12n
0681
(K12n
0681
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 11 3 10 12 7 8 5 9
Solving Sequence
8,11
10
3,7
6 5 12 2 4 1 9
c
10
c
7
c
6
c
5
c
11
c
2
c
4
c
1
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.29319 × 10
25
u
26
− 8.14562 × 10
25
u
25
+ ··· + 2.10275 × 10
26
b + 1.50746 × 10
26
,
9.51680 × 10
25
u
26
+ 6.52825 × 10
26
u
25
+ ··· + 2.10275 × 10
26
a − 7.67512 × 10
27
, u
27
+ 7u
26
+ ··· − 65u + 1i
I
u
2
= h−u
7
− 2u
6
+ 2u
5
+ 4u
4
− 2u
3
− u
2
+ b + u − 3, 2u
7
+ 2u
6
− 5u
5
− 4u
4
+ 3u
3
+ a + u + 3,
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1i
I
u
3
= h4a
2
+ 23b − 33a + 3, a
3
− 8a
2
+ 3a − 7, u − 1i
I
u
4
= hb − 2u + 1, a + u + 4, u
2
+ u − 1i
I
u
5
= hb + u, a − u − 2, u
2
+ u − 1i
* 5 irreducible components of dim
C
= 0, with total 42 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−1.29 × 10
25
u
26
− 8.15 × 10
25
u
25
+ · · · + 2.10 × 10
26
b + 1.51 ×
10
26
, 9.52 × 10
25
u
26
+ 6.53 × 10
26
u
25
+ · · · + 2.10 × 10
26
a − 7.68 ×
10
27
, u
27
+ 7u
26
+ · · · − 65u + 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
10
=
1
u
2
a
3
=
−0.452588u
26
− 3.10462u
25
+ ··· − 55.6935u + 36.5004
0.0614998u
26
+ 0.387379u
25
+ ··· + 3.33121u − 0.716900
a
7
=
−u
−u
3
+ u
a
6
=
−0.486093u
26
− 3.57968u
25
+ ··· − 55.8612u + 20.4963
0.0646653u
26
+ 0.428826u
25
+ ··· + 0.403387u − 0.357433
a
5
=
−0.550758u
26
− 4.00850u
25
+ ··· − 56.2646u + 20.8537
0.0646653u
26
+ 0.428826u
25
+ ··· + 0.403387u − 0.357433
a
12
=
−0.0697411u
26
− 0.446921u
25
+ ··· − 5.57659u + 8.05416
−0.0811376u
26
− 0.484277u
25
+ ··· + 5.23091u − 0.200858
a
2
=
−0.478778u
26
− 3.47699u
25
+ ··· − 53.7199u + 20.8543
0.0459720u
26
+ 0.264293u
25
+ ··· − 0.415707u − 0.385704
a
4
=
−0.514677u
26
− 3.50894u
25
+ ··· − 55.7920u + 36.4942
0.0496659u
26
+ 0.333051u
25
+ ··· + 5.46117u − 0.741054
a
1
=
0.0759051u
26
+ 0.518888u
25
+ ··· + 8.57256u − 7.97571
0.100287u
26
+ 0.631138u
25
+ ··· − 2.83878u + 0.163326
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
128221912491069143432166825
210275241869650933749294956
u
26
−
1069012753234217659630964039
210275241869650933749294956
u
25
+
··· −
14132876007342110386546979695
210275241869650933749294956
u −
1802365230129749281114104905
210275241869650933749294956
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
27
− 12u
26
+ ··· − 82u − 1
c
3
, c
6
u
27
+ 4u
26
+ ··· + 640u − 256
c
5
, c
11
u
27
+ 3u
26
+ ··· − 112u + 16
c
7
, c
9
, c
10
u
27
+ 7u
26
+ ··· − 65u + 1
c
8
, c
12
u
27
− 4u
26
+ ··· − 36u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
27
− 46y
26
+ ··· + 6314y −1
c
3
, c
6
y
27
− 54y
26
+ ··· + 5095424y −65536
c
5
, c
11
y
27
+ 25y
26
+ ··· + 12928y −256
c
7
, c
9
, c
10
y
27
− 15y
26
+ ··· + 4023y −1
c
8
, c
12
y
27
+ 12y
26
+ ··· + 7696y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.989617
a = 9.38049
b = 0.408460
0.561362 203.500
u = −1.035410 + 0.203452I
a = −0.415320 + 0.863420I
b = −0.33969 + 1.38997I
4.69512 + 2.40532I 3.41247 + 6.32084I
u = −1.035410 − 0.203452I
a = −0.415320 − 0.863420I
b = −0.33969 − 1.38997I
4.69512 − 2.40532I 3.41247 − 6.32084I
u = 1.035950 + 0.225932I
a = −0.376658 + 0.725229I
b = −0.061105 − 0.490914I
1.156740 + 0.801856I 6.71973 + 0.16728I
u = 1.035950 − 0.225932I
a = −0.376658 − 0.725229I
b = −0.061105 + 0.490914I
1.156740 − 0.801856I 6.71973 − 0.16728I
u = −0.231476 + 0.812947I
a = −1.010650 − 0.337430I
b = 0.147510 + 0.585443I
−2.46734 + 1.28188I 0.019660 − 0.966602I
u = −0.231476 − 0.812947I
a = −1.010650 + 0.337430I
b = 0.147510 − 0.585443I
−2.46734 − 1.28188I 0.019660 + 0.966602I
u = 0.707396
a = −4.75513
b = 0.0513464
−7.81649 57.8300
u = −1.305580 + 0.386979I
a = −0.110314 − 0.122910I
b = 0.073018 − 0.478334I
1.16826 − 5.86191I 6.68570 + 1.60407I
u = −1.305580 − 0.386979I
a = −0.110314 + 0.122910I
b = 0.073018 + 0.478334I
1.16826 + 5.86191I 6.68570 − 1.60407I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.836004 + 1.095460I
a = −0.0688478 + 0.0584922I
b = −0.25035 + 1.73182I
−7.44159 − 1.29405I −1.37770 + 1.27497I
u = −0.836004 − 1.095460I
a = −0.0688478 − 0.0584922I
b = −0.25035 − 1.73182I
−7.44159 + 1.29405I −1.37770 − 1.27497I
u = 0.558394 + 0.255485I
a = 0.39540 + 1.82247I
b = −0.99845 − 1.93414I
−0.834252 + 0.150815I 17.5464 − 6.6365I
u = 0.558394 − 0.255485I
a = 0.39540 − 1.82247I
b = −0.99845 + 1.93414I
−0.834252 − 0.150815I 17.5464 + 6.6365I
u = −1.016510 + 0.945163I
a = 1.046130 − 0.262688I
b = −0.340400 − 0.098336I
−13.16990 − 3.53439I −0.41212 + 2.09406I
u = −1.016510 − 0.945163I
a = 1.046130 + 0.262688I
b = −0.340400 + 0.098336I
−13.16990 + 3.53439I −0.41212 − 2.09406I
u = −1.16800 + 0.91505I
a = 1.271790 − 0.556686I
b = −0.35535 − 1.77081I
−6.34499 − 6.08931I −0.21731 + 3.73020I
u = −1.16800 − 0.91505I
a = 1.271790 + 0.556686I
b = −0.35535 + 1.77081I
−6.34499 + 6.08931I −0.21731 − 3.73020I
u = 0.491518
a = −0.797650
b = 0.376982
0.859867 11.9670
u = 0.06761 + 1.51979I
a = −0.239446 + 0.151266I
b = 0.28102 − 2.41058I
18.5782 + 5.9421I −1.31050 − 2.20591I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.06761 − 1.51979I
a = −0.239446 − 0.151266I
b = 0.28102 + 2.41058I
18.5782 − 5.9421I −1.31050 + 2.20591I
u = −1.60127
a = 2.03965
b = 3.57318
7.98804 43.0640
u = −1.55855 + 0.65683I
a = −1.03603 + 1.21486I
b = 0.57720 + 2.23959I
−15.7574 − 13.5083I 0.77209 + 5.37939I
u = −1.55855 − 0.65683I
a = −1.03603 − 1.21486I
b = 0.57720 − 2.23959I
−15.7574 + 13.5083I 0.77209 − 5.37939I
u = 1.68805 + 0.77291I
a = 0.823135 + 0.943566I
b = −0.10758 + 2.58327I
−16.0044 + 2.3695I 0
u = 1.68805 − 0.77291I
a = 0.823135 − 0.943566I
b = −0.10758 − 2.58327I
−16.0044 − 2.3695I 0
u = 0.0157914
a = 35.5742
b = −0.661597
−1.12664 −9.59770
7
II. I
u
2
= h−u
7
− 2u
6
+ 2u
5
+ 4u
4
− 2u
3
− u
2
+ b + u − 3, 2u
7
+ 2u
6
− 5u
5
−
4u
4
+ 3u
3
+ a + u + 3, u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
10
=
1
u
2
a
3
=
−2u
7
− 2u
6
+ 5u
5
+ 4u
4
− 3u
3
− u − 3
u
7
+ 2u
6
− 2u
5
− 4u
4
+ 2u
3
+ u
2
− u + 3
a
7
=
−u
−u
3
+ u
a
6
=
−u
−u
3
+ u
a
5
=
u
3
− 2u
−u
3
+ u
a
12
=
u
6
− 3u
4
+ 2u
2
+ 1
−u
6
+ 2u
4
− u
2
a
2
=
−2u
7
− 2u
6
+ 5u
5
+ 4u
4
− 4u
3
+ u − 3
u
7
+ 2u
6
− 2u
5
− 4u
4
+ 3u
3
+ u
2
− 2u + 3
a
4
=
−2u
7
− 2u
6
+ 5u
5
+ 4u
4
− 3u
3
− u − 3
u
7
+ 2u
6
− 2u
5
− 4u
4
+ 2u
3
+ u
2
− u + 3
a
1
=
−u
3
+ 2u
u
3
− u
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 21u
7
+ 38u
6
− 48u
5
− 85u
4
+ 39u
3
+ 27u
2
− 5u + 58
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
6
u
8
c
4
(u + 1)
8
c
5
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
7
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
8
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
9
, c
10
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
11
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
12
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
6
y
8
c
5
, c
11
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
7
, c
9
, c
10
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
8
, c
12
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.180120 + 0.268597I
a = 1.23903 + 1.07030I
b = −0.281371 + 1.128550I
−0.604279 + 1.131230I 0.744211 + 0.553382I
u = 1.180120 − 0.268597I
a = 1.23903 − 1.07030I
b = −0.281371 − 1.128550I
−0.604279 − 1.131230I 0.744211 − 0.553382I
u = 0.108090 + 0.747508I
a = −0.188536 + 0.513699I
b = 0.208670 − 0.825203I
−3.80435 + 2.57849I −2.39106 − 4.72239I
u = 0.108090 − 0.747508I
a = −0.188536 − 0.513699I
b = 0.208670 + 0.825203I
−3.80435 − 2.57849I −2.39106 + 4.72239I
u = −1.37100
a = 0.942639
b = 0.829189
4.85780 8.45210
u = −1.334530 + 0.318930I
a = −0.271933 + 0.551071I
b = 0.284386 + 0.605794I
0.73474 − 6.44354I 0.47538 + 9.99765I
u = −1.334530 − 0.318930I
a = −0.271933 − 0.551071I
b = 0.284386 − 0.605794I
0.73474 + 6.44354I 0.47538 − 9.99765I
u = 0.463640
a = −3.49976
b = 2.74744
−0.799899 60.8910
11
III. I
u
3
= h4a
2
+ 23b − 33a + 3, a
3
− 8a
2
+ 3a − 7, u − 1i
(i) Arc colorings
a
8
=
0
1
a
11
=
1
0
a
10
=
1
1
a
3
=
a
−
4
23
a
2
+
33
23
a −
3
23
a
7
=
−1
0
a
6
=
1
23
a
2
+
9
23
a −
51
23
−
1
23
a
2
+
14
23
a −
41
23
a
5
=
2
23
a
2
−
5
23
a −
10
23
−
1
23
a
2
+
14
23
a −
41
23
a
12
=
0
−
5
23
a
2
+
47
23
a −
67
23
a
2
=
1
23
a
2
+
9
23
a −
5
23
−
1
23
a
2
+
14
23
a −
41
23
a
4
=
4
23
a
2
−
10
23
a +
3
23
−
4
23
a
2
+
33
23
a −
3
23
a
1
=
0
−
5
23
a
2
+
47
23
a −
67
23
a
9
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
75
23
a
2
+
314
23
a −
39
23
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
u
3
+ u
2
− 1
c
3
u
3
− u
2
+ 2u − 1
c
4
u
3
− u
2
+ 1
c
5
u
3
− 3u
2
+ 2u + 1
c
6
u
3
+ u
2
+ 2u + 1
c
7
(u + 1)
3
c
8
, c
12
u
3
c
9
, c
10
(u − 1)
3
c
11
u
3
+ 3u
2
+ 2u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
3
− y
2
+ 2y −1
c
3
, c
6
y
3
+ 3y
2
+ 2y −1
c
5
, c
11
y
3
− 5y
2
+ 10y −1
c
7
, c
9
, c
10
(y −1)
3
c
8
, c
12
y
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.135484 + 0.941977I
b = 0.215080 + 1.307140I
4.66906 − 2.82812I 2.98758 + 12.02771I
u = 1.00000
a = 0.135484 − 0.941977I
b = 0.215080 − 1.307140I
4.66906 + 2.82812I 2.98758 − 12.02771I
u = 1.00000
a = 7.72903
b = 0.569840
0.531480 −90.9750
15
IV. I
u
4
= hb − 2u + 1, a + u + 4, u
2
+ u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
10
=
1
−u + 1
a
3
=
−u − 4
2u − 1
a
7
=
−u
−u + 1
a
6
=
3u + 5
0
a
5
=
3u + 5
0
a
12
=
1
0
a
2
=
4u + 4
2u − 1
a
4
=
−u − 3
u − 1
a
1
=
u
u − 1
a
9
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −45
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
9
, c
10
, c
12
u
2
+ u − 1
c
4
, c
6
, c
7
c
8
u
2
− u − 1
c
5
, c
11
u
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
9
, c
10
c
12
y
2
− 3y + 1
c
5
, c
11
y
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −4.61803
b = 0.236068
−7.89568 −45.0000
u = −1.61803
a = −2.38197
b = −4.23607
7.89568 −45.0000
19
V. I
u
5
= hb + u, a − u − 2, u
2
+ u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
10
=
1
−u + 1
a
3
=
u + 2
−u
a
7
=
−u
−u + 1
a
6
=
1
0
a
5
=
1
0
a
12
=
1
0
a
2
=
2
−u
a
4
=
2u + 1
u − 1
a
1
=
u
u − 1
a
9
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
9
, c
10
, c
12
u
2
+ u − 1
c
4
, c
6
, c
7
c
8
u
2
− u − 1
c
5
, c
11
u
2
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
9
, c
10
c
12
y
2
− 3y + 1
c
5
, c
11
y
2
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 2.61803
b = −0.618034
0 0
u = −1.61803
a = 0.381966
b = 1.61803
0 0
23
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
8
)(u
2
+ u − 1)
2
(u
3
+ u
2
− 1)(u
27
− 12u
26
+ ··· − 82u − 1)
c
3
u
8
(u
2
+ u − 1)
2
(u
3
− u
2
+ 2u − 1)(u
27
+ 4u
26
+ ··· + 640u − 256)
c
4
((u + 1)
8
)(u
2
− u − 1)
2
(u
3
− u
2
+ 1)(u
27
− 12u
26
+ ··· − 82u − 1)
c
5
u
4
(u
3
− 3u
2
+ 2u + 1)
· (u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
27
+ 3u
26
+ ··· − 112u + 16)
c
6
u
8
(u
2
− u − 1)
2
(u
3
+ u
2
+ 2u + 1)(u
27
+ 4u
26
+ ··· + 640u − 256)
c
7
(u + 1)
3
(u
2
− u − 1)
2
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)
· (u
27
+ 7u
26
+ ··· − 65u + 1)
c
8
u
3
(u
2
− u − 1)
2
(u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
· (u
27
− 4u
26
+ ··· − 36u + 8)
c
9
, c
10
(u − 1)
3
(u
2
+ u − 1)
2
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
· (u
27
+ 7u
26
+ ··· − 65u + 1)
c
11
u
4
(u
3
+ 3u
2
+ 2u − 1)
· (u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
27
+ 3u
26
+ ··· − 112u + 16)
c
12
u
3
(u
2
+ u − 1)
2
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
· (u
27
− 4u
26
+ ··· − 36u + 8)
24
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
(y
2
− 3y + 1)
2
(y
3
− y
2
+ 2y −1)
· (y
27
− 46y
26
+ ··· + 6314y −1)
c
3
, c
6
y
8
(y
2
− 3y + 1)
2
(y
3
+ 3y
2
+ 2y −1)
· (y
27
− 54y
26
+ ··· + 5095424y −65536)
c
5
, c
11
y
4
(y
3
− 5y
2
+ 10y −1)
· (y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
27
+ 25y
26
+ ··· + 12928y −256)
c
7
, c
9
, c
10
(y −1)
3
(y
2
− 3y + 1)
2
· (y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
27
− 15y
26
+ ··· + 4023y −1)
c
8
, c
12
y
3
(y
2
− 3y + 1)
2
(y
8
− 3y
7
+ ··· − 4y + 1)
· (y
27
+ 12y
26
+ ··· + 7696y −64)
25
