12a
0168
(K12a
0168
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 12 1 11 10 4 8 7 6
Solving Sequence
1,7
6 12
2,5
3 4 11 8 10 9
c
6
c
12
c
5
c
2
c
4
c
11
c
7
c
10
c
9
c
1
, c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
28
+ 10u
26
+ ··· + 6u
2
+ b, −u
34
+ u
33
+ ··· + a − 8u, u
35
− 2u
34
+ ··· − u + 1i
I
u
2
= hu
5
− u
3
+ b − u, u
3
+ a,
u
15
− 5u
13
− u
12
+ 10u
11
+ 4u
10
− 6u
9
− 6u
8
− 7u
7
+ u
6
+ 11u
5
+ 5u
4
− u
3
− 3u
2
− 3u − 1i
I
u
3
= hb + 1, a − 1, u + 1i
* 3 irreducible components of dim
C
= 0, with total 51 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−u
28
+10u
26
+· · ·+6u
2
+b, −u
34
+u
33
+· · ·+a−8u, u
35
−2u
34
+· · ·−u+1i
(i) Arc colorings
a
1
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
12
=
−u
−u
3
+ u
a
2
=
u
34
− u
33
+ ··· + 20u
2
+ 8u
u
28
− 10u
26
+ ··· − 16u
3
− 6u
2
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
−u
33
+ 13u
31
+ ··· + 28u
2
+ 8u
−u
34
+ u
33
+ ··· − u + 1
a
4
=
u
34
− 13u
32
+ ··· − 7u − 1
u
34
− u
33
+ ··· + u − 1
a
11
=
u
3
− 2u
−u
3
+ u
a
8
=
u
6
− 3u
4
+ 2u
2
+ 1
−u
6
+ 2u
4
− u
2
a
10
=
u
9
− 4u
7
+ 5u
5
− 3u
−u
9
+ 3u
7
− 3u
5
+ u
a
9
=
u
12
− 5u
10
+ 9u
8
− 4u
6
− 6u
4
+ 5u
2
+ 1
−u
12
+ 4u
10
− 6u
8
+ 2u
6
+ 3u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
33
+ 4u
32
+ 26u
31
− 48u
30
− 154u
29
+ 252u
28
+ 536u
27
−
724u
26
− 1162u
25
+ 1096u
24
+ 1448u
23
− 336u
22
− 448u
21
− 1792u
20
− 1708u
19
+
2984u
18
+ 2926u
17
− 836u
16
− 1400u
15
− 2388u
14
− 1280u
13
+ 2228u
12
+ 1868u
11
+
404u
10
− 328u
9
− 1140u
8
− 596u
7
+ 60u
6
+ 196u
5
+ 236u
4
+ 100u
3
+ 28u
2
− 6u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
35
+ 20u
34
+ ··· + 3u + 1
c
2
, c
4
u
35
− 2u
34
+ ··· + 3u − 1
c
3
, c
9
u
35
+ 2u
34
+ ··· − 2u − 2
c
5
, c
6
, c
12
u
35
+ 2u
34
+ ··· − u − 1
c
7
, c
8
, c
10
c
11
u
35
− 6u
34
+ ··· + 8u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
35
− 8y
34
+ ··· + 35y −1
c
2
, c
4
y
35
− 20y
34
+ ··· + 3y −1
c
3
, c
9
y
35
− 6y
34
+ ··· + 8y −4
c
5
, c
6
, c
12
y
35
− 28y
34
+ ··· − 13y −1
c
7
, c
8
, c
10
c
11
y
35
+ 42y
34
+ ··· − 136y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.029628 + 0.934102I
a = −3.61963 + 0.54565I
b = 4.19510 − 0.71037I
−14.7947 − 8.3253I −0.91975 + 5.31258I
u = −0.029628 − 0.934102I
a = −3.61963 − 0.54565I
b = 4.19510 + 0.71037I
−14.7947 + 8.3253I −0.91975 − 5.31258I
u = 0.004692 + 0.923942I
a = 3.64319 + 0.66960I
b = −4.10450 − 1.08656I
−14.9356 + 1.5707I −1.236325 − 0.670783I
u = 0.004692 − 0.923942I
a = 3.64319 − 0.66960I
b = −4.10450 + 1.08656I
−14.9356 − 1.5707I −1.236325 + 0.670783I
u = −1.140910 + 0.226464I
a = 0.0410600 + 0.0527107I
b = −0.796518 + 0.135470I
1.16280 − 0.99874I 6.67808 + 0.23087I
u = −1.140910 − 0.226464I
a = 0.0410600 − 0.0527107I
b = −0.796518 − 0.135470I
1.16280 + 0.99874I 6.67808 − 0.23087I
u = −1.211370 + 0.063006I
a = −0.024848 + 0.395141I
b = −0.93931 − 1.68300I
2.16189 − 1.54722I 7.54202 + 4.01814I
u = −1.211370 − 0.063006I
a = −0.024848 − 0.395141I
b = −0.93931 + 1.68300I
2.16189 + 1.54722I 7.54202 − 4.01814I
u = −0.139390 + 0.744521I
a = −2.63405 + 0.70052I
b = 1.81184 − 0.03604I
−4.64613 − 6.27110I 0.28899 + 7.66392I
u = −0.139390 − 0.744521I
a = −2.63405 − 0.70052I
b = 1.81184 + 0.03604I
−4.64613 + 6.27110I 0.28899 − 7.66392I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.233890 + 0.279365I
a = −0.14304 + 1.43122I
b = 2.18803 − 1.44880I
−1.47413 − 4.67753I 3.26707 + 4.95430I
u = −1.233890 − 0.279365I
a = −0.14304 − 1.43122I
b = 2.18803 + 1.44880I
−1.47413 + 4.67753I 3.26707 − 4.95430I
u = 1.276390 + 0.259219I
a = −0.047693 + 0.240818I
b = 0.432237 + 0.227863I
2.43611 + 5.45580I 9.29163 − 6.05568I
u = 1.276390 − 0.259219I
a = −0.047693 − 0.240818I
b = 0.432237 − 0.227863I
2.43611 − 5.45580I 9.29163 + 6.05568I
u = 1.302250 + 0.054350I
a = 0.287821 − 0.006470I
b = 0.723841 + 0.741201I
6.06693 + 0.68963I 14.6543 − 0.4108I
u = 1.302250 − 0.054350I
a = 0.287821 + 0.006470I
b = 0.723841 − 0.741201I
6.06693 − 0.68963I 14.6543 + 0.4108I
u = 1.314130 + 0.126505I
a = 0.477565 + 0.637528I
b = −0.035059 − 0.928052I
5.18133 + 4.97391I 11.84206 − 7.53779I
u = 1.314130 − 0.126505I
a = 0.477565 − 0.637528I
b = −0.035059 + 0.928052I
5.18133 − 4.97391I 11.84206 + 7.53779I
u = 0.028586 + 0.673943I
a = 2.75743 + 1.25369I
b = −1.39032 − 1.02302I
−5.31953 + 1.23761I −2.13192 − 0.75441I
u = 0.028586 − 0.673943I
a = 2.75743 − 1.25369I
b = −1.39032 + 1.02302I
−5.31953 − 1.23761I −2.13192 + 0.75441I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.306990 + 0.302770I
a = 0.54892 + 1.54089I
b = −2.14001 − 0.30607I
−0.13672 + 10.00530I 5.74688 − 9.45823I
u = 1.306990 − 0.302770I
a = 0.54892 − 1.54089I
b = −2.14001 + 0.30607I
−0.13672 − 10.00530I 5.74688 + 9.45823I
u = −1.274370 + 0.449256I
a = 0.464706 + 0.179729I
b = −0.289894 − 0.612925I
−7.05845 − 1.54689I 5.29634 + 0.63495I
u = −1.274370 − 0.449256I
a = 0.464706 − 0.179729I
b = −0.289894 + 0.612925I
−7.05845 + 1.54689I 5.29634 − 0.63495I
u = −1.292980 + 0.447168I
a = −0.48290 + 2.40208I
b = 4.44596 + 0.24017I
−10.90240 − 6.46399I 2.13256 + 3.64968I
u = −1.292980 − 0.447168I
a = −0.48290 − 2.40208I
b = 4.44596 − 0.24017I
−10.90240 + 6.46399I 2.13256 − 3.64968I
u = 1.300000 + 0.439361I
a = −0.464718 + 0.254925I
b = 0.131131 − 0.562319I
−6.86104 + 8.18007I 5.63732 − 5.18856I
u = 1.300000 − 0.439361I
a = −0.464718 − 0.254925I
b = 0.131131 + 0.562319I
−6.86104 − 8.18007I 5.63732 + 5.18856I
u = 1.313670 + 0.446415I
a = 0.61332 + 2.40054I
b = −4.34015 + 0.59580I
−10.6081 + 13.2523I 2.64420 − 8.00654I
u = 1.313670 − 0.446415I
a = 0.61332 − 2.40054I
b = −4.34015 − 0.59580I
−10.6081 − 13.2523I 2.64420 + 8.00654I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.605125
a = 0.150256
b = −0.724538
0.878305 11.8450
u = −0.357155 + 0.473523I
a = −1.032500 + 0.775837I
b = −0.077318 + 0.333065I
0.06389 − 3.04882I 6.31761 + 9.14792I
u = −0.357155 − 0.473523I
a = −1.032500 − 0.775837I
b = −0.077318 − 0.333065I
0.06389 + 3.04882I 6.31761 − 9.14792I
u = 0.135558 + 0.221933I
a = 0.04024 + 3.17604I
b = 0.547219 − 0.274542I
−1.63800 + 0.52732I −3.97358 − 0.67184I
u = 0.135558 − 0.221933I
a = 0.04024 − 3.17604I
b = 0.547219 + 0.274542I
−1.63800 − 0.52732I −3.97358 + 0.67184I
8
II. I
u
2
= hu
5
− u
3
+ b − u, u
3
+ a, u
15
− 5u
13
+ · · · − 3u − 1i
(i) Arc colorings
a
1
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
12
=
−u
−u
3
+ u
a
2
=
−u
3
−u
5
+ u
3
+ u
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
−u
−u
3
+ u
a
4
=
u
4
− u
2
+ 1
u
6
− 2u
4
+ u
2
a
11
=
u
3
− 2u
−u
3
+ u
a
8
=
u
6
− 3u
4
+ 2u
2
+ 1
−u
6
+ 2u
4
− u
2
a
10
=
u
9
− 4u
7
+ 5u
5
− 3u
−u
9
+ 3u
7
− 3u
5
+ u
a
9
=
u
12
− 5u
10
+ 9u
8
− 4u
6
− 6u
4
+ 5u
2
+ 1
−u
12
+ 4u
10
− 6u
8
+ 2u
6
+ 3u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
12
− 16u
10
− 4u
9
+ 24u
8
+ 12u
7
− 12u
5
− 28u
4
− 8u
3
+ 16u
2
+ 12u + 14
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
+ 10u
14
+ ··· + 3u + 1
c
2
, c
4
, c
5
c
6
, c
12
u
15
− 5u
13
+ ··· − 3u + 1
c
3
, c
9
(u
5
− u
4
+ u
2
+ u − 1)
3
c
7
, c
8
, c
10
c
11
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
3
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
− 10y
14
+ ··· + 23y −1
c
2
, c
4
, c
5
c
6
, c
12
y
15
− 10y
14
+ ··· + 3y −1
c
3
, c
9
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)
3
c
7
, c
8
, c
10
c
11
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
3
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.016489 + 0.918115I
a = −0.041693 + 0.773164I
b = 0.083746 − 0.505303I
−10.95830 − 3.33174I 2.08126 + 2.36228I
u = −0.016489 − 0.918115I
a = −0.041693 − 0.773164I
b = 0.083746 + 0.505303I
−10.95830 + 3.33174I 2.08126 − 2.36228I
u = −1.088950 + 0.365332I
a = 0.85527 − 1.25089I
b = −2.03946 − 0.38065I
−1.81981 + 2.21397I 3.11432 − 4.22289I
u = −1.088950 − 0.365332I
a = 0.85527 + 1.25089I
b = −2.03946 + 0.38065I
−1.81981 − 2.21397I 3.11432 + 4.22289I
u = 1.16504
a = −1.58132
b = 0.600011
0.882183 11.6090
u = 1.193940 + 0.276748I
a = −1.42761 − 1.16230I
b = 1.46394 − 1.07220I
−1.81981 + 2.21397I 3.11432 − 4.22289I
u = 1.193940 − 0.276748I
a = −1.42761 + 1.16230I
b = 1.46394 + 1.07220I
−1.81981 − 2.21397I 3.11432 + 4.22289I
u = −0.104987 + 0.642080I
a = −0.128690 + 0.243477I
b = 0.108165 + 0.318259I
−1.81981 − 2.21397I 3.11432 + 4.22289I
u = −0.104987 − 0.642080I
a = −0.128690 − 0.243477I
b = 0.108165 − 0.318259I
−1.81981 + 2.21397I 3.11432 − 4.22289I
u = −1.269280 + 0.467945I
a = 1.21110 − 2.15922I
b = −3.35937 − 1.81738I
−10.95830 + 3.33174I 2.08126 − 2.36228I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.269280 − 0.467945I
a = 1.21110 + 2.15922I
b = −3.35937 + 1.81738I
−10.95830 − 3.33174I 2.08126 + 2.36228I
u = 1.285770 + 0.450170I
a = −1.34395 − 2.14145I
b = 3.15926 − 2.07048I
−10.95830 + 3.33174I 2.08126 − 2.36228I
u = 1.285770 − 0.450170I
a = −1.34395 + 2.14145I
b = 3.15926 + 2.07048I
−10.95830 − 3.33174I 2.08126 + 2.36228I
u = −0.582519 + 0.134108I
a = 0.166235 − 0.134108I
b = −0.716289 + 0.199149I
0.882183 11.60884 + 0.I
u = −0.582519 − 0.134108I
a = 0.166235 + 0.134108I
b = −0.716289 − 0.199149I
0.882183 11.60884 + 0.I
13
III. I
u
3
= hb + 1, a − 1, u + 1i
(i) Arc colorings
a
1
=
0
−1
a
7
=
1
0
a
6
=
1
1
a
12
=
1
0
a
2
=
1
−1
a
5
=
0
1
a
3
=
1
0
a
4
=
1
0
a
11
=
1
0
a
8
=
1
0
a
10
=
1
0
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
u − 1
c
3
, c
7
, c
8
c
9
, c
10
, c
11
u
c
4
, c
5
, c
6
u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
12
y − 1
c
3
, c
7
, c
8
c
9
, c
10
, c
11
y
16
(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
0 0
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
15
+ 10u
14
+ ··· + 3u + 1)(u
35
+ 20u
34
+ ··· + 3u + 1)
c
2
(u − 1)(u
15
− 5u
13
+ ··· − 3u + 1)(u
35
− 2u
34
+ ··· + 3u − 1)
c
3
, c
9
u(u
5
− u
4
+ u
2
+ u − 1)
3
(u
35
+ 2u
34
+ ··· − 2u − 2)
c
4
(u + 1)(u
15
− 5u
13
+ ··· − 3u + 1)(u
35
− 2u
34
+ ··· + 3u − 1)
c
5
, c
6
(u + 1)(u
15
− 5u
13
+ ··· − 3u + 1)(u
35
+ 2u
34
+ ··· − u − 1)
c
7
, c
8
, c
10
c
11
u(u
5
− u
4
+ ··· + 3u − 1)
3
(u
35
− 6u
34
+ ··· + 8u − 4)
c
12
(u − 1)(u
15
− 5u
13
+ ··· − 3u + 1)(u
35
+ 2u
34
+ ··· − u − 1)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y − 1)(y
15
− 10y
14
+ ··· + 23y − 1)(y
35
− 8y
34
+ ··· + 35y − 1)
c
2
, c
4
(y − 1)(y
15
− 10y
14
+ ··· + 3y − 1)(y
35
− 20y
34
+ ··· + 3y − 1)
c
3
, c
9
y(y
5
− y
4
+ ··· + 3y − 1)
3
(y
35
− 6y
34
+ ··· + 8y − 4)
c
5
, c
6
, c
12
(y − 1)(y
15
− 10y
14
+ ··· + 3y − 1)(y
35
− 28y
34
+ ··· − 13y − 1)
c
7
, c
8
, c
10
c
11
y(y
5
+ 7y
4
+ ··· + 3y − 1)
3
(y
35
+ 42y
34
+ ··· − 136y − 16)
19
