4 -3 3 -1
31112
-3 5 -1
3 -4 5 -5 5 -3 2 -1
41112
-6 3
1 -2 3 -5 6 -6 6 -4 3 -1
51112
-3 7 -1
3 -4 6 -7 7 -6 5 -3 2 -1
61112
-8 3
1 -2 3 -5 6 -7 8 -7 6 -4 3 -1
71112
-3 9 -1
3 -4 6 -7 8 -8 7 -6 5 -3 2 -1
81112
-10 3
1 -2 3 -5 6 -7 8 -8 8 -7 6 -4 3 -1
91112
-3 11
-1 3 -4 6 -7 8 -8 8 -8 7 -6 5 -3
2 -1
101112
-12 3
1 -2 3 -5 6 -7 8 -8 8 -8 8 -7 6 -4
3 -1
111112
-3 13 -1 3
-4 6 -7 8 -8 8 -8 8 -8 7 -6 5 -3
2 -1
41113
-6 4 -1
3 -5 7 -8 8 -7 6 -3 2 -1
61113
-9 3 -1
3 -5 7 -9 10 -10 9 -7 6 -3 2 -1
81113
101113
-15 1 -1
3 -5 7 -9 10 -11 11 -11 11 -10 9 -7 6 -3 2 -1
41114
-8 3
1 -2 3 -6 8 -9 10 -9
8 -5 3 -1
51114
-6 6 -1
3 -5 8 -10 11 -11 10 -8 6
-3 2 -1
61114
-10 3
1 -2 3 -6 8 -10 12 -12 12 -10
8 -5 3 -1
71114
-6 8 -1
3 -5 8 -10 12 -13 13 -12 10 -8
6 -3 2 -1
81114
-12 3
1 -2 3 -6 8 -10 12 -13 14 -13
12 -10 8 -5 3 -1
91114
-6 10 -1 3 -5
8 -10 12 -13 14 -14 13 -12 10 -8 6
-3 2 -1
61115
-9 5 -1
3 -5 8 -11 13 -14 14 -13 11 -8 6 -3 2
-1
81115
-12 4 -1
3 -5 8 -11 13 -15 16 -16 15 -13 11 -8 6 -3 2 -1
61116
-12 3
1 -2 3 -6 8 -11 14 -15 16 -15
14 -11 8 -5 3 -1
71116
-9 7 -1
3 -5 8 -11 14 -16 17 -17 16 -14 11
-8 6 -3 2 -1
Symmetry groups: D4
if p = q; otherwise D2.
Symmetry type: reversible.
Signatures:
0 if
p
= q = 0 (mod 2);
q-2
if
p = 1 (mod 2) and q = 0 (mod 2);
p-2
if
q = 1 (mod 2) and p = 0 (mod 2);
Unknotting numbers:
u(p1112)
= 1;
u(p1113) = p/2; u(p1114) =2;
u(p111q)
= min (u((p-3)1(q-1)),
u((p-1)1(q-3))
+ 1
hence:
u((2m)111(2n))
= n;
u((2m)111(2n+1))
= m;
u((2m+1)111(2n))
= n.
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