Knot family: pq1r
(r = 0(mod
2) or p = r = 1 (mod 2);
pq1r
= r1pq)
Notation:
| 2212 |
76 |
| 3212 |
811 |
|
2312 |
88 |
| 4212 |
912 |
|
2412 |
98 |
|
2214 |
911 |
| 5212 |
107 |
|
2512 |
1034 |
|
| 4312 |
1012 |
|
3412 |
1021 |
|
| 3213 |
913 |
| 3214 |
1016 |
|
2314 |
1015 |
|
| 3313 |
1022 |
|
Dowker codes:
3212
8 12 14 10 2 6 4
5212
10 14 16 18 12 2 8 4 6
7212
12 16 18 20 22 14 2 10 4 6 8
9212
14 18 20 22 24 26 16 2 12 4 6 8 10
11212
16 20 22 24 26 28 30 18 2 14 4 6 8 10 12
3312
10 14 16 12 2 8 6 4
3412
10 16 18 14 12 2 8 6 4
3512
12 18 20 16 14 2 10 8 6 4
3612
12 20 22 18 16 14 2 10 8 6 4
3712
14 22 24 20 18 16 2 12 10 8
6 4
3812
14 24 26 22 20 18 16 2 12 10
8 6 4
3912
16 26 28 24 22 20 18 2 14 12 10
8 6 4
31012
16 28 30 26 24 22 20 18 2 14 12 10 8 6 4
31112
18 30 32 28 26 24 22 20 2 16 14 12 10 8 6 4
5312
12 16 18 20 14 2 10 8 4 6
7312
14 18 20 22 24 16 2 12 10 4 6 8
9312
16 20 22 24 26 28 18 2 14 12 4 6 8 10
11312
18 22 24 26 28 30 32 20 2 16 14 4 6 8 10 12
5412
12 18 20 22 16 14 2 10 8 4 6
7412
14 20 22 24 26 18 16 2 12 10 4 6 8
9412
16 22 24 26 28 30 20 18 2 14 12
4 6 8 10
5512
14 20 22 24 18 16 2 12 10 8 4 6
5612
14 22 24 26 20 18 16 2 12 10 8 4 6
5712
16 24 26 28 22 20 18 2 14 12 10 8 4 6
5812
16 26 28 30 24 22 20 18 2 14 12 10 8 4 6
5912
18 28 30 32 26 24 22 20 2 16 14 12 10 8 4 6
7512
16 22 24 26 28 20 18 2 14 12 10 4 6 8
9512
18 24 26 28 30 32 22 20 2 16 14 12 4 6 8 10
7612
16 24 26 28 30 22 20 18 2 14 12 10 4 6 8
7712
18 26 28 30 32 24 22 20 2 16 14 12 10 4 6 8
3313
12 10 14 18 16 2 4 6 8
4313
14 16 12 20 18 6 4 2 10 8
5313
14 12 16 22 20 18 2 4 6 10 8
6313
16 18 14 24 22 20 6 4 2 12 10 8
7313
16 14 18 26 24 22 20 2 4 6 12 10 8
8313
18 20 16 28 26 24 22 6 4 2 14 12 10 8
9313
18 16 20 30 28 26 24 22 2 4 6 14 12 10 8
10313
20 22 18 32 30 28 26 24 6 4 2 16 14 12 10 8
353
14 12 16 18 22 20 2 4 6 8 10
373
16 14 18 20 22 26 24 2 4 6 8 10 12
393
18 16 20 22 24 26 30 28 2 4 6 8 10 12 14
4413
16 18 14 12 20 22 6 4 2 8 10
6413
18 20 16 14 22 24 26 6 4 2 8 10 12
8413
20 22 18 16 24 26 28 30 6 4 2 8 10 12 14
4513
18 20 16 14 24 22 8 6 4 2 12 10
4613
20 22 18 16 14 24 26 8 6 4 2 10 12
4713
22 24 20 18 16 28 26 10 8 6 4 2 14 12
4813
24 26 22 20 18 16 28 30 10 8 6 4 2 12 14
4913
26 28 24 22 20 18 32 30 12 10 8 6 4 2 16 14
5513
16 14 18 20 26 24 22 2 4 6 8 12 10
6513
20 22 18 16 28 26 24 8 6 4 2 14 12 10
7513
18 16 20 22 30 28 26 24 2 4 6 8 14 12 10
8513
22 24 20 18 32 30 28 26 8 6 4 2 16 14 12 10
5713
18 16 20 22 24 30 28 26 2 4 6 8 10 14 12
6613
22 24 20 18 16 26 28 30 8 6 4 2 10 12 14
6713
24 26 22 20 18 32 30 28 10 8 6 4 2 16 14 12
5414
14 16 22 24 26 20 18 2 4 12 10 6 8
7414
16 18 24 26 28 30 22 20 2 4 14 12 6 8 10
5514
18 16 24 26 28 22 20 4 2 14 12 10 6 8
7514
20 18 26 28 30 32 24 22 4 2 16 14 12 6 8 10
5614
16 18 26 28 30 24 22 20 2 4 14 12 10 6 8
5714
20 18 28 30 32 26 24 22 4 2 16 14 12 10 6 8
5515
20 18 16 22 24 30 28 26 4 2 6 8 10 14 12
6515
22 24 26 20 18 32 30 28 10 8 6 2 4 16 14 12
Alexander polynomials:
3212
[5 -4 2
5212
[5 -5 4 -2
7212
[5 -5 5 -4 2
9212
[5 -5 5 -5 4 -2
11212
[5 -5 5 -5 5 -4 2
3312
[7 -6 2
3412
[9 -7 3
3512
[11 -9 3
3612
[13 -10 4
3712
[15 -12 4
3812
[17 -13 5
3912
[19 -15 5
31012
[21 -16 6
31112
[23 -18 6
5312
[7 -7 6 -2
7312
[7 -7 7 -6 2
9312
[7 -7 7 -7 6 -2
11312
[7 -7 7 -7 7 -6 2
5412
[9 -9 7 -3
7412
[9 -9 9 -7 3
9412
[9 -9 9 -9 7 -3
5512
[11 -11 9 -3
5612
[13 -13 10 -4
5712
[15 -15 12 -4
5812
[17 -17 13 -5
5912
[19 -19 15 -5
7512
[11 -11 11 -9 3
9512
[11 -11 11 -11 9 -3
7612
[13 -13 13 -10 4
7712
[15 -15 15 -12 4
3313
[ 9 -8 4
4313
[13 -11 4
5313
[15 -13 6
6313
[19 -16 6
7313
[21 -18 8
8313
[25 -21 8
9313
[27 -23 10
10313
[31 -26 10
353
[9 -9 8 -4
373
[9 -9 9 -8 4
393
[9 -9 9 -9 8 -4
4413
[13 -11 7 -3
6413
[13 -13 11 -7 3
8413
[13 -13 13 -11 7 -3
4513
[21 -17 6
4613
[19 -16 10 -4
4713
[29 -23 8
4813
[25 -21 13 -5
4913
[37 -29 10
5513
[15 -15 13 -6
6513
[31 -25 9
7513
[21 -21 18 -8
8513
[41 -33 12
5713
[15 -15 15 -13 6
6613
[19 -19 16 -10 4
6713
[43 -34 12
5414
[17 -15 11 -7 3
7414
[17 -17 15 -11 7 -3
5514
[21 -21 17 -6
7514
[21 -21 21 -17 6
5614
[25 -22 16 -10 4
5714
[29 -29 23 -8
5515
[25 -25 21 -9
6515
[31 -31 25 -9
| D((2k)(2l)1(2m))
= kl-(3kl+k+1)t + |
|
| + (4kl+2k+1) |
2m
å
i = 2 |
(-t)i-(3kl+k+1)t2m+1+klt2m+2 |
|
| D((2k+1)(2l)1(2m))
= (km+m)-(6km-2k-m+4)t + |
|
| + (2k+1)(2m+1) |
2m
å
i = 2 |
(-t)i-(6km-2k-m+4)t2m+1+(km+m)t2m+2 |
|
| D((2k)(2l+1)1(2m))
= (l+1) |
2k-1
å
i = 0 |
(2i+1)(-t)i+ |
|
| + (4kl+4k+1) |
2m
å
i = 2k |
(-t)i + (l+1) |
2k-1
å
i = 0 |
(2i+1)(-t)2k+2m-i,
k £ m |
|
| D((2k)(2l+1)1(2m))
= (l+1) |
2m
å
i = 0 |
(2i+1)(-t)i + |
|
| + (2ml+2m+2l+1) |
2k-1
å
i = 2m+1 |
(-t)i + (l+1) |
2m
å
i = 0 |
(2i+1)(-t)2k+2m-i,
k >
m |
|
| D((2k+1)(2l)1(2m+1))
= (l+1)(m+1)-(3lm+3l+2m+1)t + |
|
| + (4lm+4l+2m+1) |
2k
å
i = 2 |
(-t)i-(3lm+3l+2m+1)t2k+1+(l+1)(m+1)t2k+2 |
|
| D((2k+1)(2l+1)1(2m+1))
= (l+1) |
2k
å
i = 0 |
(2i+1)(-t)i+ |
|
| + (4kl+4k+2l+3) |
2m+1
å
i = 2k+1 |
(-t)i + (l+1) |
2k
å
i = 0 |
(2i+1)(-t)2k+2m+1-i,
k £ m |
|
| D((2k+1)(2l+1)1(2m+1))
= (l+1) |
2m+1
å
i = 0 |
(2i+1)(-t)i + |
|
| + (4lm+4l+4m+3) |
2k
å
i = 2m+2 |
(-t)i + (l+1) |
2k
å
i = 0 |
(2i+1)(-t)2k+2m+1-i,
k >
m |
|
Jones polynomials:
322
2 9
1 -1 3 -3 3 -3 2 -1
522
3 12 1 -1
3 -3 4 -5 4 -3 2 -1
722
4 15 1 -1
3 -3 4 -5 5 -5 4 -3 2 -1
922
5 18 1 -1
3 -3 4 -5 5 -5 5 -5 4 -3 2 -1
1122
6 21 1 -1
3 -3 4 -5 5 -5 5 -5 5 -5 4 -3 2 -1
332
-7 1 1 -2
3 -4 4 -4 3 -1 1
342
2 11 1 -1 3 -4
5 -5 4 -3 2 -1
352
-9 1 1 -2
3 -4 5 -6 5 -4 3 -1 1
362
2 13 1 -1 3 -4
5 -6 6 -5 4 -3 2 -1
372
-11 1
1 -2 3 -4 5 -6 6 -6 5 -4 3 -1 1
382
2 15
1 -1 3 -4 5 -6 6 -6 6 -5 4 -3 2 -1
392
-13 1
1 -2 3 -4 5 -6 6 -6 6 -6 5 -4 3 -1
1
3102
2 17 1 -1 3 -4
5 -6 6 -6 6 -6 6 -5 4 -3 2 -1
3112
-15 1 1 -2
3 -4 5 -6 6 -6 6 -6 6 -6 5 -4 3 -1
1
532
-10 0 1 -2
3 -5 6 -6 5 -4 3 -1 1
732
-13 -1 1 -2 3 -5
6 -7 7 -6 5 -4 3 -1 1
932
-16 -2 1 -2 3 -5
6 -7 7 -7 7 -6 5 -4 3 -1 1
1132
-19 -3 1 -2 3 -5
6 -7 7 -7 7 -7 7 -6 5 -4 3 -1 1
542
3 14 1 -1 3 -4
6 -7 7 -7 5 -3 2 -1
742
4 17 1 -1 3 -4
6 -7 8 -9 8 -7 5 -3 2 -1
942
5 20
1 -1 3 -4 6 -7 8 -9 9 -9 8 -7 5 -3
2 -1
552
-12 0 1 -2
3 -5 7 -8 8 -8 6 -4 3 -1
1
562
3 16 1 -1 3 -4
6 -8 9 -9 8. -7 5 -3 2 -1
572
-14 0 1 -2
3 -5 7 -8 9 -10 9 -8 6 -4 3 -1
1
582
3 18 1 -1 3 -4
6 -8 9 -10 10 -9 8 -7 5 -3 2 -1
592
-16 0 1 -2
3 -5 7 -8 9 -10 10 -10 9 -8 6 -4 3 -1
1
752
-15 -1 1 -2 3 -5
7 -9 10 -10 9 -8 6 -4 3 -1 1
952
-18 -2 1 -2 3 -5
7 -9 10 -11 11 -10 9 -8 6 -4 3 -1 1
762
4 19 1 -1 3 -4 6
-8 10 -11 11 -11 9 -7 5 -3 2 -1
772
-17 -1 1 -2 3 -5
7 -9 11 -12 12 -12 10 -8 6 -4 3 -1 1
333
2 11 1 -2 4 -5
6 -5 5 -3 1 -1
433
-3 7 1 -1
3 -5 6 -7 7 -6 4 -2
1
533
2 13 1 -2 4 -6
8 -8 8 -6 5 -3 1 -1
633
-5 7 1 -1
3 -5 6 -8 9 -9 8 -6
4 -2 1
733
2 15 1 -2 4 -6
8 -9 10 -9 8 -6 5 -3 1
-1
833
-7 7 1 -1
3 -5 6 -8 9 -10 10 -9 8 -6 4
-2 1
933
2 17 1 -2 4 -6
8 -9 10 -10 10 -9 8 -6 5 -3 1 -1
1033
-9 7 1 -1
3 -5 6 -8 9 -10 10 -10 10 -9 8 -6 4 -2
1
353
3 14 1 -2 4 -5
7 -8 8 -6 5 -3 1 -1
373
4 17 1 -2 4 -5
7 -8 9 -9 8 -6 5 -3 1 -1
393
5 20 1 -2 4 -5
7 -8 9 -9 9 -9 8 -6 5 -3 1 -1
443
3 14 1 -1 3 -5
7 -8 9 -8 6 -4 2
-1
643
4 17 1 -1 3 -5
7 -9 11 -11 11 -9 6 -4 2 -1
843
5 20 1 -1 3 -5
7 -9 11 -12 13 -12 11 -9 6 -4 2 -1
453
-3 9 1 -1 3
-5 7 -9 10 -10 8 -6 4 -2
1
463
3 16 1 -1 3 -5
7 -9 11 -11 10 -8 6 -4 2 -1
473
-3 11 1 -1 3 -5 7
-9 11 -12 11 -10 8 -6 4 -2 1
483
3 18 1 -1 3 -5
7 -9 11 -12 12 -11 10 -8 6 -4 2 -1
493
-3 13 1 -1 3 -5
7 -9 11 -12 12 -12 11 -10 8 -6 4 -2 1
553
3 16 1 -2 4 -6
9 -11 12 -11 10 -7 5 -3 1 -1
653
-5 9 1 -1
3 -5 7 -10 12 -13 13 -12 9 -6 4 -2
1
753
3 18 1 -2 4 -6
9 -12 14 -14 14 -12 10 -7 5 -3 1 -1
853
-7 9 1 -1
3 -5 7 -10 12 -14 15 -15 14 -12 9 -6 4 -2 1
573
4 19 1 -2 4 -6
9 -11 13 -14 14 -12 10 -7 5 -3 1 -1
663
4 19 1 -1 3 -5
7 -10 13 -14 15 -14 12 -9 6 -4 2 -1
673
-5 11 1 -1 3 -5
7 -10 13 -15 16 -16 14 -12 9 -6 4 -2 1
544
4 17 1 -1 3 -5
8 -10 12 -13 12 -10 7 -4 2 -1
744
5 20 1 -1 3 -5
8 -10 13 -15 15 -15 13 -10 7 -4 2 -1
554
-12 2 1 -2
4 -7 10 -13 15 -15 13 -11 8 -5 3 -1 1
754
-15 1 1 -2
4 -7 10 -14 17 -18 18 -17 14 -11 8 -5 3 -1 1
564
4 19 1 -1 3 -5
8 -11 14 -16 17 -16 13 -10 7 -4 2 -1
574
-14 2 1 -2
4 -7 10 -13 16 -18 18 -17 14 -11 8 -5 3 -1 1
555
3 18 1 -2 4 -7 11
-14 17 -18 17 -14 12 -8 5 -3 1 -1
655
-4 12 1 -1 3 -5
8 -12 15 -18 20 -20 18 -15 11 -7 4 -2 1
Symmetry groups: D2
Symmetry type: reversible.
Signatures:
|p-r|
if q = 1 (mod 2);
p+1 if p = r = 1 (mod 2)
and q = 0 (mod 2);
q if q = r = 0 (mod 2)
and p = 1 (mod 2);
r if p = q = r = 0 (mod 2).
Unknotting numbers:
u((2k)(2l)1(2m)) = min(k,
l-m)+m if l ³
m
u((2k)(2l)1(2m)) = m
if l<m
u((2k+1)(2l)1(2m)) = l
if l ³ m
u((2k+1)(2l)1(2m)) = min(k+1,m-l)+l
if l< m
u((2k)(2l+1)1(2m)) = l+1
if l = m, k = 1
u((2k)(2l+1)1(2m)) = k+l-1
if l = m, k ¹ 1
u((2k)(2l+1)1(2m)) = k+m
if l > m
u((2k)(2l+1)1(2m)) = l+1
if k = m, l < m
u((2k)(2l+1)1(2m)) = k+l-m
if l <
m
< k, k+l ¹
m
u((2k)(2l+1)1(2m)) = l+1
if k+l = m
u((2k)(2l+1)1(2m)) = |k+l-m|+l
if k+l ¹ m, l<
m
u((2k+1)(2l)1(2m+1)) = min(k+l+1,
k+m+1)
u((2k+1)(2l+1)1(2m+1)) =
l+1 if m = k+l
u((2k+1)(2l+1)1(2m+1)) = k+l
if l ³ m
u((2k+1)(2l+1)1(2m+1)) = |m-k-l|+l
if m ¹ k+l, <
m
|