12n
0641
(K12n
0641
)
A knot diagram
1
Linearized knot diagam
3 8 10 8 11 12 2 4 1 7 6 4
Solving Sequence
2,8 3,10
4 5 1 7 11 9 12 6
c
2
c
3
c
4
c
1
c
7
c
10
c
9
c
12
c
6
c
5
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h8.49827 × 10
54
u
47
+ 2.37919 × 10
54
u
46
+ ··· + 1.70744 × 10
55
b + 8.16269 × 10
54
,
2.71230 × 10
55
u
47
+ 6.45531 × 10
54
u
46
+ ··· + 1.70744 × 10
55
a − 5.14701 × 10
56
, u
48
− u
47
+ ··· + 55u − 1i
I
u
2
= h−u
14
− 2u
13
+ 3u
12
+ 8u
11
− 6u
10
− 21u
9
+ 5u
8
+ 33u
7
− u
6
− 36u
5
− 2u
4
+ 22u
3
+ u
2
+ b − 8u,
− 7u
14
− 6u
13
+ ··· + a + 9,
u
15
+ u
14
− 4u
13
− 5u
12
+ 10u
11
+ 14u
10
− 15u
9
− 25u
8
+ 16u
7
+ 28u
6
− 11u
5
− 19u
4
+ 5u
3
+ 7u
2
− u − 1i
I
u
3
= hb + 1, a + 1, u + 1i
* 3 irreducible components of dim
C
= 0, with total 64 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
= h8.50 × 10
54
u
47
+ 2.38 × 10
54
u
46
+ · · · + 1.71 × 10
55
b + 8.16 × 10
54
, 2.71 ×
10
55
u
47
+6.46×10
54
u
46
+· · ·+1.71×10
55
a−5.15×10
56
, u
48
−u
47
+· · ·+55u−1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
10
=
−1.58851u
47
− 0.378069u
46
+ ··· + 195.933u + 30.1446
−0.497719u
47
− 0.139342u
46
+ ··· + 94.5690u − 0.478065
a
4
=
−2.00213u
47
+ 1.12164u
46
+ ··· + 231.824u − 45.9301
0.532079u
47
− 0.731391u
46
+ ··· + 30.3920u − 2.14822
a
5
=
−2.00213u
47
+ 1.12164u
46
+ ··· + 231.824u − 45.9301
0.836676u
47
− 0.793694u
46
+ ··· − 16.0332u − 1.26772
a
1
=
−u
2
+ 1
−u
4
a
7
=
u
u
a
11
=
−0.852685u
47
− 0.569587u
46
+ ··· + 123.900u + 31.4741
0.238109u
47
− 0.330861u
46
+ ··· + 22.5362u + 0.851455
a
9
=
−1.07589u
47
− 0.438793u
46
+ ··· + 131.390u + 30.0699
−0.529673u
47
− 0.199681u
46
+ ··· + 108.525u − 0.735248
a
12
=
−0.634463u
47
− 0.0674825u
46
+ ··· − 11.2930u + 67.9453
−0.184246u
47
+ 0.409138u
46
+ ··· − 14.0906u + 2.82587
a
6
=
−0.0720669u
47
+ 0.836777u
46
+ ··· + 39.2316u − 83.6537
0.845245u
47
− 0.416255u
46
+ ··· − 32.8739u − 2.52326
(ii) Obstruction class = −1
(iii) Cusp Shapes = 5.23094u
47
− 3.23434u
46
+ ··· − 404.902u + 24.1944
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
48
+ 15u
47
+ ··· + 2837u + 1
c
2
, c
7
u
48
+ u
47
+ ··· − 55u − 1
c
3
u
48
+ u
47
+ ··· − 14u + 1
c
4
, c
8
u
48
− 2u
47
+ ··· + 14560u + 15853
c
5
, c
6
, c
11
u
48
+ 2u
47
+ ··· − 12u − 1
c
9
u
48
+ u
47
+ ··· + 719u − 293
c
10
u
48
− 3u
47
+ ··· + 10344u + 649
c
12
u
48
+ 4u
47
+ ··· − 10508u − 1369
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
48
+ 53y
47
+ ··· − 7937733y + 1
c
2
, c
7
y
48
− 15y
47
+ ··· − 2837y + 1
c
3
y
48
+ 5y
47
+ ··· − 56y + 1
c
4
, c
8
y
48
− 64y
47
+ ··· − 8556537210y + 251317609
c
5
, c
6
, c
11
y
48
− 50y
47
+ ··· − 66y + 1
c
9
y
48
− 61y
47
+ ··· − 3226039y + 85849
c
10
y
48
− 39y
47
+ ··· − 45579572y + 421201
c
12
y
48
− 60y
47
+ ··· − 53809914y + 1874161
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.780489 + 0.693178I
a = 0.705695 − 0.209382I
b = 0.862042 + 0.604842I
−0.43674 − 2.43309I 6.83450 + 4.39555I
u = 0.780489 − 0.693178I
a = 0.705695 + 0.209382I
b = 0.862042 − 0.604842I
−0.43674 + 2.43309I 6.83450 − 4.39555I
u = 0.529535 + 0.795568I
a = 1.41726 + 0.21633I
b = 0.20894 + 1.50824I
7.15738 + 1.35453I 12.13081 − 1.57989I
u = 0.529535 − 0.795568I
a = 1.41726 − 0.21633I
b = 0.20894 − 1.50824I
7.15738 − 1.35453I 12.13081 + 1.57989I
u = −1.052950 + 0.007832I
a = −0.707699 + 0.058294I
b = −0.965617 + 0.004183I
1.65094 + 0.00093I 6.00000 + 0.11555I
u = −1.052950 − 0.007832I
a = −0.707699 − 0.058294I
b = −0.965617 − 0.004183I
1.65094 − 0.00093I 6.00000 − 0.11555I
u = 1.05847
a = 1.90687
b = 3.40742
8.27045 10.2350
u = 0.912791 + 0.183084I
a = −1.080850 − 0.514657I
b = −1.221840 + 0.446864I
−0.81244 − 3.97179I 3.30801 + 3.48283I
u = 0.912791 − 0.183084I
a = −1.080850 + 0.514657I
b = −1.221840 − 0.446864I
−0.81244 + 3.97179I 3.30801 − 3.48283I
u = 0.918294 + 0.651222I
a = −0.957067 − 0.032883I
b = −0.745662 − 0.216607I
−1.09728 − 2.73985I 8.74885 + 1.99555I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.918294 − 0.651222I
a = −0.957067 + 0.032883I
b = −0.745662 + 0.216607I
−1.09728 + 2.73985I 8.74885 − 1.99555I
u = −1.064190 + 0.512825I
a = 1.060480 − 0.792956I
b = 0.694905 − 1.218350I
−0.91409 + 4.62733I 6.00000 − 9.84335I
u = −1.064190 − 0.512825I
a = 1.060480 + 0.792956I
b = 0.694905 + 1.218350I
−0.91409 − 4.62733I 6.00000 + 9.84335I
u = −0.798516 + 0.871376I
a = −1.77948 + 0.85153I
b = 0.462172 + 1.321260I
14.6213 + 0.5837I 11.01767 + 0.I
u = −0.798516 − 0.871376I
a = −1.77948 − 0.85153I
b = 0.462172 − 1.321260I
14.6213 − 0.5837I 11.01767 + 0.I
u = 0.846297 + 0.834975I
a = −0.598082 − 0.641634I
b = −0.31678 − 1.81032I
7.53406 − 1.71339I 6.00000 + 0.I
u = 0.846297 − 0.834975I
a = −0.598082 + 0.641634I
b = −0.31678 + 1.81032I
7.53406 + 1.71339I 6.00000 + 0.I
u = −0.717176 + 0.997613I
a = 0.584667 − 0.723750I
b = −0.218697 − 1.389430I
7.96025 − 3.20779I 0
u = −0.717176 − 0.997613I
a = 0.584667 + 0.723750I
b = −0.218697 + 1.389430I
7.96025 + 3.20779I 0
u = −0.734348 + 0.989945I
a = −0.626011 − 0.351788I
b = −0.795948 + 0.818156I
4.61963 + 5.43773I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.734348 − 0.989945I
a = −0.626011 + 0.351788I
b = −0.795948 − 0.818156I
4.61963 − 5.43773I 0
u = 0.958511 + 0.797758I
a = 1.47360 + 0.69247I
b = 0.338863 + 1.303150I
7.17999 − 4.40128I 0
u = 0.958511 − 0.797758I
a = 1.47360 − 0.69247I
b = 0.338863 − 1.303150I
7.17999 + 4.40128I 0
u = 1.177050 + 0.420372I
a = −0.180221 − 0.366873I
b = −0.0135532 + 0.0338086I
−1.66489 − 2.36125I 0
u = 1.177050 − 0.420372I
a = −0.180221 + 0.366873I
b = −0.0135532 − 0.0338086I
−1.66489 + 2.36125I 0
u = 1.076140 + 0.640039I
a = −1.46685 − 0.95325I
b = −0.59898 − 2.24608I
5.49499 − 6.77666I 0
u = 1.076140 − 0.640039I
a = −1.46685 + 0.95325I
b = −0.59898 + 2.24608I
5.49499 + 6.77666I 0
u = −0.724809 + 1.033950I
a = 1.064180 + 0.400574I
b = 0.486196 − 0.793628I
4.91559 + 3.36660I 0
u = −0.724809 − 1.033950I
a = 1.064180 − 0.400574I
b = 0.486196 + 0.793628I
4.91559 − 3.36660I 0
u = −0.712486
a = −1.97788
b = −2.16627
2.80010 −4.83520
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.014230 + 0.800845I
a = 0.675996 − 0.593941I
b = 0.53182 − 2.37364I
13.9471 + 5.6591I 0
u = −1.014230 − 0.800845I
a = 0.675996 + 0.593941I
b = 0.53182 + 2.37364I
13.9471 − 5.6591I 0
u = −0.694037 + 0.060458I
a = 1.24961 − 1.05080I
b = 0.696219 + 0.640293I
−4.12135 + 0.22643I 0.431456 + 0.910944I
u = −0.694037 − 0.060458I
a = 1.24961 + 1.05080I
b = 0.696219 − 0.640293I
−4.12135 − 0.22643I 0.431456 − 0.910944I
u = −0.414437 + 0.505333I
a = −0.950768 + 0.151291I
b = −0.409586 + 0.582781I
0.970583 − 0.357083I 10.29975 + 3.20108I
u = −0.414437 − 0.505333I
a = −0.950768 − 0.151291I
b = −0.409586 − 0.582781I
0.970583 + 0.357083I 10.29975 − 3.20108I
u = −1.086260 + 0.818808I
a = −1.40570 + 0.48518I
b = −0.89489 + 1.58340I
6.79477 + 9.83464I 0
u = −1.086260 − 0.818808I
a = −1.40570 − 0.48518I
b = −0.89489 − 1.58340I
6.79477 − 9.83464I 0
u = 0.602545 + 0.150612I
a = −0.74027 + 2.02291I
b = −0.480320 − 0.677791I
0.44759 − 3.90440I 6.22150 + 4.06117I
u = 0.602545 − 0.150612I
a = −0.74027 − 2.02291I
b = −0.480320 + 0.677791I
0.44759 + 3.90440I 6.22150 − 4.06117I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.587564
a = 6.68300
b = 2.00737
10.3493 −7.38360
u = 0.748414 + 1.199110I
a = −0.608875 − 0.775844I
b = 0.92843 − 1.37321I
14.9212 + 6.6040I 0
u = 0.748414 − 1.199110I
a = −0.608875 + 0.775844I
b = 0.92843 + 1.37321I
14.9212 − 6.6040I 0
u = 1.17568 + 0.88106I
a = 1.43185 + 0.33117I
b = 1.26045 + 2.00959I
13.4534 − 14.0241I 0
u = 1.17568 − 0.88106I
a = 1.43185 − 0.33117I
b = 1.26045 − 2.00959I
13.4534 + 14.0241I 0
u = −1.40098 + 0.67383I
a = 0.295089 + 0.214824I
b = 0.930979 + 0.758020I
2.75175 + 3.73669I 0
u = −1.40098 − 0.67383I
a = 0.295089 − 0.214824I
b = 0.930979 − 0.758020I
2.75175 − 3.73669I 0
u = 0.0188398
a = 33.6749
b = 1.27318
6.34803 16.5030
9
II.
I
u
2
= h−u
14
−2u
13
+· · ·+b−8u, −7u
14
−6u
13
+· · ·+a+9, u
15
+u
14
+· · ·−u−1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
10
=
7u
14
+ 6u
13
+ ··· + 25u − 9
u
14
+ 2u
13
+ ··· − u
2
+ 8u
a
4
=
u
14
− 5u
12
+ ··· + 2u − 7
−4u
14
− 4u
13
+ ··· − 5u − 1
a
5
=
u
14
− 5u
12
+ ··· + 2u − 7
−4u
14
− 4u
13
+ ··· − 6u
2
− 5u
a
1
=
−u
2
+ 1
−u
4
a
7
=
u
u
a
11
=
6u
14
+ 6u
13
+ ··· + 21u − 7
2u
13
+ 2u
12
+ ··· + 4u + 2
a
9
=
6u
14
+ 5u
13
+ ··· + 19u − 8
u
14
+ 2u
13
+ ··· − u
2
+ 8u
a
12
=
−u
14
+ 5u
12
+ ··· − 2u + 8
5u
14
+ 4u
13
+ ··· + 15u
2
+ 8u
a
6
=
−8u
14
− 2u
13
+ ··· − 16u + 18
−6u
14
− u
13
+ ··· − 8u + 9
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12u
14
+ 58u
12
+ 12u
11
− 169u
10
− 44u
9
+ 317u
8
+ 107u
7
−
433u
6
− 113u
5
+ 392u
4
+ 65u
3
− 214u
2
− 6u + 58
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
− 9u
14
+ ··· + 15u − 1
c
2
u
15
+ u
14
+ ··· − u − 1
c
3
u
15
+ 4u
13
+ ··· − 3u
2
− 1
c
4
u
15
+ 3u
13
+ ··· + 4u
2
+ 1
c
5
, c
6
u
15
− 8u
13
+ ··· − 12u
3
− 1
c
7
u
15
− u
14
+ ··· − u + 1
c
8
u
15
+ 3u
13
+ ··· − 4u
2
− 1
c
9
u
15
− u
14
+ ··· − u + 1
c
10
u
15
− 8u
12
+ ··· − 3u
2
+ 1
c
11
u
15
− 8u
13
+ ··· − 12u
3
+ 1
c
12
u
15
− u
13
+ ··· + 4u
2
+ 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
+ 11y
14
+ ··· + 31y −1
c
2
, c
7
y
15
− 9y
14
+ ··· + 15y −1
c
3
y
15
+ 8y
14
+ ··· − 6y −1
c
4
, c
8
y
15
+ 6y
14
+ ··· − 8y −1
c
5
, c
6
, c
11
y
15
− 16y
14
+ ··· + 12y
2
− 1
c
9
y
15
− 15y
14
+ ··· + 9y −1
c
10
y
15
− 22y
13
+ ··· + 6y −1
c
12
y
15
− 2y
14
+ ··· − 8y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.894768 + 0.381504I
a = 0.561192 − 0.613047I
b = −0.133265 + 0.484278I
−3.89584 + 1.58539I 2.11826 − 3.74447I
u = −0.894768 − 0.381504I
a = 0.561192 + 0.613047I
b = −0.133265 − 0.484278I
−3.89584 − 1.58539I 2.11826 + 3.74447I
u = 1.002760 + 0.378711I
a = −0.492433 − 0.387786I
b = −0.380671 + 0.657450I
−0.33265 − 5.51163I 4.94815 + 7.40556I
u = 1.002760 − 0.378711I
a = −0.492433 + 0.387786I
b = −0.380671 − 0.657450I
−0.33265 + 5.51163I 4.94815 − 7.40556I
u = 0.800420 + 0.321515I
a = −0.433427 − 0.987439I
b = 0.703055 + 0.417003I
0.48032 + 2.56256I 7.50726 + 0.32927I
u = 0.800420 − 0.321515I
a = −0.433427 + 0.987439I
b = 0.703055 − 0.417003I
0.48032 − 2.56256I 7.50726 − 0.32927I
u = −0.712036
a = −1.36567
b = −2.13667
5.68620 3.34500
u = 1.083910 + 0.722936I
a = −0.725633 − 0.017802I
b = −0.738745 − 0.392095I
−1.62262 − 3.23030I −0.70757 + 9.71898I
u = 1.083910 − 0.722936I
a = −0.725633 + 0.017802I
b = −0.738745 + 0.392095I
−1.62262 + 3.23030I −0.70757 − 9.71898I
u = −0.945359 + 0.963665I
a = 0.814338 + 0.249756I
b = 0.603655 − 0.873044I
3.65766 + 5.39923I 5.36277 − 6.12141I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.945359 − 0.963665I
a = 0.814338 − 0.249756I
b = 0.603655 + 0.873044I
3.65766 − 5.39923I 5.36277 + 6.12141I
u = 0.620323
a = 2.57729
b = 2.00007
3.14873 19.3840
u = −1.268280 + 0.578450I
a = 0.548163 − 0.056389I
b = 1.152480 − 0.111993I
1.86244 + 1.54512I 7.16780 − 3.17961I
u = −1.268280 − 0.578450I
a = 0.548163 + 0.056389I
b = 1.152480 + 0.111993I
1.86244 − 1.54512I 7.16780 + 3.17961I
u = −0.465662
a = −7.75602
b = −2.27641
10.6056 24.4780
14
III. I
u
3
= hb + 1, a + 1, u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
−1
a
3
=
1
1
a
10
=
−1
−1
a
4
=
1
1
a
5
=
1
0
a
1
=
0
−1
a
7
=
−1
−1
a
11
=
−1
−1
a
9
=
−1
−2
a
12
=
1
0
a
6
=
0
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
12
u + 1
c
2
, c
4
, c
5
c
6
, c
7
, c
8
c
9
, c
11
u − 1
c
3
, c
10
u
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
8
, c
9
, c
11
c
12
y −1
c
3
, c
10
y
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = −1.00000
1.64493 6.00000
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
15
− 9u
14
+ ··· + 15u − 1)(u
48
+ 15u
47
+ ··· + 2837u + 1)
c
2
(u − 1)(u
15
+ u
14
+ ··· − u − 1)(u
48
+ u
47
+ ··· − 55u − 1)
c
3
u(u
15
+ 4u
13
+ ··· − 3u
2
− 1)(u
48
+ u
47
+ ··· − 14u + 1)
c
4
(u − 1)(u
15
+ 3u
13
+ ··· + 4u
2
+ 1)(u
48
− 2u
47
+ ··· + 14560u + 15853)
c
5
, c
6
(u − 1)(u
15
− 8u
13
+ ··· − 12u
3
− 1)(u
48
+ 2u
47
+ ··· − 12u − 1)
c
7
(u − 1)(u
15
− u
14
+ ··· − u + 1)(u
48
+ u
47
+ ··· − 55u − 1)
c
8
(u − 1)(u
15
+ 3u
13
+ ··· − 4u
2
− 1)(u
48
− 2u
47
+ ··· + 14560u + 15853)
c
9
(u − 1)(u
15
− u
14
+ ··· − u + 1)(u
48
+ u
47
+ ··· + 719u − 293)
c
10
u(u
15
− 8u
12
+ ··· − 3u
2
+ 1)(u
48
− 3u
47
+ ··· + 10344u + 649)
c
11
(u − 1)(u
15
− 8u
13
+ ··· − 12u
3
+ 1)(u
48
+ 2u
47
+ ··· − 12u − 1)
c
12
(u + 1)(u
15
− u
13
+ ··· + 4u
2
+ 1)(u
48
+ 4u
47
+ ··· − 10508u − 1369)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
15
+ 11y
14
+ ··· + 31y −1)(y
48
+ 53y
47
+ ··· − 7937733y + 1)
c
2
, c
7
(y −1)(y
15
− 9y
14
+ ··· + 15y −1)(y
48
− 15y
47
+ ··· − 2837y + 1)
c
3
y(y
15
+ 8y
14
+ ··· − 6y −1)(y
48
+ 5y
47
+ ··· − 56y + 1)
c
4
, c
8
(y −1)(y
15
+ 6y
14
+ ··· − 8y −1)
· (y
48
− 64y
47
+ ··· − 8556537210y + 251317609)
c
5
, c
6
, c
11
(y −1)(y
15
− 16y
14
+ ··· + 12y
2
− 1)(y
48
− 50y
47
+ ··· − 66y + 1)
c
9
(y −1)(y
15
− 15y
14
+ ··· + 9y −1)
· (y
48
− 61y
47
+ ··· − 3226039y + 85849)
c
10
y(y
15
− 22y
13
+ ··· + 6y −1)
· (y
48
− 39y
47
+ ··· − 45579572y + 421201)
c
12
(y −1)(y
15
− 2y
14
+ ··· − 8y −1)
· (y
48
− 60y
47
+ ··· − 53809914y + 1874161)
20
