12n
0685
(K12n
0685
)
A knot diagram
1
Linearized knot diagam
4 5 6 8 10 1 12 2 5 6 3 7
Solving Sequence
1,6 4,7
3 12 8 11 10 5 2 9
c
6
c
3
c
12
c
7
c
11
c
10
c
5
c
2
c
8
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−2.68320 × 10
40
u
53
+ 4.36845 × 10
39
u
52
+ ··· + 1.09407 × 10
40
b − 1.26785 × 10
41
,
1.64738 × 10
41
u
53
− 5.57914 × 10
40
u
52
+ ··· + 1.09407 × 10
40
a + 4.47020 × 10
41
, u
54
+ 27u
52
+ ··· + 12u + 1i
I
u
2
= hu
12
+ u
11
+ 6u
10
+ 5u
9
+ 13u
8
+ 9u
7
+ 10u
6
+ 5u
5
− 3u
4
− 2u
3
− 6u
2
+ b − 2u,
− u
12
− 6u
10
− 13u
8
− 9u
6
+ 6u
4
− u
3
+ 9u
2
+ a − 2u,
u
15
+ u
14
+ 8u
13
+ 7u
12
+ 25u
11
+ 19u
10
+ 36u
9
+ 23u
8
+ 17u
7
+ 8u
6
− 12u
5
− 6u
4
− 12u
3
− 3u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 69 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
= h−2.68 × 10
40
u
53
+ 4.37 × 10
39
u
52
+ · · · + 1.09 × 10
40
b − 1.27 ×
10
41
, 1.65 × 10
41
u
53
− 5.58 × 10
40
u
52
+ · · · + 1.09 × 10
40
a + 4.47 ×
10
41
, u
54
+ 27u
52
+ · · · + 12u + 1i
(i) Arc colorings
a
1
=
0
u
a
6
=
1
0
a
4
=
−15.0573u
53
+ 5.09943u
52
+ ··· − 402.388u − 40.8584
2.45249u
53
− 0.399285u
52
+ ··· + 88.5992u + 11.5884
a
7
=
1
−u
2
a
3
=
−12.6048u
53
+ 4.70015u
52
+ ··· − 313.789u − 29.2701
2.45249u
53
− 0.399285u
52
+ ··· + 88.5992u + 11.5884
a
12
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
11
=
0.314507u
53
− 1.02928u
52
+ ··· + 9.85086u + 7.65563
3.56268u
53
− 1.46466u
52
+ ··· + 94.5090u + 11.9366
a
10
=
−3.24818u
53
+ 0.435380u
52
+ ··· − 84.6581u − 4.28102
3.56268u
53
− 1.46466u
52
+ ··· + 94.5090u + 11.9366
a
5
=
−15.6451u
53
+ 5.62013u
52
+ ··· − 401.641u − 38.9539
2.72593u
53
− 0.398787u
52
+ ··· + 92.7573u + 11.9546
a
2
=
−13.4990u
53
+ 5.55580u
52
+ ··· − 340.986u − 35.3036
5.19761u
53
− 1.33305u
52
+ ··· + 151.350u + 17.6227
a
9
=
0.950637u
53
+ 0.741154u
52
+ ··· + 8.38831u + 2.44017
−3.60099u
53
+ 1.54873u
52
+ ··· − 107.000u − 11.4663
(ii) Obstruction class = −1
(iii) Cusp Shapes = −8.59125u
53
+ 0.873911u
52
+ ··· − 257.258u − 36.1424
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
54
− u
53
+ ··· − 23705u − 2291
c
2
u
54
+ 4u
53
+ ··· + 28u − 1
c
3
u
54
− u
53
+ ··· − 2159u + 239
c
4
u
54
− 6u
52
+ ··· + 22u + 1
c
5
, c
9
, c
10
u
54
+ u
53
+ ··· + 11u − 1
c
6
, c
7
, c
12
u
54
+ 27u
52
+ ··· + 12u + 1
c
8
u
54
+ u
53
+ ··· + u + 1
c
11
u
54
− 2u
53
+ ··· + 36164u − 1709
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
54
− 35y
53
+ ··· − 169277117y + 5248681
c
2
y
54
+ 54y
53
+ ··· + 60y + 1
c
3
y
54
− 53y
53
+ ··· + 93385y + 57121
c
4
y
54
− 12y
53
+ ··· − 440y + 1
c
5
, c
9
, c
10
y
54
− y
53
+ ··· − 45y + 1
c
6
, c
7
, c
12
y
54
+ 54y
53
+ ··· − 74y + 1
c
8
y
54
− 49y
53
+ ··· − 151y + 1
c
11
y
54
+ 62y
53
+ ··· − 555348524y + 2920681
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.727247 + 0.655546I
a = −1.120070 − 0.569467I
b = 1.45756 − 0.28228I
−7.16637 + 5.36550I 3.29115 − 2.38015I
u = −0.727247 − 0.655546I
a = −1.120070 + 0.569467I
b = 1.45756 + 0.28228I
−7.16637 − 5.36550I 3.29115 + 2.38015I
u = −0.817874 + 0.466543I
a = −1.01459 − 1.44161I
b = 1.54231 + 0.48478I
−6.57271 − 10.55460I 4.62377 + 7.07679I
u = −0.817874 − 0.466543I
a = −1.01459 + 1.44161I
b = 1.54231 − 0.48478I
−6.57271 + 10.55460I 4.62377 − 7.07679I
u = 0.658331 + 0.610282I
a = 0.991278 − 0.834575I
b = −1.55170 + 0.23948I
−7.17018 + 2.76260I 2.77811 − 3.18993I
u = 0.658331 − 0.610282I
a = 0.991278 + 0.834575I
b = −1.55170 − 0.23948I
−7.17018 − 2.76260I 2.77811 + 3.18993I
u = 0.782295 + 0.434217I
a = 1.34381 − 1.23021I
b = −1.407260 + 0.014903I
−6.54490 + 2.06360I 3.73772 − 2.56730I
u = 0.782295 − 0.434217I
a = 1.34381 + 1.23021I
b = −1.407260 − 0.014903I
−6.54490 − 2.06360I 3.73772 + 2.56730I
u = 0.873178
a = −0.654136
b = 0.754805
1.61182 1.21940
u = 0.600533 + 0.615475I
a = 0.088416 + 1.268790I
b = 0.943653 − 0.397797I
−0.31286 + 3.64145I 11.2572 − 9.4738I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.600533 − 0.615475I
a = 0.088416 − 1.268790I
b = 0.943653 + 0.397797I
−0.31286 − 3.64145I 11.2572 + 9.4738I
u = 0.110651 + 1.211630I
a = 0.649484 + 0.213764I
b = 0.742245 + 0.814719I
−2.38520 + 2.82306I 0
u = 0.110651 − 1.211630I
a = 0.649484 − 0.213764I
b = 0.742245 − 0.814719I
−2.38520 − 2.82306I 0
u = −0.142943 + 1.223340I
a = 0.983994 − 0.047102I
b = −0.904696 + 0.730444I
−4.86587 + 0.83794I 0
u = −0.142943 − 1.223340I
a = 0.983994 + 0.047102I
b = −0.904696 − 0.730444I
−4.86587 − 0.83794I 0
u = −0.722991
a = −1.44649
b = 0.416019
5.96932 18.1570
u = −0.591356 + 0.394727I
a = 1.15354 + 1.47575I
b = −1.372070 − 0.284950I
−2.42244 − 1.86344I 0.32896 + 2.97816I
u = −0.591356 − 0.394727I
a = 1.15354 − 1.47575I
b = −1.372070 + 0.284950I
−2.42244 + 1.86344I 0.32896 − 2.97816I
u = −0.284927 + 1.273200I
a = −0.610811 − 0.705027I
b = 0.378777 + 0.245213I
2.02546 − 3.64647I 0
u = −0.284927 − 1.273200I
a = −0.610811 + 0.705027I
b = 0.378777 − 0.245213I
2.02546 + 3.64647I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.425694 + 1.274300I
a = −0.137225 + 0.299114I
b = 0.804888 + 0.001080I
−2.34924 + 4.63206I 0
u = 0.425694 − 1.274300I
a = −0.137225 − 0.299114I
b = 0.804888 − 0.001080I
−2.34924 − 4.63206I 0
u = 0.079610 + 1.366620I
a = 0.059053 − 0.444591I
b = −0.069313 + 0.938702I
−3.77402 + 1.92762I 0
u = 0.079610 − 1.366620I
a = 0.059053 + 0.444591I
b = −0.069313 − 0.938702I
−3.77402 − 1.92762I 0
u = 0.630083
a = −0.310787
b = 0.798544
1.05920 8.70240
u = −0.027362 + 1.385500I
a = 0.562065 + 0.539030I
b = 1.62034 − 0.54932I
−1.87045 − 0.53073I 0
u = −0.027362 − 1.385500I
a = 0.562065 − 0.539030I
b = 1.62034 + 0.54932I
−1.87045 + 0.53073I 0
u = −0.057418 + 1.395130I
a = −1.54371 + 0.64324I
b = −1.136840 − 0.424786I
−5.83103 − 3.79502I 0
u = −0.057418 − 1.395130I
a = −1.54371 − 0.64324I
b = −1.136840 + 0.424786I
−5.83103 + 3.79502I 0
u = −0.16335 + 1.41281I
a = −0.556055 + 0.832668I
b = −0.72716 − 1.67965I
−5.98732 − 6.70530I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.16335 − 1.41281I
a = −0.556055 − 0.832668I
b = −0.72716 + 1.67965I
−5.98732 + 6.70530I 0
u = −0.493438 + 0.257877I
a = 0.35961 + 2.32754I
b = −0.373419 − 1.315190I
−0.60246 − 4.33501I 8.99729 + 10.44789I
u = −0.493438 − 0.257877I
a = 0.35961 − 2.32754I
b = −0.373419 + 1.315190I
−0.60246 + 4.33501I 8.99729 − 10.44789I
u = 0.509006 + 0.161109I
a = −0.289495 − 0.461991I
b = 0.494322 + 0.360163I
0.947117 + 0.129909I 11.27686 − 2.22644I
u = 0.509006 − 0.161109I
a = −0.289495 + 0.461991I
b = 0.494322 − 0.360163I
0.947117 − 0.129909I 11.27686 + 2.22644I
u = −0.22783 + 1.45064I
a = −0.329271 + 1.292780I
b = −1.72420 − 0.54667I
−8.34733 − 4.89875I 0
u = −0.22783 − 1.45064I
a = −0.329271 − 1.292780I
b = −1.72420 + 0.54667I
−8.34733 + 4.89875I 0
u = 0.07299 + 1.48329I
a = 0.151391 + 1.225570I
b = −0.143907 − 0.422731I
−8.28129 + 3.22679I 0
u = 0.07299 − 1.48329I
a = 0.151391 − 1.225570I
b = −0.143907 + 0.422731I
−8.28129 − 3.22679I 0
u = 0.069990 + 0.509753I
a = 1.19972 + 2.08736I
b = −0.070424 + 0.287433I
−1.81357 + 2.38364I 0.95087 − 1.41179I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.069990 − 0.509753I
a = 1.19972 − 2.08736I
b = −0.070424 − 0.287433I
−1.81357 − 2.38364I 0.95087 + 1.41179I
u = 0.29461 + 1.50032I
a = −0.017831 − 1.339470I
b = −1.42401 + 0.25866I
−12.8030 + 6.0112I 0
u = 0.29461 − 1.50032I
a = −0.017831 + 1.339470I
b = −1.42401 − 0.25866I
−12.8030 − 6.0112I 0
u = −0.30047 + 1.51505I
a = 0.318429 − 1.306700I
b = 1.68707 + 0.60140I
−12.9927 − 14.6415I 0
u = −0.30047 − 1.51505I
a = 0.318429 + 1.306700I
b = 1.68707 − 0.60140I
−12.9927 + 14.6415I 0
u = 0.20551 + 1.53545I
a = −0.261245 − 0.741640I
b = −1.86481 + 0.32365I
−14.2076 + 5.8827I 0
u = 0.20551 − 1.53545I
a = −0.261245 + 0.741640I
b = −1.86481 − 0.32365I
−14.2076 − 5.8827I 0
u = 0.22302 + 1.54513I
a = 0.600789 + 0.941334I
b = 1.199140 − 0.475695I
−7.40823 + 6.78161I 0
u = 0.22302 − 1.54513I
a = 0.600789 − 0.941334I
b = 1.199140 + 0.475695I
−7.40823 − 6.78161I 0
u = −0.20084 + 1.57551I
a = 0.122450 − 0.759219I
b = 1.53708 − 0.00469I
−14.6210 + 2.0031I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.20084 − 1.57551I
a = 0.122450 + 0.759219I
b = 1.53708 + 0.00469I
−14.6210 − 2.0031I 0
u = −0.320973 + 0.010257I
a = 1.47897 + 4.46204I
b = −0.824971 − 0.660602I
−1.13474 − 2.72736I 9.52970 + 1.05898I
u = −0.320973 − 0.010257I
a = 1.47897 − 4.46204I
b = −0.824971 + 0.660602I
−1.13474 + 2.72736I 9.52970 − 1.05898I
u = −0.132701
a = 4.04601
b = 1.40543
2.79911 −5.45500
10
II.
I
u
2
= hu
12
+u
11
+· · ·+b−2u, −u
12
−6u
10
+· · ·+a−2u, u
15
+u
14
+· · ·−3u
2
+1i
(i) Arc colorings
a
1
=
0
u
a
6
=
1
0
a
4
=
u
12
+ 6u
10
+ 13u
8
+ 9u
6
− 6u
4
+ u
3
− 9u
2
+ 2u
−u
12
− u
11
+ ··· + 6u
2
+ 2u
a
7
=
1
−u
2
a
3
=
−u
11
− 5u
9
− 9u
7
− u
6
− 5u
5
− 3u
4
+ 3u
3
− 3u
2
+ 4u
−u
12
− u
11
+ ··· + 6u
2
+ 2u
a
12
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
11
=
−u
13
− 2u
12
+ ··· + u − 3
−u
13
− u
12
+ ··· − 2u − 1
a
10
=
−u
12
− u
11
+ ··· + 3u − 2
−u
13
− u
12
+ ··· − 2u − 1
a
5
=
−u
11
− 5u
9
− 9u
7
− 2u
6
− 6u
5
− 6u
4
+ u
3
− 5u
2
+ 3u + 1
u
14
+ u
13
+ ··· + 4u
2
+ 2u
a
2
=
−2u
14
− 3u
13
+ ··· + 5u − 3
2u
13
+ u
12
+ ··· + u
2
+ 1
a
9
=
u
13
− u
12
+ ··· + 6u − 2
−2u
13
− 2u
12
+ ··· − u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7u
13
− 6u
12
− 49u
11
− 38u
10
− 130u
9
− 90u
8
− 150u
7
− 87u
6
−
39u
5
− 10u
4
+ 58u
3
+ 27u
2
+ 31u + 8
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
− 12u
14
+ ··· + 15u − 1
c
2
u
15
+ 3u
14
+ ··· + 4u
2
− 1
c
3
u
15
− 6u
13
+ ··· + 3u − 1
c
4
u
15
− u
14
+ ··· − 6u
2
+ 1
c
5
u
15
− 4u
13
+ ··· + u − 1
c
6
, c
7
u
15
+ u
14
+ ··· − 3u
2
+ 1
c
8
u
15
− 6u
13
+ ··· − u − 1
c
9
, c
10
u
15
− 4u
13
+ ··· + u + 1
c
11
u
15
− u
14
+ ··· + 4u
2
− 1
c
12
u
15
− u
14
+ ··· + 3u
2
− 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
− 2y
14
+ ··· + 49y −1
c
2
y
15
+ 7y
14
+ ··· + 8y −1
c
3
y
15
− 12y
14
+ ··· + 11y −1
c
4
y
15
− 15y
14
+ ··· + 12y −1
c
5
, c
9
, c
10
y
15
− 8y
14
+ ··· − 3y −1
c
6
, c
7
, c
12
y
15
+ 15y
14
+ ··· + 6y −1
c
8
y
15
− 12y
14
+ ··· + 15y −1
c
11
y
15
+ 3y
14
+ ··· + 8y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.917074
a = 0.317754
b = −0.706611
2.01181 23.5320
u = −0.074215 + 1.225200I
a = 1.37536 + 0.51706I
b = −0.477263 + 0.644746I
−4.31514 + 2.06106I 2.80971 − 3.49368I
u = −0.074215 − 1.225200I
a = 1.37536 − 0.51706I
b = −0.477263 − 0.644746I
−4.31514 − 2.06106I 2.80971 + 3.49368I
u = 0.738441
a = −1.88795
b = 0.822162
5.31354 5.05140
u = −0.391682 + 1.206530I
a = −0.448299 + 0.314891I
b = −0.633852 − 0.204044I
−1.63448 − 4.71343I 9.81376 + 7.25544I
u = −0.391682 − 1.206530I
a = −0.448299 − 0.314891I
b = −0.633852 + 0.204044I
−1.63448 + 4.71343I 9.81376 − 7.25544I
u = 0.310878 + 1.284290I
a = −0.579681 + 0.974594I
b = 0.784151 − 0.173085I
1.30709 + 3.78442I 1.00032 − 4.22940I
u = 0.310878 − 1.284290I
a = −0.579681 − 0.974594I
b = 0.784151 + 0.173085I
1.30709 − 3.78442I 1.00032 + 4.22940I
u = 0.117562 + 1.341930I
a = 0.493543 + 0.042097I
b = 1.57940 + 0.48851I
−1.30643 + 1.67719I 7.37794 − 3.24911I
u = 0.117562 − 1.341930I
a = 0.493543 − 0.042097I
b = 1.57940 − 0.48851I
−1.30643 − 1.67719I 7.37794 + 3.24911I
14
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.19173 + 1.46876I
a = −0.602052 + 1.218750I
b = −1.25573 − 0.76427I
−7.60875 − 5.65349I 0.73377 + 5.33315I
u = −0.19173 − 1.46876I
a = −0.602052 − 1.218750I
b = −1.25573 + 0.76427I
−7.60875 + 5.65349I 0.73377 − 5.33315I
u = −0.356463 + 0.351194I
a = −0.23894 + 3.19582I
b = −0.810143 − 0.625914I
−1.51467 − 3.34740I 2.47459 + 9.57162I
u = −0.356463 − 0.351194I
a = −0.23894 − 3.19582I
b = −0.810143 + 0.625914I
−1.51467 + 3.34740I 2.47459 − 9.57162I
u = 0.349922
a = −0.429663
b = 1.51132
3.08020 23.9960
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
15
− 12u
14
+ ··· + 15u − 1)(u
54
− u
53
+ ··· − 23705u − 2291)
c
2
(u
15
+ 3u
14
+ ··· + 4u
2
− 1)(u
54
+ 4u
53
+ ··· + 28u − 1)
c
3
(u
15
− 6u
13
+ ··· + 3u − 1)(u
54
− u
53
+ ··· − 2159u + 239)
c
4
(u
15
− u
14
+ ··· − 6u
2
+ 1)(u
54
− 6u
52
+ ··· + 22u + 1)
c
5
(u
15
− 4u
13
+ ··· + u − 1)(u
54
+ u
53
+ ··· + 11u − 1)
c
6
, c
7
(u
15
+ u
14
+ ··· − 3u
2
+ 1)(u
54
+ 27u
52
+ ··· + 12u + 1)
c
8
(u
15
− 6u
13
+ ··· − u − 1)(u
54
+ u
53
+ ··· + u + 1)
c
9
, c
10
(u
15
− 4u
13
+ ··· + u + 1)(u
54
+ u
53
+ ··· + 11u − 1)
c
11
(u
15
− u
14
+ ··· + 4u
2
− 1)(u
54
− 2u
53
+ ··· + 36164u − 1709)
c
12
(u
15
− u
14
+ ··· + 3u
2
− 1)(u
54
+ 27u
52
+ ··· + 12u + 1)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
15
− 2y
14
+ ··· + 49y −1)
· (y
54
− 35y
53
+ ··· − 169277117y + 5248681)
c
2
(y
15
+ 7y
14
+ ··· + 8y −1)(y
54
+ 54y
53
+ ··· + 60y + 1)
c
3
(y
15
− 12y
14
+ ··· + 11y −1)(y
54
− 53y
53
+ ··· + 93385y + 57121)
c
4
(y
15
− 15y
14
+ ··· + 12y −1)(y
54
− 12y
53
+ ··· − 440y + 1)
c
5
, c
9
, c
10
(y
15
− 8y
14
+ ··· − 3y −1)(y
54
− y
53
+ ··· − 45y + 1)
c
6
, c
7
, c
12
(y
15
+ 15y
14
+ ··· + 6y −1)(y
54
+ 54y
53
+ ··· − 74y + 1)
c
8
(y
15
− 12y
14
+ ··· + 15y −1)(y
54
− 49y
53
+ ··· − 151y + 1)
c
11
(y
15
+ 3y
14
+ ··· + 8y −1)
· (y
54
+ 62y
53
+ ··· − 555348524y + 2920681)
17
