12a
0876
(K12a
0876
)
A knot diagram
1
Linearized knot diagam
4 6 7 10 2 3 11 12 5 1 8 9
Solving Sequence
2,5
6 3
7,10
4 1 11 9 12 8
c
5
c
2
c
6
c
4
c
1
c
10
c
9
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
13
− 8u
11
+ 23u
9
− u
8
− 28u
7
+ 5u
6
+ 14u
5
− 7u
4
− 4u
3
+ 2u
2
+ b − u − 1, −u
8
+ 5u
6
− 7u
4
+ 2u
2
+ a − 1,
u
14
+ u
13
− 8u
12
− 7u
11
+ 24u
10
+ 16u
9
− 34u
8
− 11u
7
+ 26u
6
− 2u
5
− 13u
4
+ u
3
+ 2u
2
− 3u − 1i
I
u
2
= h2u
41
+ 2u
40
+ ··· − u
2
+ b, −3u
41
− 4u
40
+ ··· + a + 1, u
42
+ 2u
41
+ ··· + u + 1i
I
u
3
= hb, a + 1, u
2
− u − 1i
I
u
4
= hb, a + u − 2, u
2
− u − 1i
* 4 irreducible components of dim
C
= 0, with total 60 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
= hu
13
−8u
11
+· · ·+b −1, −u
8
+5u
6
−7u
4
+2u
2
+a−1, u
14
+u
13
+· · ·−3u −1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
−u
13
+ 8u
11
+ ··· + u + 1
a
4
=
u
3
− 2u
u
5
− 3u
3
+ u
a
1
=
u
7
− 4u
5
+ 4u
3
u
9
− 5u
7
+ 7u
5
− 2u
3
+ u
a
11
=
u
11
− 6u
9
+ u
8
+ 12u
7
− 5u
6
− 8u
5
+ 7u
4
− 2u
2
+ 1
u
11
− 6u
9
+ u
8
+ 12u
7
− 5u
6
− 9u
5
+ 7u
4
+ 2u
3
− 2u
2
+ u + 1
a
9
=
−u
13
+ 8u
11
+ ··· + u + 2
−u
13
+ 8u
11
+ ··· + u + 1
a
12
=
u
13
− 7u
11
+ ··· − u − 1
u
13
− 7u
11
+ ··· − u − 1
a
8
=
−u
12
+ 6u
10
− u
9
− 12u
8
+ 5u
7
+ 8u
6
− 7u
5
+ 2u
3
− u
2
− u + 1
−u
12
+ 6u
10
− u
9
− 12u
8
+ 5u
7
+ 9u
6
− 7u
5
− 3u
4
+ 2u
3
+ u
2
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
13
+14u
11
−2u
10
−32u
9
+16u
8
+20u
7
−40u
6
+14u
5
+30u
4
−18u
3
+2u
2
+12u −12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
14
− 3u
13
+ ··· − 3u − 1
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
11
, c
12
u
14
+ u
13
+ ··· − 3u − 1
c
4
, c
9
u
14
− 5u
13
+ ··· − 8u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
14
+ 7y
13
+ ··· − 29y + 1
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
11
, c
12
y
14
− 17y
13
+ ··· − 13y + 1
c
4
, c
9
y
14
+ 5y
13
+ ··· − 64y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.680303 + 0.531876I
a = −1.24132 + 1.66316I
b = −0.572300 − 1.150040I
1.04590 − 7.72861I −13.6329 + 9.6019I
u = 0.680303 − 0.531876I
a = −1.24132 − 1.66316I
b = −0.572300 + 1.150040I
1.04590 + 7.72861I −13.6329 − 9.6019I
u = −0.593521 + 0.378079I
a = 0.054243 − 0.515402I
b = −0.834976 − 0.363198I
−1.38750 + 2.49320I −16.0799 − 7.8719I
u = −0.593521 − 0.378079I
a = 0.054243 + 0.515402I
b = −0.834976 + 0.363198I
−1.38750 − 2.49320I −16.0799 + 7.8719I
u = 0.303532 + 0.566158I
a = 0.62968 − 1.83164I
b = −0.251015 + 1.107770I
3.30391 + 0.11980I −7.23583 + 2.81079I
u = 0.303532 − 0.566158I
a = 0.62968 + 1.83164I
b = −0.251015 − 1.107770I
3.30391 − 0.11980I −7.23583 − 2.81079I
u = −1.45549 + 0.12558I
a = 0.560799 + 0.786391I
b = 0.204002 − 1.257830I
−8.06473 + 4.40167I −16.0274 − 3.4872I
u = −1.45549 − 0.12558I
a = 0.560799 − 0.786391I
b = 0.204002 + 1.257830I
−8.06473 − 4.40167I −16.0274 + 3.4872I
u = 1.48768
a = 0.650213
b = 1.05020
−12.8678 −19.3260
u = 1.58880 + 0.12925I
a = −0.470390 + 0.414195I
b = −1.052550 + 0.599886I
−16.3668 − 6.3822I −20.6641 + 3.1830I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.58880 − 0.12925I
a = −0.470390 − 0.414195I
b = −1.052550 − 0.599886I
−16.3668 + 6.3822I −20.6641 − 3.1830I
u = −1.61264 + 0.16202I
a = −1.29225 − 0.74678I
b = −0.754602 + 1.160230I
−14.5457 + 12.9375I −19.3006 − 6.7062I
u = −1.61264 − 0.16202I
a = −1.29225 + 0.74678I
b = −0.754602 − 1.160230I
−14.5457 − 12.9375I −19.3006 + 6.7062I
u = −0.309637
a = 0.868272
b = 0.472677
−0.638892 −14.7920
6
II.
I
u
2
= h2u
41
+2u
40
+ · · ·−u
2
+ b, −3u
41
−4u
40
+ · · ·+a+1, u
42
+2u
41
+ · · ·+u+1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
3u
41
+ 4u
40
+ ··· − 6u − 1
−2u
41
− 2u
40
+ ··· − 6u
3
+ u
2
a
4
=
u
3
− 2u
u
5
− 3u
3
+ u
a
1
=
u
7
− 4u
5
+ 4u
3
u
9
− 5u
7
+ 7u
5
− 2u
3
+ u
a
11
=
u
40
− u
39
+ ··· − 5u − 1
−3u
41
− 3u
40
+ ··· − 5u
3
+ 2u
2
a
9
=
u
41
+ 2u
40
+ ··· − 6u − 1
−2u
41
− 2u
40
+ ··· − 6u
3
+ u
2
a
12
=
−u
41
− 2u
40
+ ··· − 2u
2
+ u
−u
14
+ 8u
12
+ ··· + u
2
+ 2u
a
8
=
2u
41
+ 3u
40
+ ··· − u + 1
u
41
+ u
40
+ ··· + 7u
3
− 2u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
40
+ 3u
39
+ ··· + 18u
2
− 15
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
42
− 12u
41
+ ··· + 53u + 31
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
11
, c
12
u
42
+ 2u
41
+ ··· + u + 1
c
4
, c
9
(u
21
+ 2u
20
+ ··· + 5u + 2)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
42
− 12y
41
+ ··· + 34825y + 961
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
11
, c
12
y
42
− 48y
41
+ ··· − 11y + 1
c
4
, c
9
(y
21
+ 10y
20
+ ··· − 15y −4)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.952260 + 0.275050I
a = −0.237674 + 0.141836I
b = 0.469429 + 1.026280I
−8.66083 − 3.12379I −18.1716 + 1.7818I
u = −0.952260 − 0.275050I
a = −0.237674 − 0.141836I
b = 0.469429 − 1.026280I
−8.66083 + 3.12379I −18.1716 − 1.7818I
u = −0.892649 + 0.147229I
a = 0.102532 − 0.122821I
b = −0.268462 − 0.851142I
−1.46292 − 1.33471I −14.9864 + 4.7477I
u = −0.892649 − 0.147229I
a = 0.102532 + 0.122821I
b = −0.268462 + 0.851142I
−1.46292 + 1.33471I −14.9864 − 4.7477I
u = 0.723689 + 0.540993I
a = 1.26266 − 1.59018I
b = 0.677487 + 1.162350I
−6.63828 − 10.29320I −16.6887 + 8.0442I
u = 0.723689 − 0.540993I
a = 1.26266 + 1.59018I
b = 0.677487 − 1.162350I
−6.63828 + 10.29320I −16.6887 − 8.0442I
u = 0.622201 + 0.517014I
a = 1.19658 − 1.76438I
b = 0.436892 + 1.122040I
2.37193 − 3.84440I −10.04174 + 4.38533I
u = 0.622201 − 0.517014I
a = 1.19658 + 1.76438I
b = 0.436892 − 1.122040I
2.37193 + 3.84440I −10.04174 − 4.38533I
u = −0.650081 + 0.454963I
a = −0.175433 + 0.522705I
b = 0.982337 + 0.491258I
−8.77344 + 4.23823I −18.3836 − 4.9951I
u = −0.650081 − 0.454963I
a = −0.175433 − 0.522705I
b = 0.982337 − 0.491258I
−8.77344 − 4.23823I −18.3836 + 4.9951I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.452901 + 0.570594I
a = −0.88626 + 1.82784I
b = −0.052121 − 1.208570I
−1.92244 − 1.93968I −12.20153 + 3.66263I
u = 0.452901 − 0.570594I
a = −0.88626 − 1.82784I
b = −0.052121 + 1.208570I
−1.92244 + 1.93968I −12.20153 − 3.66263I
u = 0.631403 + 0.254962I
a = 1.91563 − 2.30921I
b = 0.397322 + 0.594617I
−10.10480 − 0.64503I −18.1436 + 8.7498I
u = 0.631403 − 0.254962I
a = 1.91563 + 2.30921I
b = 0.397322 − 0.594617I
−10.10480 + 0.64503I −18.1436 − 8.7498I
u = 0.178370 + 0.653047I
a = 0.50210 − 1.62752I
b = −0.580700 + 1.149510I
−5.02807 + 6.26735I −13.39857 − 3.31929I
u = 0.178370 − 0.653047I
a = 0.50210 + 1.62752I
b = −0.580700 − 1.149510I
−5.02807 − 6.26735I −13.39857 + 3.31929I
u = 0.543160 + 0.383707I
a = −1.24755 + 2.15130I
b = −0.268462 − 0.851142I
−1.46292 − 1.33471I −14.9864 + 4.7477I
u = 0.543160 − 0.383707I
a = −1.24755 − 2.15130I
b = −0.268462 + 0.851142I
−1.46292 + 1.33471I −14.9864 − 4.7477I
u = 0.227652 + 0.609745I
a = −0.53434 + 1.72086I
b = 0.436892 − 1.122040I
2.37193 + 3.84440I −10.04174 − 4.38533I
u = 0.227652 − 0.609745I
a = −0.53434 − 1.72086I
b = 0.436892 + 1.122040I
2.37193 − 3.84440I −10.04174 + 4.38533I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.409680 + 0.044575I
a = −0.180441 − 0.797743I
b = −0.052121 + 1.208570I
−1.92244 + 1.93968I 0
u = −1.409680 − 0.044575I
a = −0.180441 + 0.797743I
b = −0.052121 − 1.208570I
−1.92244 − 1.93968I 0
u = −0.226402 + 0.475799I
a = −0.056046 − 1.133970I
b = −0.885131 + 0.313438I
−7.56038 − 0.95789I −15.2596 − 1.5508I
u = −0.226402 − 0.475799I
a = −0.056046 + 1.133970I
b = −0.885131 − 0.313438I
−7.56038 + 0.95789I −15.2596 + 1.5508I
u = 1.56042 + 0.06421I
a = −0.462955 + 0.220565I
b = −0.885131 + 0.313438I
−7.56038 − 0.95789I 0
u = 1.56042 − 0.06421I
a = −0.462955 − 0.220565I
b = −0.885131 − 0.313438I
−7.56038 + 0.95789I 0
u = −1.56589 + 0.11035I
a = 1.08354 + 1.11273I
b = 0.469429 − 1.026280I
−8.66083 + 3.12379I 0
u = −1.56589 − 0.11035I
a = 1.08354 − 1.11273I
b = 0.469429 + 1.026280I
−8.66083 − 3.12379I 0
u = 1.57529 + 0.10471I
a = 0.470499 − 0.343906I
b = 0.982337 − 0.491258I
−8.77344 − 4.23823I 0
u = 1.57529 − 0.10471I
a = 0.470499 + 0.343906I
b = 0.982337 + 0.491258I
−8.77344 + 4.23823I 0
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.359452 + 0.212787I
a = 0.467612 + 0.591791I
b = 0.596034
−0.606975 −13.01685 + 0.I
u = −0.359452 − 0.212787I
a = 0.467612 − 0.591791I
b = 0.596034
−0.606975 −13.01685 + 0.I
u = −1.57630 + 0.14785I
a = −1.13809 − 0.86977I
b = −0.580700 + 1.149510I
−5.02807 + 6.26735I 0
u = −1.57630 − 0.14785I
a = −1.13809 + 0.86977I
b = −0.580700 − 1.149510I
−5.02807 − 6.26735I 0
u = −1.58753 + 0.08171I
a = −1.29868 − 1.37872I
b = −0.475070 + 0.853809I
−17.7147 + 1.9468I 0
u = −1.58753 − 0.08171I
a = −1.29868 + 1.37872I
b = −0.475070 − 0.853809I
−17.7147 − 1.9468I 0
u = −1.59635 + 0.15789I
a = 1.22706 + 0.79548I
b = 0.677487 − 1.162350I
−6.63828 + 10.29320I 0
u = −1.59635 − 0.15789I
a = 1.22706 − 0.79548I
b = 0.677487 + 1.162350I
−6.63828 − 10.29320I 0
u = 1.63726 + 0.03700I
a = 0.176213 − 0.298794I
b = 0.397322 − 0.594617I
−10.10480 + 0.64503I 0
u = 1.63726 − 0.03700I
a = 0.176213 + 0.298794I
b = 0.397322 + 0.594617I
−10.10480 − 0.64503I 0
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.66424 + 0.05653I
a = −0.186954 + 0.414396I
b = −0.475070 + 0.853809I
−17.7147 + 1.9468I 0
u = 1.66424 − 0.05653I
a = −0.186954 − 0.414396I
b = −0.475070 − 0.853809I
−17.7147 − 1.9468I 0
14
III. I
u
3
= hb, a + 1, u
2
− u − 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u + 1
a
3
=
−u
−u − 1
a
7
=
−u
−u
a
10
=
−1
0
a
4
=
1
0
a
1
=
u
u
a
11
=
−u − 2
−u − 1
a
9
=
−1
0
a
12
=
2u
u
a
8
=
2u + 1
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −20
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
, c
10
u
2
+ u − 1
c
4
, c
9
u
2
c
5
, c
6
, c
11
c
12
u
2
− u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
4
, c
9
y
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −1.00000
b = 0
−1.97392 −20.0000
u = 1.61803
a = −1.00000
b = 0
−17.7653 −20.0000
18
IV. I
u
4
= hb, a + u − 2, u
2
− u − 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u + 1
a
3
=
−u
−u − 1
a
7
=
−u
−u
a
10
=
−u + 2
0
a
4
=
1
0
a
1
=
u
u
a
11
=
−u + 3
1
a
9
=
−u + 2
0
a
12
=
3u − 3
u
a
8
=
2u − 4
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −15
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
, c
10
u
2
+ u − 1
c
4
, c
9
u
2
c
5
, c
6
, c
11
c
12
u
2
− u − 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
4
, c
9
y
2
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 2.61803
b = 0
−9.86960 −15.0000
u = 1.61803
a = 0.381966
b = 0
−9.86960 −15.0000
22
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
10
((u
2
+ u − 1)
2
)(u
14
− 3u
13
+ ··· − 3u − 1)(u
42
− 12u
41
+ ··· + 53u + 31)
c
2
, c
3
, c
7
c
8
((u
2
+ u − 1)
2
)(u
14
+ u
13
+ ··· − 3u − 1)(u
42
+ 2u
41
+ ··· + u + 1)
c
4
, c
9
u
4
(u
14
− 5u
13
+ ··· − 8u + 4)(u
21
+ 2u
20
+ ··· + 5u + 2)
2
c
5
, c
6
, c
11
c
12
((u
2
− u − 1)
2
)(u
14
+ u
13
+ ··· − 3u − 1)(u
42
+ 2u
41
+ ··· + u + 1)
23
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
10
((y
2
− 3y + 1)
2
)(y
14
+ 7y
13
+ ··· − 29y + 1)
· (y
42
− 12y
41
+ ··· + 34825y + 961)
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
11
, c
12
((y
2
− 3y + 1)
2
)(y
14
− 17y
13
+ ··· − 13y + 1)
· (y
42
− 48y
41
+ ··· − 11y + 1)
c
4
, c
9
y
4
(y
14
+ 5y
13
+ ··· − 64y + 16)(y
21
+ 10y
20
+ ··· − 15y −4)
2
24
