12a
0165
(K12a
0165
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 12 11 10 4 8 7 1 6
Solving Sequence
4,8
9 10
1,3
2 5 7 11 6 12
c
8
c
9
c
3
c
1
c
4
c
7
c
10
c
6
c
12
c
2
, c
5
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h4u
18
+ 9u
17
+ ··· + b − 5, 3u
18
+ 3u
17
+ ··· + 2a − 5u, u
19
+ 3u
18
+ ··· − 2u − 2i
I
u
2
= hu
15
a + 3u
16
+ ··· − a + 1, −u
14
a − u
15
+ ··· − a + 2, u
17
− u
16
+ ··· + u − 1i
I
v
1
= ha, b − 1, v + 1i
* 3 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
= h4u
18
+9u
17
+· · ·+b−5, 3u
18
+3u
17
+· · ·+2a−5u, u
19
+3u
18
+· · ·−2u−2i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
10
=
u
2
+ 1
u
2
a
1
=
−
3
2
u
18
−
3
2
u
17
+ ··· +
1
2
u
2
+
5
2
u
−4u
18
− 9u
17
+ ··· + 5u + 5
a
3
=
u
u
3
+ u
a
2
=
−
1
2
u
18
−
1
2
u
17
+ ··· +
1
2
u
2
+
3
2
u
−u
18
− 2u
17
+ ··· + 2u + 1
a
5
=
1
2
u
18
+
3
2
u
17
+ ··· −
1
2
u − 1
−u
18
− u
16
+ ··· + 2u − 1
a
7
=
u
4
+ u
2
+ 1
u
4
a
11
=
u
6
+ u
4
+ 2u
2
+ 1
u
6
+ u
2
a
6
=
u
8
+ u
6
+ 3u
4
+ 2u
2
+ 1
u
8
+ 2u
4
a
12
=
1
2
u
18
+
1
2
u
17
+ ··· −
1
2
u + 1
u
18
+ 2u
17
+ ··· − u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
18
+ 2u
16
− 6u
15
+ 4u
14
− 12u
13
+ 2u
12
− 28u
11
− 2u
10
−
24u
9
− 2u
8
− 28u
7
− 12u
6
− 8u
5
− 12u
4
− 6u
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
19
+ 11u
18
+ ··· + 3u + 1
c
2
, c
4
, c
5
c
12
u
19
− u
18
+ ··· − u + 1
c
3
, c
8
u
19
− 3u
18
+ ··· − 2u + 2
c
6
, c
7
, c
9
c
10
u
19
− 3u
18
+ ··· − 11u
2
+ 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
19
− 3y
18
+ ··· + 11y −1
c
2
, c
4
, c
5
c
12
y
19
− 11y
18
+ ··· + 3y −1
c
3
, c
8
y
19
+ 3y
18
+ ··· + 11y
2
− 4
c
6
, c
7
, c
9
c
10
y
19
+ 23y
18
+ ··· + 88y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.347446 + 0.933456I
a = 0.171985 − 0.194773I
b = 0.969269 + 0.134564I
0.10929 + 6.34273I −3.77049 − 10.42741I
u = −0.347446 − 0.933456I
a = 0.171985 + 0.194773I
b = 0.969269 − 0.134564I
0.10929 − 6.34273I −3.77049 + 10.42741I
u = 0.532414 + 0.771389I
a = 0.511523 + 0.882342I
b = 0.007510 + 0.817176I
−0.41325 − 2.04302I −2.62456 + 3.49236I
u = 0.532414 − 0.771389I
a = 0.511523 − 0.882342I
b = 0.007510 − 0.817176I
−0.41325 + 2.04302I −2.62456 − 3.49236I
u = 0.838741 + 0.661261I
a = −0.960098 − 0.906788I
b = −0.480763 − 0.879178I
−6.95459 + 5.11431I −11.79581 − 4.98965I
u = 0.838741 − 0.661261I
a = −0.960098 + 0.906788I
b = −0.480763 + 0.879178I
−6.95459 − 5.11431I −11.79581 + 4.98965I
u = −0.009736 + 0.866710I
a = −0.023776 + 0.565154I
b = −0.632690 + 0.365808I
1.96169 − 1.46588I 2.04992 + 4.47072I
u = −0.009736 − 0.866710I
a = −0.023776 − 0.565154I
b = −0.632690 − 0.365808I
1.96169 + 1.46588I 2.04992 − 4.47072I
u = 0.674488 + 0.956724I
a = −0.561349 − 1.206360I
b = −0.041367 − 1.175230I
−5.94319 − 10.65650I −9.44590 + 9.96875I
u = 0.674488 − 0.956724I
a = −0.561349 + 1.206360I
b = −0.041367 + 1.175230I
−5.94319 + 10.65650I −9.44590 − 9.96875I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.722854 + 0.279055I
a = −0.268290 − 0.582437I
b = −0.042347 + 0.263428I
−2.21379 − 2.62773I −8.93554 + 6.59868I
u = −0.722854 − 0.279055I
a = −0.268290 + 0.582437I
b = −0.042347 − 0.263428I
−2.21379 + 2.62773I −8.93554 − 6.59868I
u = −0.902022 + 0.935041I
a = −1.22720 + 1.31378I
b = 0.59682 + 2.85556I
−9.32638 + 3.32620I −5.56917 − 2.29363I
u = −0.902022 − 0.935041I
a = −1.22720 − 1.31378I
b = 0.59682 − 2.85556I
−9.32638 − 3.32620I −5.56917 + 2.29363I
u = −0.954578 + 0.916463I
a = 2.01023 − 1.04776I
b = 0.44541 − 3.24116I
−17.2005 − 6.5526I −12.03748 + 3.63371I
u = −0.954578 − 0.916463I
a = 2.01023 + 1.04776I
b = 0.44541 + 3.24116I
−17.2005 + 6.5526I −12.03748 − 3.63371I
u = −0.914047 + 0.984596I
a = 0.93853 − 2.07020I
b = −1.52424 − 3.36626I
−16.9728 + 13.4124I −11.60113 − 8.01307I
u = −0.914047 − 0.984596I
a = 0.93853 + 2.07020I
b = −1.52424 + 3.36626I
−16.9728 − 13.4124I −11.60113 + 8.01307I
u = 0.610080
a = 0.816899
b = 0.404798
−1.23828 −6.53970
6
II.
I
u
2
= hu
15
a+3u
16
+· · ·−a+1, −u
14
a−u
15
+· · ·−a+2, u
17
−u
16
+· · ·+u−1i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
10
=
u
2
+ 1
u
2
a
1
=
a
−
1
2
u
15
a −
3
2
u
16
+ ··· +
1
2
a −
1
2
a
3
=
u
u
3
+ u
a
2
=
1
2
u
16
a −
1
2
u
16
+ ··· +
3
2
a +
3
2
u
−u
15
a − 3u
16
+ ··· + a − 1
a
5
=
−
1
2
u
15
a −
3
2
u
16
+ ··· −
1
2
a −
1
2
−
1
2
u
15
a −
3
2
u
16
+ ··· +
1
2
a −
1
2
a
7
=
u
4
+ u
2
+ 1
u
4
a
11
=
u
6
+ u
4
+ 2u
2
+ 1
u
6
+ u
2
a
6
=
u
8
+ u
6
+ 3u
4
+ 2u
2
+ 1
u
8
+ 2u
4
a
12
=
−
1
2
u
15
a −
1
2
u
16
+ ··· +
3
2
a +
3
2
−
1
2
u
16
a − 3u
16
+ ··· + a −
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
15
+ 4u
14
− 8u
13
+ 4u
12
− 28u
11
+ 20u
10
− 36u
9
+ 16u
8
−
56u
7
+ 28u
6
− 40u
5
+ 16u
4
− 28u
3
+ 16u
2
− 12u − 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
34
+ 21u
33
+ ··· + 12u + 1
c
2
, c
4
, c
5
c
12
u
34
− u
33
+ ··· − 6u
2
+ 1
c
3
, c
8
(u
17
+ u
16
+ ··· + u + 1)
2
c
6
, c
7
, c
9
c
10
(u
17
− 3u
16
+ ··· − 3u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
34
− 17y
33
+ ··· − 140y + 1
c
2
, c
4
, c
5
c
12
y
34
− 21y
33
+ ··· − 12y + 1
c
3
, c
8
(y
17
+ 3y
16
+ ··· − 3y −1)
2
c
6
, c
7
, c
9
c
10
(y
17
+ 23y
16
+ ··· + 9y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.672243 + 0.786311I
a = −1.082100 − 0.898592I
b = −1.31901 − 1.59889I
−7.18216 − 2.50454I −12.07700 + 3.85927I
u = 0.672243 + 0.786311I
a = −1.00326 − 1.62412I
b = 0.608646 − 0.931855I
−7.18216 − 2.50454I −12.07700 + 3.85927I
u = 0.672243 − 0.786311I
a = −1.082100 + 0.898592I
b = −1.31901 + 1.59889I
−7.18216 + 2.50454I −12.07700 − 3.85927I
u = 0.672243 − 0.786311I
a = −1.00326 + 1.62412I
b = 0.608646 + 0.931855I
−7.18216 + 2.50454I −12.07700 − 3.85927I
u = −0.706998 + 0.642933I
a = −0.807968 + 0.702661I
b = −0.524414 + 1.168210I
−3.89702 − 1.19537I −8.59794 + 0.58854I
u = −0.706998 + 0.642933I
a = 0.67143 − 1.25663I
b = −0.057241 − 0.574590I
−3.89702 − 1.19537I −8.59794 + 0.58854I
u = −0.706998 − 0.642933I
a = −0.807968 − 0.702661I
b = −0.524414 − 1.168210I
−3.89702 + 1.19537I −8.59794 − 0.58854I
u = −0.706998 − 0.642933I
a = 0.67143 + 1.25663I
b = −0.057241 + 0.574590I
−3.89702 + 1.19537I −8.59794 − 0.58854I
u = −0.616947 + 0.891729I
a = 0.670001 − 0.916287I
b = 0.668676 − 1.015290I
−3.09054 + 6.12281I −6.33796 − 6.84601I
u = −0.616947 + 0.891729I
a = −0.599055 + 1.125390I
b = 0.432694 + 1.039890I
−3.09054 + 6.12281I −6.33796 − 6.84601I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.616947 − 0.891729I
a = 0.670001 + 0.916287I
b = 0.668676 + 1.015290I
−3.09054 − 6.12281I −6.33796 + 6.84601I
u = −0.616947 − 0.891729I
a = −0.599055 − 1.125390I
b = 0.432694 − 1.039890I
−3.09054 − 6.12281I −6.33796 + 6.84601I
u = 0.208716 + 0.869278I
a = 0.072000 + 0.778055I
b = 0.256802 + 0.282630I
1.42740 − 2.28997I 0.30509 + 4.71022I
u = 0.208716 + 0.869278I
a = −0.172514 + 0.206222I
b = −1.016220 + 0.354488I
1.42740 − 2.28997I 0.30509 + 4.71022I
u = 0.208716 − 0.869278I
a = 0.072000 − 0.778055I
b = 0.256802 − 0.282630I
1.42740 + 2.28997I 0.30509 − 4.71022I
u = 0.208716 − 0.869278I
a = −0.172514 − 0.206222I
b = −1.016220 − 0.354488I
1.42740 + 2.28997I 0.30509 − 4.71022I
u = 0.929005 + 0.919626I
a = 1.29872 + 1.27388I
b = −0.39727 + 3.05185I
−13.30230 + 1.56927I −8.91940 − 0.65050I
u = 0.929005 + 0.919626I
a = −1.96214 − 1.14150I
b = −0.17796 − 3.10658I
−13.30230 + 1.56927I −8.91940 − 0.65050I
u = 0.929005 − 0.919626I
a = 1.29872 − 1.27388I
b = −0.39727 − 3.05185I
−13.30230 − 1.56927I −8.91940 + 0.65050I
u = 0.929005 − 0.919626I
a = −1.96214 + 1.14150I
b = −0.17796 + 3.10658I
−13.30230 − 1.56927I −8.91940 + 0.65050I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.920829 + 0.944574I
a = 1.09183 − 2.00211I
b = −1.03641 − 3.68580I
−17.2424 + 3.3872I −12.08288 − 2.32417I
u = −0.920829 + 0.944574I
a = 2.02633 − 1.22733I
b = −0.07151 − 3.28042I
−17.2424 + 3.3872I −12.08288 − 2.32417I
u = −0.920829 − 0.944574I
a = 1.09183 + 2.00211I
b = −1.03641 + 3.68580I
−17.2424 − 3.3872I −12.08288 + 2.32417I
u = −0.920829 − 0.944574I
a = 2.02633 + 1.22733I
b = −0.07151 + 3.28042I
−17.2424 − 3.3872I −12.08288 + 2.32417I
u = 0.905075 + 0.964023I
a = 1.23229 + 1.39113I
b = −0.84889 + 2.93758I
−13.1567 − 8.3174I −8.64033 + 5.18877I
u = 0.905075 + 0.964023I
a = −0.98957 − 1.99266I
b = 1.20467 − 3.37460I
−13.1567 − 8.3174I −8.64033 + 5.18877I
u = 0.905075 − 0.964023I
a = 1.23229 − 1.39113I
b = −0.84889 − 2.93758I
−13.1567 + 8.3174I −8.64033 − 5.18877I
u = 0.905075 − 0.964023I
a = −0.98957 + 1.99266I
b = 1.20467 + 3.37460I
−13.1567 + 8.3174I −8.64033 − 5.18877I
u = −0.231740 + 0.588876I
a = 0.776006 + 0.665775I
b = 1.69903 + 1.15771I
−3.00025 + 0.92655I −6.49670 − 7.34204I
u = −0.231740 + 0.588876I
a = −0.09148 − 2.42444I
b = −0.307065 + 0.314242I
−3.00025 + 0.92655I −6.49670 − 7.34204I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.231740 − 0.588876I
a = 0.776006 − 0.665775I
b = 1.69903 − 1.15771I
−3.00025 − 0.92655I −6.49670 + 7.34204I
u = −0.231740 − 0.588876I
a = −0.09148 + 2.42444I
b = −0.307065 − 0.314242I
−3.00025 − 0.92655I −6.49670 + 7.34204I
u = 0.522950
a = 1.21130
b = 0.271850
−1.19234 −8.30570
u = 0.522950
a = 0.527647
b = 0.499095
−1.19234 −8.30570
13
III. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
4
=
−1
0
a
8
=
1
0
a
9
=
1
0
a
10
=
1
0
a
1
=
0
1
a
3
=
−1
0
a
2
=
−1
1
a
5
=
0
−1
a
7
=
1
0
a
11
=
1
0
a
6
=
1
0
a
12
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
11
u − 1
c
3
, c
6
, c
7
c
8
, c
9
, c
10
u
c
4
, c
12
u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
11
, c
12
y − 1
c
3
, c
6
, c
7
c
8
, c
9
, c
10
y
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
11
(u − 1)(u
19
+ 11u
18
+ ··· + 3u + 1)(u
34
+ 21u
33
+ ··· + 12u + 1)
c
2
, c
5
(u − 1)(u
19
− u
18
+ ··· − u + 1)(u
34
− u
33
+ ··· − 6u
2
+ 1)
c
3
, c
8
u(u
17
+ u
16
+ ··· + u + 1)
2
(u
19
− 3u
18
+ ··· − 2u + 2)
c
4
, c
12
(u + 1)(u
19
− u
18
+ ··· − u + 1)(u
34
− u
33
+ ··· − 6u
2
+ 1)
c
6
, c
7
, c
9
c
10
u(u
17
− 3u
16
+ ··· − 3u + 1)
2
(u
19
− 3u
18
+ ··· − 11u
2
+ 4)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
11
(y − 1)(y
19
− 3y
18
+ ··· + 11y − 1)(y
34
− 17y
33
+ ··· − 140y + 1)
c
2
, c
4
, c
5
c
12
(y − 1)(y
19
− 11y
18
+ ··· + 3y − 1)(y
34
− 21y
33
+ ··· − 12y + 1)
c
3
, c
8
y(y
17
+ 3y
16
+ ··· − 3y − 1)
2
(y
19
+ 3y
18
+ ··· + 11y
2
− 4)
c
6
, c
7
, c
9
c
10
y(y
17
+ 23y
16
+ ··· + 9y − 1)
2
(y
19
+ 23y
18
+ ··· + 88y − 16)
19
