12n
0709
(K12n
0709
)
A knot diagram
1
Linearized knot diagam
4 5 6 9 11 12 1 4 5 2 3 7
Solving Sequence
5,11 3,6
12 7 2 10 9 4 1 8
c
5
c
11
c
6
c
2
c
10
c
9
c
4
c
1
c
7
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h4.82508 × 10
67
u
41
− 8.04373 × 10
66
u
40
+ ··· + 3.17613 × 10
68
b + 7.49719 × 10
68
,
7.68981 × 10
68
u
41
+ 6.83382 × 10
67
u
40
+ ··· + 3.17613 × 10
68
a + 4.36130 × 10
69
, u
42
− 5u
40
+ ··· + 12u − 1i
I
u
2
= h2u
11
+ 2u
10
− 9u
9
− 9u
8
+ 17u
7
+ 11u
6
− 20u
5
− 5u
4
+ 14u
3
+ 5u
2
+ b − 3u,
3u
11
+ 4u
10
− 13u
9
− 19u
8
+ 23u
7
+ 30u
6
− 27u
5
− 26u
4
+ 21u
3
+ 22u
2
+ a − 5u − 6,
u
12
+ u
11
− 5u
10
− 5u
9
+ 11u
8
+ 8u
7
− 15u
6
− 6u
5
+ 13u
4
+ 4u
3
− 6u
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 54 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
=
h4.83×10
67
u
41
−8.04×10
66
u
40
+· · ·+3.18×10
68
b+7.50×10
68
, 7.69×10
68
u
41
+
6.83 × 10
67
u
40
+ · · · + 3.18 × 10
68
a + 4.36 × 10
69
, u
42
− 5u
40
+ · · · + 12u − 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−2.42113u
41
− 0.215162u
40
+ ··· + 47.7375u − 13.7315
−0.151917u
41
+ 0.0253256u
40
+ ··· + 4.43664u − 2.36048
a
6
=
1
u
2
a
12
=
1.40883u
41
− 0.283645u
40
+ ··· − 31.6002u + 11.3295
0.451792u
41
− 0.229337u
40
+ ··· + 0.897332u + 1.83430
a
7
=
−1.66145u
41
− 0.892224u
40
+ ··· + 72.0897u − 18.1638
0.208120u
41
− 0.395827u
40
+ ··· + 17.3792u − 4.31286
a
2
=
−2.26921u
41
− 0.240488u
40
+ ··· + 43.3008u − 11.3710
−0.151917u
41
+ 0.0253256u
40
+ ··· + 4.43664u − 2.36048
a
10
=
0.699121u
41
− 0.0570474u
40
+ ··· − 24.8728u + 7.43404
0.257913u
41
+ 0.00273987u
40
+ ··· − 5.62471u + 2.06112
a
9
=
0.957035u
41
− 0.0543075u
40
+ ··· − 30.4975u + 9.49516
0.257913u
41
+ 0.00273987u
40
+ ··· − 5.62471u + 2.06112
a
4
=
−2.53284u
41
− 0.147464u
40
+ ··· + 43.4617u − 11.5862
−0.325011u
41
+ 0.129277u
40
+ ··· + 3.51256u − 2.29278
a
1
=
−2.13070u
41
− 0.169775u
40
+ ··· + 38.9427u − 12.7081
−0.812419u
41
+ 0.186428u
40
+ ··· + 8.64257u − 3.43916
a
8
=
−0.374567u
41
+ 0.0574880u
40
+ ··· + 18.7871u + 0.721860
0.269350u
41
+ 0.303079u
40
+ ··· − 12.4553u + 2.13098
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0.117845u
41
− 0.341047u
40
+ ··· − 3.69676u + 5.23841
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
42
− 2u
41
+ ··· + 7155u + 1759
c
2
u
42
− 20u
40
+ ··· − 4u + 1
c
3
u
42
− 3u
40
+ ··· + 36u − 7
c
4
, c
8
, c
9
u
42
− u
41
+ ··· + 125u − 43
c
5
u
42
− 5u
40
+ ··· + 12u − 1
c
6
, c
7
, c
12
u
42
− 2u
41
+ ··· − 9u + 1
c
10
u
42
+ u
41
+ ··· + 17u + 1
c
11
u
42
− 2u
41
+ ··· − 260u − 29
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
42
+ 84y
41
+ ··· − 28548659y + 3094081
c
2
y
42
− 40y
41
+ ··· − 132y + 1
c
3
y
42
− 6y
41
+ ··· − 246y + 49
c
4
, c
8
, c
9
y
42
+ 5y
41
+ ··· + 3123y + 1849
c
5
y
42
− 10y
41
+ ··· − 102y + 1
c
6
, c
7
, c
12
y
42
− 56y
41
+ ··· − 185y + 1
c
10
y
42
+ 41y
41
+ ··· − 193y + 1
c
11
y
42
− 26y
41
+ ··· − 54028y + 841
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.826491 + 0.442043I
a = 0.44718 + 1.76659I
b = 0.70153 + 1.70890I
7.45710 + 5.86314I −2.65176 − 8.52020I
u = −0.826491 − 0.442043I
a = 0.44718 − 1.76659I
b = 0.70153 − 1.70890I
7.45710 − 5.86314I −2.65176 + 8.52020I
u = −0.922502
a = −1.78773
b = −0.148859
−1.70873 −6.53080
u = 0.242616 + 0.880583I
a = 0.16751 + 1.61694I
b = 0.219203 − 0.132487I
10.17820 − 3.58420I 3.84598 + 2.12866I
u = 0.242616 − 0.880583I
a = 0.16751 − 1.61694I
b = 0.219203 + 0.132487I
10.17820 + 3.58420I 3.84598 − 2.12866I
u = 0.890844 + 0.643047I
a = 0.563817 − 1.208620I
b = 1.35559 − 0.46608I
2.11218 − 2.53038I 6.12123 + 0.14610I
u = 0.890844 − 0.643047I
a = 0.563817 + 1.208620I
b = 1.35559 + 0.46608I
2.11218 + 2.53038I 6.12123 − 0.14610I
u = −0.876142
a = 0.172308
b = −1.12353
−2.38946 −2.90900
u = 0.721992 + 0.460484I
a = 0.164016 − 1.355880I
b = 0.239818 − 1.211100I
−0.02917 − 3.92711I −3.97147 + 9.92723I
u = 0.721992 − 0.460484I
a = 0.164016 + 1.355880I
b = 0.239818 + 1.211100I
−0.02917 + 3.92711I −3.97147 − 9.92723I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.720687 + 0.899817I
a = 0.273465 + 0.765302I
b = 1.53565 + 0.40748I
5.72444 + 3.42674I 2.80097 − 3.58941I
u = −0.720687 − 0.899817I
a = 0.273465 − 0.765302I
b = 1.53565 − 0.40748I
5.72444 − 3.42674I 2.80097 + 3.58941I
u = −0.780727 + 0.131252I
a = 0.82386 + 2.32713I
b = 1.26635 + 0.92629I
7.74135 + 4.15125I −0.41061 − 1.82266I
u = −0.780727 − 0.131252I
a = 0.82386 − 2.32713I
b = 1.26635 − 0.92629I
7.74135 − 4.15125I −0.41061 + 1.82266I
u = −0.615064 + 0.357683I
a = 0.070719 + 0.713992I
b = −0.245314 + 0.610196I
−1.089030 + 0.732853I −6.79767 − 2.20104I
u = −0.615064 − 0.357683I
a = 0.070719 − 0.713992I
b = −0.245314 − 0.610196I
−1.089030 − 0.732853I −6.79767 + 2.20104I
u = 1.30953
a = −0.929642
b = −0.351451
−6.86880 −18.9610
u = −1.083940 + 0.809437I
a = 0.675081 + 1.062750I
b = 1.284690 + 0.182998I
4.64441 + 2.89183I 1.76368 − 2.87517I
u = −1.083940 − 0.809437I
a = 0.675081 − 1.062750I
b = 1.284690 − 0.182998I
4.64441 − 2.89183I 1.76368 + 2.87517I
u = 0.804703 + 1.139140I
a = 0.374165 − 0.503272I
b = 1.67797 − 0.36893I
15.2391 − 4.3076I 2.80868 + 2.57557I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.804703 − 1.139140I
a = 0.374165 + 0.503272I
b = 1.67797 + 0.36893I
15.2391 + 4.3076I 2.80868 − 2.57557I
u = 1.411490 + 0.009834I
a = −0.221337 + 0.166065I
b = −0.899307 − 0.199819I
5.57535 + 0.53866I 0. + 2.21094I
u = 1.411490 − 0.009834I
a = −0.221337 − 0.166065I
b = −0.899307 + 0.199819I
5.57535 − 0.53866I 0. − 2.21094I
u = −0.003662 + 0.550660I
a = 1.13441 − 1.56878I
b = 0.175353 − 0.047964I
1.34218 + 1.37911I 2.89590 − 0.80863I
u = −0.003662 − 0.550660I
a = 1.13441 + 1.56878I
b = 0.175353 + 0.047964I
1.34218 − 1.37911I 2.89590 + 0.80863I
u = −0.90754 + 1.17188I
a = 0.055593 − 0.788021I
b = −1.155080 − 0.321541I
1.72011 + 4.36399I 0
u = −0.90754 − 1.17188I
a = 0.055593 + 0.788021I
b = −1.155080 + 0.321541I
1.72011 − 4.36399I 0
u = 1.10237 + 1.02421I
a = −0.255654 + 1.035640I
b = −1.41241 + 0.50973I
4.99599 − 9.83709I 0
u = 1.10237 − 1.02421I
a = −0.255654 − 1.035640I
b = −1.41241 − 0.50973I
4.99599 + 9.83709I 0
u = −1.53577
a = −0.547612
b = −0.588263
−4.41753 5.34160
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.02878 + 1.16133I
a = −0.276463 + 0.398264I
b = −1.234350 − 0.070649I
5.40084 + 1.95358I 0
u = 1.02878 − 1.16133I
a = −0.276463 − 0.398264I
b = −1.234350 + 0.070649I
5.40084 − 1.95358I 0
u = 1.24059 + 0.94227I
a = 0.677354 − 0.959216I
b = 1.305130 − 0.002881I
13.8696 − 3.3208I 0
u = 1.24059 − 0.94227I
a = 0.677354 + 0.959216I
b = 1.305130 + 0.002881I
13.8696 + 3.3208I 0
u = −0.81469 + 1.33843I
a = −0.401599 − 0.501693I
b = −1.42135 + 0.14198I
15.8064 − 4.9457I 0
u = −0.81469 − 1.33843I
a = −0.401599 + 0.501693I
b = −1.42135 − 0.14198I
15.8064 + 4.9457I 0
u = −1.25187 + 0.99541I
a = −0.477673 − 1.039360I
b = −1.61618 − 0.53409I
14.3267 + 13.2133I 0
u = −1.25187 − 0.99541I
a = −0.477673 + 1.039360I
b = −1.61618 + 0.53409I
14.3267 − 13.2133I 0
u = 0.332125 + 0.172454I
a = −0.91615 − 3.36326I
b = 0.882687 − 0.380208I
0.97861 − 1.95574I −3.65064 + 3.03665I
u = 0.332125 − 0.172454I
a = −0.91615 + 3.36326I
b = 0.882687 + 0.380208I
0.97861 + 1.95574I −3.65064 − 3.03665I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.373759
a = 1.40517
b = −1.39216
−3.18526 8.01790
u = 0.109443
a = −8.06908
b = −1.71570
3.71239 4.37580
9
II.
I
u
2
= h2u
11
+2u
10
+· · ·+b−3u, 3u
11
+4u
10
+· · ·+a−6, u
12
+u
11
+· · ·−u+1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−3u
11
− 4u
10
+ ··· + 5u + 6
−2u
11
− 2u
10
+ ··· − 5u
2
+ 3u
a
6
=
1
u
2
a
12
=
−3u
11
− 5u
10
+ ··· + 9u + 5
2u
11
+ 2u
10
+ ··· − u − 3
a
7
=
−4u
11
− 6u
10
+ ··· + 4u + 11
−3u
11
− 4u
10
+ ··· − 14u
2
+ 3
a
2
=
−u
11
− 2u
10
+ ··· + 2u + 6
−2u
11
− 2u
10
+ ··· − 5u
2
+ 3u
a
10
=
−5u
11
− 6u
10
+ ··· + 14u + 7
−u
10
− u
9
+ 4u
8
+ 4u
7
− 7u
6
− 4u
5
+ 8u
4
+ 2u
3
− 5u
2
− 2u + 1
a
9
=
−5u
11
− 7u
10
+ ··· + 12u + 8
−u
10
− u
9
+ 4u
8
+ 4u
7
− 7u
6
− 4u
5
+ 8u
4
+ 2u
3
− 5u
2
− 2u + 1
a
4
=
−2u
11
− 3u
10
+ ··· + 4u + 7
−u
11
− u
10
+ 5u
9
+ 5u
8
− 10u
7
− 7u
6
+ 12u
5
+ 3u
4
− 9u
3
− 3u
2
+ 2u
a
1
=
4u
11
+ 7u
10
+ ··· − 8u − 5
−u
11
+ 6u
9
+ 2u
8
− 12u
7
− 2u
6
+ 11u
5
− 2u
4
− 7u
3
− 4u
2
+ u + 1
a
8
=
−3u
11
− 4u
10
+ ··· + 6u + 5
u
11
+ 2u
10
− 3u
9
− 8u
8
+ 3u
7
+ 12u
6
− 3u
5
− 12u
4
+ u
3
+ 9u
2
− 2
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −22u
11
−25u
10
+99u
9
+115u
8
−193u
7
−165u
6
+241u
5
+111u
4
−185u
3
−73u
2
+54u+6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
+ u
11
+ ··· + 4u + 1
c
2
u
12
+ u
11
+ ··· − 7u − 1
c
3
u
12
− 3u
11
+ ··· + u − 1
c
4
u
12
− 4u
10
− u
9
+ u
8
+ u
7
+ 5u
6
+ 4u
5
+ u
4
− 2u
3
− 2u
2
− 2u − 1
c
5
u
12
+ u
11
+ ··· − u + 1
c
6
, c
7
u
12
+ u
11
+ ··· − 2u − 1
c
8
, c
9
u
12
− 4u
10
+ u
9
+ u
8
− u
7
+ 5u
6
− 4u
5
+ u
4
+ 2u
3
− 2u
2
+ 2u − 1
c
10
u
12
+ 2u
11
+ 2u
10
+ 2u
9
− u
8
− 4u
7
− 5u
6
− u
5
− u
4
+ u
3
+ 4u
2
− 1
c
11
u
12
− 3u
11
+ ··· − 11u + 1
c
12
u
12
− u
11
+ ··· + 2u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
+ 7y
11
+ ··· − 66y + 1
c
2
y
12
− 13y
11
+ ··· − 75y + 1
c
3
y
12
− 11y
11
+ ··· + 3y + 1
c
4
, c
8
, c
9
y
12
− 8y
11
+ 18y
10
+ y
9
− 35y
8
+ 5y
7
+ 29y
6
− 2y
5
− y
4
− 2y
3
− 6y
2
+ 1
c
5
y
12
− 11y
11
+ ··· − 13y + 1
c
6
, c
7
, c
12
y
12
− 17y
11
+ ··· − 16y + 1
c
10
y
12
− 6y
10
− 2y
9
− y
8
− 2y
7
+ 29y
6
+ 5y
5
− 35y
4
+ y
3
+ 18y
2
− 8y + 1
c
11
y
12
− 11y
11
+ ··· − 47y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.758819 + 0.689182I
a = 0.17020 − 1.46697I
b = 1.078870 − 0.519683I
1.60765 − 3.05777I −1.33050 + 7.30606I
u = 0.758819 − 0.689182I
a = 0.17020 + 1.46697I
b = 1.078870 + 0.519683I
1.60765 + 3.05777I −1.33050 − 7.30606I
u = −0.771425 + 0.444552I
a = 0.78709 + 2.28125I
b = 1.15458 + 1.18981I
8.30950 + 5.26934I 4.00276 − 5.66957I
u = −0.771425 − 0.444552I
a = 0.78709 − 2.28125I
b = 1.15458 − 1.18981I
8.30950 − 5.26934I 4.00276 + 5.66957I
u = 1.29947
a = −1.51655
b = −1.10874
0.486710 −0.332780
u = −1.33762
a = −1.04653
b = −0.605429
−6.53216 5.49390
u = 0.656374
a = −0.765708
b = −2.08682
3.11330 −10.0660
u = −0.603038
a = 0.280338
b = −1.46953
−3.50332 −18.5260
u = 1.43791
a = −0.393742
b = 0.122601
−4.98629 −9.47130
u = 0.481939
a = 2.60883
b = −0.696593
−0.845589 3.83140
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.45491 + 0.63338I
a = 0.459395 + 0.118062I
b = 1.188800 − 0.172635I
6.08613 − 1.26753I 8.36359 + 3.51289I
u = −1.45491 − 0.63338I
a = 0.459395 − 0.118062I
b = 1.188800 + 0.172635I
6.08613 + 1.26753I 8.36359 − 3.51289I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
12
+ u
11
+ ··· + 4u + 1)(u
42
− 2u
41
+ ··· + 7155u + 1759)
c
2
(u
12
+ u
11
+ ··· − 7u − 1)(u
42
− 20u
40
+ ··· − 4u + 1)
c
3
(u
12
− 3u
11
+ ··· + u − 1)(u
42
− 3u
40
+ ··· + 36u − 7)
c
4
(u
12
− 4u
10
− u
9
+ u
8
+ u
7
+ 5u
6
+ 4u
5
+ u
4
− 2u
3
− 2u
2
− 2u − 1)
· (u
42
− u
41
+ ··· + 125u − 43)
c
5
(u
12
+ u
11
+ ··· − u + 1)(u
42
− 5u
40
+ ··· + 12u − 1)
c
6
, c
7
(u
12
+ u
11
+ ··· − 2u − 1)(u
42
− 2u
41
+ ··· − 9u + 1)
c
8
, c
9
(u
12
− 4u
10
+ u
9
+ u
8
− u
7
+ 5u
6
− 4u
5
+ u
4
+ 2u
3
− 2u
2
+ 2u − 1)
· (u
42
− u
41
+ ··· + 125u − 43)
c
10
(u
12
+ 2u
11
+ 2u
10
+ 2u
9
− u
8
− 4u
7
− 5u
6
− u
5
− u
4
+ u
3
+ 4u
2
− 1)
· (u
42
+ u
41
+ ··· + 17u + 1)
c
11
(u
12
− 3u
11
+ ··· − 11u + 1)(u
42
− 2u
41
+ ··· − 260u − 29)
c
12
(u
12
− u
11
+ ··· + 2u − 1)(u
42
− 2u
41
+ ··· − 9u + 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
12
+ 7y
11
+ ··· − 66y + 1)
· (y
42
+ 84y
41
+ ··· − 28548659y + 3094081)
c
2
(y
12
− 13y
11
+ ··· − 75y + 1)(y
42
− 40y
41
+ ··· − 132y + 1)
c
3
(y
12
− 11y
11
+ ··· + 3y + 1)(y
42
− 6y
41
+ ··· − 246y + 49)
c
4
, c
8
, c
9
(y
12
− 8y
11
+ 18y
10
+ y
9
− 35y
8
+ 5y
7
+ 29y
6
− 2y
5
− y
4
− 2y
3
− 6y
2
+ 1)
· (y
42
+ 5y
41
+ ··· + 3123y + 1849)
c
5
(y
12
− 11y
11
+ ··· − 13y + 1)(y
42
− 10y
41
+ ··· − 102y + 1)
c
6
, c
7
, c
12
(y
12
− 17y
11
+ ··· − 16y + 1)(y
42
− 56y
41
+ ··· − 185y + 1)
c
10
(y
12
− 6y
10
− 2y
9
− y
8
− 2y
7
+ 29y
6
+ 5y
5
− 35y
4
+ y
3
+ 18y
2
− 8y + 1)
· (y
42
+ 41y
41
+ ··· − 193y + 1)
c
11
(y
12
− 11y
11
+ ··· − 47y + 1)(y
42
− 26y
41
+ ··· − 54028y + 841)
16
