12a
0701
(K12a
0701
)
A knot diagram
1
Linearized knot diagam
3 8 6 10 1 11 2 12 4 9 7 5
Solving Sequence
2,7
8
3,11
12 9 1 6 4 5 10
c
7
c
2
c
11
c
8
c
1
c
6
c
3
c
5
c
10
c
4
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= ⟨−7.70053 × 10
53
u
39
− 1.74213 × 10
54
u
38
+ ··· + 2.50493 × 10
54
b − 1.01102 × 10
54
,
5.17594 × 10
54
u
39
+ 1.38121 × 10
55
u
38
+ ··· + 1.08547 × 10
55
a − 8.78741 × 10
53
,
5u
40
+ 15u
39
+ ··· + 4u + 13⟩
I
u
2
= ⟨2u
31
a + 3u
31
+ ··· − 2a + 4, 2u
30
a + 16u
31
+ ··· + 2a − 15, u
32
− u
31
+ ··· − 2u + 1⟩
I
u
3
= ⟨−u
3
+ b, u
3
+ u
2
+ 2a − 2u, u
4
− u
2
+ 1⟩
I
u
4
= ⟨b, a − 1, u
4
− u
3
+ 1⟩
I
u
5
= ⟨b + 1, −u
3
+ a − 1, u
4
− u
3
+ 1⟩
I
u
6
= ⟨b, a − 1, u + 1⟩
I
u
7
= ⟨b + 1, a, u + 1⟩
I
u
8
= ⟨−u
3
+ b, u
3
+ 2a − 2u − 1, u
4
− u
2
+ 1⟩
I
u
9
= ⟨b + 1, u
5
a − u
5
− u
3
a + 2u
3
+ au − u + 1⟩
* 8 irreducible components of dim
C
= 0, with total 122 representations.
* 1 irreducible components of dim
C
= 1
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
= ⟨−7.70 × 10
53
u
39
− 1.74 × 10
54
u
38
+ · · · + 2.50 × 10
54
b − 1.01 ×
10
54
, 5.18 × 10
54
u
39
+ 1.38 × 10
55
u
38
+ · · · + 1.09 × 10
55
a − 8.79 ×
10
53
, 5u
40
+ 15u
39
+ · · · + 4u + 13⟩
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
11
=
−0.476839u
39
− 1.27246u
38
+ ··· − 5.39284u + 0.0809549
0.307415u
39
+ 0.695481u
38
+ ··· + 2.07333u + 0.403613
a
12
=
−0.169424u
39
− 0.576975u
38
+ ··· − 3.31951u + 0.484568
0.307415u
39
+ 0.695481u
38
+ ··· + 2.07333u + 0.403613
a
9
=
−0.211994u
39
− 0.480895u
38
+ ··· − 3.96053u − 0.213115
0.0500582u
39
+ 0.116432u
38
+ ··· + 0.995645u + 0.307144
a
1
=
u
3
u
5
− u
3
+ u
a
6
=
−0.191581u
39
− 0.280282u
38
+ ··· + 5.99057u + 1.63083
−0.0736869u
39
− 0.292199u
38
+ ··· − 0.484607u − 0.887781
a
4
=
0.231719u
39
+ 0.602871u
38
+ ··· + 1.98498u − 0.495273
0.0632015u
39
+ 0.154197u
38
+ ··· + 0.0133278u + 0.385259
a
5
=
−0.245279u
39
− 0.480339u
38
+ ··· + 6.44253u + 1.18353
−0.133010u
39
− 0.382392u
38
+ ··· − 0.608099u − 1.13846
a
10
=
−0.660060u
39
− 1.59385u
38
+ ··· − 8.11569u − 0.968106
0.246464u
39
+ 0.516908u
38
+ ··· + 1.74196u + 0.398054
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0.355272u
39
+ 2.47238u
38
+ ··· + 1.40321u + 8.32948
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
25(25u
40
+ 355u
39
+ ··· − 2766u + 169)
c
2
, c
7
5(5u
40
− 15u
39
+ ··· − 4u + 13)
c
3
, c
8
64(64u
40
+ 192u
39
+ ··· + 80u + 25)
c
4
, c
9
5(5u
40
− 15u
39
+ ··· − 58u + 13)
c
5
, c
6
, c
11
c
12
u
40
+ 4u
39
+ ··· + 36u + 4
c
10
25(25u
40
− 455u
39
+ ··· + 718u + 169)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
625(625y
40
+ 14425y
39
+ ··· − 2166706y + 28561)
c
2
, c
7
25(25y
40
− 355y
39
+ ··· + 2766y + 169)
c
3
, c
8
4096(4096y
40
− 53248y
39
+ ··· − 13450y + 625)
c
4
, c
9
25(25y
40
− 455y
39
+ ··· + 718y + 169)
c
5
, c
6
, c
11
c
12
y
40
− 12y
39
+ ··· − 696y + 16
c
10
625(625y
40
+ 4425y
39
+ ··· − 2689202y + 28561)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.970531 + 0.178964I
a = −0.833678 + 1.090430I
b = 0.657380 + 0.842879I
−4.36949 + 3.79276I −0.84937 − 5.47947I
u = −0.970531 − 0.178964I
a = −0.833678 − 1.090430I
b = 0.657380 − 0.842879I
−4.36949 − 3.79276I −0.84937 + 5.47947I
u = −0.800492 + 0.625954I
a = −1.186250 − 0.317798I
b = 0.13126 + 1.45702I
−0.377065 − 0.505610I 13.55208 + 2.04018I
u = −0.800492 − 0.625954I
a = −1.186250 + 0.317798I
b = 0.13126 − 1.45702I
−0.377065 + 0.505610I 13.55208 − 2.04018I
u = −0.916490 + 0.502951I
a = −0.335321 + 0.936812I
b = 0.164319 + 0.087454I
0.08890 + 4.10799I 9.67304 − 7.45476I
u = −0.916490 − 0.502951I
a = −0.335321 − 0.936812I
b = 0.164319 − 0.087454I
0.08890 − 4.10799I 9.67304 + 7.45476I
u = 0.957016 + 0.438264I
a = 0.286645 + 0.087328I
b = 0.282628 + 0.561245I
−1.45001 − 1.63863I 1.206223 + 0.491984I
u = 0.957016 − 0.438264I
a = 0.286645 − 0.087328I
b = 0.282628 − 0.561245I
−1.45001 + 1.63863I 1.206223 − 0.491984I
u = −0.911045 + 0.633331I
a = −0.896481 − 0.513913I
b = −0.33799 + 1.45617I
−0.73084 + 5.44478I 10.81898 − 9.02994I
u = −0.911045 − 0.633331I
a = −0.896481 + 0.513913I
b = −0.33799 − 1.45617I
−0.73084 − 5.44478I 10.81898 + 9.02994I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.951215 + 0.571492I
a = 0.650883 − 0.383013I
b = 0.428609 + 1.127700I
−2.13110 − 1.53175I 4.45404 + 4.45582I
u = 0.951215 − 0.571492I
a = 0.650883 + 0.383013I
b = 0.428609 − 1.127700I
−2.13110 + 1.53175I 4.45404 − 4.45582I
u = 0.873906 + 0.143183I
a = 1.14313 + 1.17285I
b = −0.567731 + 0.944214I
−3.47850 + 1.11172I 0.88815 − 1.52267I
u = 0.873906 − 0.143183I
a = 1.14313 − 1.17285I
b = −0.567731 − 0.944214I
−3.47850 − 1.11172I 0.88815 + 1.52267I
u = 0.719139 + 0.511234I
a = 1.309510 + 0.000140I
b = −0.326936 + 1.143020I
−1.35289 − 2.89894I 8.70495 − 0.09481I
u = 0.719139 − 0.511234I
a = 1.309510 − 0.000140I
b = −0.326936 − 1.143020I
−1.35289 + 2.89894I 8.70495 + 0.09481I
u = −0.588353 + 0.956868I
a = −1.60068 + 0.27140I
b = 1.37609 − 0.52776I
8.8274 − 13.1342I 12.15484 + 6.65872I
u = −0.588353 − 0.956868I
a = −1.60068 − 0.27140I
b = 1.37609 + 0.52776I
8.8274 + 13.1342I 12.15484 − 6.65872I
u = 0.618330 + 0.966474I
a = 1.58166 + 0.29564I
b = −1.298800 − 0.465176I
6.35181 + 7.20509I 10.12101 − 3.51952I
u = 0.618330 − 0.966474I
a = 1.58166 − 0.29564I
b = −1.298800 + 0.465176I
6.35181 − 7.20509I 10.12101 + 3.51952I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.338231 + 1.117800I
a = −1.43704 + 0.00820I
b = 1.193230 + 0.235279I
7.17464 + 7.33287I 14.9640 − 8.5081I
u = −0.338231 − 1.117800I
a = −1.43704 − 0.00820I
b = 1.193230 − 0.235279I
7.17464 − 7.33287I 14.9640 + 8.5081I
u = −0.602564 + 1.040950I
a = −1.52308 + 0.24652I
b = 1.351000 − 0.264366I
12.76580 − 3.62003I 16.0601 + 2.4487I
u = −0.602564 − 1.040950I
a = −1.52308 − 0.24652I
b = 1.351000 + 0.264366I
12.76580 + 3.62003I 16.0601 − 2.4487I
u = −1.248820 + 0.245961I
a = −0.407145 + 0.707475I
b = 0.988408 + 0.451580I
−1.76098 + 5.75545I 3.78884 − 7.25461I
u = −1.248820 − 0.245961I
a = −0.407145 − 0.707475I
b = 0.988408 − 0.451580I
−1.76098 − 5.75545I 3.78884 + 7.25461I
u = 1.287890 + 0.155559I
a = 0.197100 + 0.675611I
b = −1.172550 + 0.479770I
1.16532 − 11.28520I 7.60677 + 9.06404I
u = 1.287890 − 0.155559I
a = 0.197100 − 0.675611I
b = −1.172550 − 0.479770I
1.16532 + 11.28520I 7.60677 − 9.06404I
u = −1.108950 + 0.734483I
a = 1.61641 − 1.35965I
b = −1.38090 − 0.60199I
7.2067 + 19.3213I 10.0095 − 10.7971I
u = −1.108950 − 0.734483I
a = 1.61641 + 1.35965I
b = −1.38090 + 0.60199I
7.2067 − 19.3213I 10.0095 + 10.7971I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.100690 + 0.748356I
a = −1.62760 − 1.23899I
b = 1.31857 − 0.55246I
4.8369 − 13.4697I 7.95384 + 7.41378I
u = 1.100690 − 0.748356I
a = −1.62760 + 1.23899I
b = 1.31857 + 0.55246I
4.8369 + 13.4697I 7.95384 − 7.41378I
u = −0.535060 + 0.369159I
a = 0.693924 + 0.131388I
b = −0.374703 + 0.056422I
1.022250 − 0.157854I 11.13565 + 1.03898I
u = −0.535060 − 0.369159I
a = 0.693924 − 0.131388I
b = −0.374703 − 0.056422I
1.022250 + 0.157854I 11.13565 − 1.03898I
u = −1.129960 + 0.773268I
a = 1.39080 − 1.14075I
b = −1.36313 − 0.39095I
11.1026 + 10.1596I 13.5699 − 6.4088I
u = −1.129960 − 0.773268I
a = 1.39080 + 1.14075I
b = −1.36313 + 0.39095I
11.1026 − 10.1596I 13.5699 + 6.4088I
u = 1.060210 + 0.886721I
a = −1.48433 − 0.62286I
b = 1.071830 − 0.283744I
2.70642 − 8.04384I 9.5123 + 11.1353I
u = 1.060210 − 0.886721I
a = −1.48433 + 0.62286I
b = 1.071830 + 0.283744I
2.70642 + 8.04384I 9.5123 − 11.1353I
u = 0.082088 + 0.218378I
a = 1.80768 − 2.55183I
b = −0.140579 + 0.756809I
−1.53964 − 2.22484I 2.67513 + 4.18107I
u = 0.082088 − 0.218378I
a = 1.80768 + 2.55183I
b = −0.140579 − 0.756809I
−1.53964 + 2.22484I 2.67513 − 4.18107I
8
II. I
u
2
=
⟨2u
31
a+3u
31
+· · ·−2a+4, 2u
30
a+16u
31
+· · ·+2a−15, u
32
−u
31
+· · ·−2u+1⟩
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
11
=
a
−2u
31
a − 3u
31
+ ··· + 2a − 4
a
12
=
−2u
31
a − 3u
31
+ ··· + 3a − 4
−2u
31
a − 3u
31
+ ··· + 2a − 4
a
9
=
u
31
a +
9
2
u
31
+ ··· − 9a −
21
2
2u
31
a + u
31
+ ··· − 4a − 3
a
1
=
u
3
u
5
− u
3
+ u
a
6
=
3u
31
a − 4u
31
+ ··· + 4a + 12
−u
31
a − 2u
31
+ ··· + 2a + 3
a
4
=
u
31
a −
13
2
u
31
+ ··· − 3a +
35
2
2u
31
a − u
31
+ ··· + a + 9
a
5
=
3u
31
a − 4u
31
+ ··· + 4a + 11
1
a
10
=
−3u
31
a −
21
2
u
31
+ ··· − a −
5
2
u
−2u
31
a − 4u
31
+ ··· + 3u − 6
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
30
− 20u
28
+ 4u
27
+ 68u
26
− 20u
25
− 156u
24
+ 68u
23
+ 276u
22
− 160u
21
− 380u
20
+
292u
19
+ 404u
18
− 428u
17
− 328u
16
+ 504u
15
+ 160u
14
− 496u
13
+ 8u
12
+ 392u
11
−
124u
10
− 252u
9
+ 156u
8
+ 120u
7
− 116u
6
− 28u
5
+ 64u
4
− 4u
3
− 16u
2
+ 12u + 10
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
32
+ 11u
31
+ ··· + 2u + 1)
2
c
2
, c
7
(u
32
+ u
31
+ ··· + 2u + 1)
2
c
3
, c
8
4(4u
64
+ 56u
63
+ ··· + 9.68814 × 10
7
u + 1.07057 × 10
7
)
c
4
, c
9
(u
32
+ u
31
+ ··· − u
2
+ 1)
2
c
5
, c
6
, c
11
c
12
u
64
+ 4u
63
+ ··· + 9164u + 2061
c
10
(u
32
− 15u
31
+ ··· − 2u + 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
32
+ 21y
31
+ ··· + 2y + 1)
2
c
2
, c
7
(y
32
− 11y
31
+ ··· − 2y + 1)
2
c
3
, c
8
16
· (16y
64
− 592y
63
+ ··· − 2588329366713886y + 114613061651001)
c
4
, c
9
(y
32
− 15y
31
+ ··· − 2y + 1)
2
c
5
, c
6
, c
11
c
12
y
64
− 44y
63
+ ··· − 8369050y + 4247721
c
10
(y
32
+ 5y
31
+ ··· + 2y + 1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.613006 + 0.792175I
a = −0.580332 + 0.498725I
b = −0.046189 − 1.144740I
4.35959 + 7.30693I 10.17644 − 4.86883I
u = 0.613006 + 0.792175I
a = −1.55904 − 0.63002I
b = 1.43094 + 0.54420I
4.35959 + 7.30693I 10.17644 − 4.86883I
u = 0.613006 − 0.792175I
a = −0.580332 − 0.498725I
b = −0.046189 + 1.144740I
4.35959 − 7.30693I 10.17644 + 4.86883I
u = 0.613006 − 0.792175I
a = −1.55904 + 0.63002I
b = 1.43094 − 0.54420I
4.35959 − 7.30693I 10.17644 + 4.86883I
u = 0.674958 + 0.742403I
a = −0.231789 + 0.247033I
b = −0.519927 − 0.890443I
6.92548 + 0.05779I 13.67435 + 0.61686I
u = 0.674958 + 0.742403I
a = −1.89732 − 0.65378I
b = 1.47394 + 0.15920I
6.92548 + 0.05779I 13.67435 + 0.61686I
u = 0.674958 − 0.742403I
a = −0.231789 − 0.247033I
b = −0.519927 + 0.890443I
6.92548 − 0.05779I 13.67435 − 0.61686I
u = 0.674958 − 0.742403I
a = −1.89732 + 0.65378I
b = 1.47394 − 0.15920I
6.92548 − 0.05779I 13.67435 − 0.61686I
u = −0.600521 + 0.762759I
a = 0.636612 + 0.360731I
b = 0.015937 − 0.912614I
2.30027 − 2.26361I 6.98106 + 0.67006I
u = −0.600521 + 0.762759I
a = 1.58656 − 0.49003I
b = −1.282540 + 0.447749I
2.30027 − 2.26361I 6.98106 + 0.67006I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.600521 − 0.762759I
a = 0.636612 − 0.360731I
b = 0.015937 + 0.912614I
2.30027 + 2.26361I 6.98106 − 0.67006I
u = −0.600521 − 0.762759I
a = 1.58656 + 0.49003I
b = −1.282540 − 0.447749I
2.30027 + 2.26361I 6.98106 − 0.67006I
u = 0.849583 + 0.407230I
a = −0.12171 + 4.60078I
b = 1.097710 + 0.061840I
3.18087 − 4.15286I 5.98714 + 7.18864I
u = 0.849583 + 0.407230I
a = −5.27629 + 1.13566I
b = −0.896174 + 0.115665I
3.18087 − 4.15286I 5.98714 + 7.18864I
u = 0.849583 − 0.407230I
a = −0.12171 − 4.60078I
b = 1.097710 − 0.061840I
3.18087 + 4.15286I 5.98714 − 7.18864I
u = 0.849583 − 0.407230I
a = −5.27629 − 1.13566I
b = −0.896174 − 0.115665I
3.18087 + 4.15286I 5.98714 − 7.18864I
u = 1.093530 + 0.032199I
a = −0.303475 + 0.746533I
b = 0.432660 + 0.694262I
−3.44018 − 1.36697I 0.099351 + 0.550230I
u = 1.093530 + 0.032199I
a = 0.132205 − 0.469626I
b = 0.906765 − 0.547724I
−3.44018 − 1.36697I 0.099351 + 0.550230I
u = 1.093530 − 0.032199I
a = −0.303475 − 0.746533I
b = 0.432660 − 0.694262I
−3.44018 + 1.36697I 0.099351 − 0.550230I
u = 1.093530 − 0.032199I
a = 0.132205 + 0.469626I
b = 0.906765 + 0.547724I
−3.44018 + 1.36697I 0.099351 − 0.550230I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.098860 + 0.059621I
a = 0.432040 + 0.929418I
b = −0.258929 + 0.813001I
−1.66914 + 6.50568I 3.03082 − 5.51070I
u = −1.098860 + 0.059621I
a = −0.314191 − 0.418292I
b = −1.079870 − 0.537053I
−1.66914 + 6.50568I 3.03082 − 5.51070I
u = −1.098860 − 0.059621I
a = 0.432040 − 0.929418I
b = −0.258929 − 0.813001I
−1.66914 − 6.50568I 3.03082 + 5.51070I
u = −1.098860 − 0.059621I
a = −0.314191 + 0.418292I
b = −1.079870 + 0.537053I
−1.66914 − 6.50568I 3.03082 + 5.51070I
u = 0.858258 + 0.694285I
a = 1.40848 + 0.49567I
b = −1.337880 − 0.399067I
5.89812 − 2.66625I 9.77705 + 3.31297I
u = 0.858258 + 0.694285I
a = −1.74760 − 1.42262I
b = 1.220450 − 0.526531I
5.89812 − 2.66625I 9.77705 + 3.31297I
u = 0.858258 − 0.694285I
a = 1.40848 − 0.49567I
b = −1.337880 + 0.399067I
5.89812 + 2.66625I 9.77705 − 3.31297I
u = 0.858258 − 0.694285I
a = −1.74760 + 1.42262I
b = 1.220450 + 0.526531I
5.89812 + 2.66625I 9.77705 − 3.31297I
u = −0.828553 + 0.741140I
a = −0.890956 + 0.613775I
b = 1.37093 − 0.68976I
8.99039 − 0.95663I 14.3549 + 0.9762I
u = −0.828553 + 0.741140I
a = 1.87999 − 1.13465I
b = −1.51210 − 0.51307I
8.99039 − 0.95663I 14.3549 + 0.9762I
14
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.828553 − 0.741140I
a = −0.890956 − 0.613775I
b = 1.37093 + 0.68976I
8.99039 + 0.95663I 14.3549 − 0.9762I
u = −0.828553 − 0.741140I
a = 1.87999 + 1.13465I
b = −1.51210 + 0.51307I
8.99039 + 0.95663I 14.3549 − 0.9762I
u = −0.891994 + 0.729689I
a = −1.31090 + 0.92685I
b = 1.59800 − 0.38325I
8.79813 + 6.53878I 13.6140 − 6.9915I
u = −0.891994 + 0.729689I
a = 1.58473 − 1.11655I
b = −1.25531 − 0.81498I
8.79813 + 6.53878I 13.6140 − 6.9915I
u = −0.891994 − 0.729689I
a = −1.31090 − 0.92685I
b = 1.59800 + 0.38325I
8.79813 − 6.53878I 13.6140 + 6.9915I
u = −0.891994 − 0.729689I
a = 1.58473 + 1.11655I
b = −1.25531 + 0.81498I
8.79813 − 6.53878I 13.6140 + 6.9915I
u = −1.022970 + 0.630121I
a = −0.200460 + 0.184858I
b = 0.180956 − 0.447265I
0.25603 + 5.05352I 3.88531 − 5.31459I
u = −1.022970 + 0.630121I
a = −1.35085 + 1.29776I
b = 0.971944 + 0.280768I
0.25603 + 5.05352I 3.88531 − 5.31459I
u = −1.022970 − 0.630121I
a = −0.200460 − 0.184858I
b = 0.180956 + 0.447265I
0.25603 − 5.05352I 3.88531 + 5.31459I
u = −1.022970 − 0.630121I
a = −1.35085 − 1.29776I
b = 0.971944 − 0.280768I
0.25603 − 5.05352I 3.88531 + 5.31459I
15
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.997643 + 0.681461I
a = −0.717630 − 0.372056I
b = 0.298652 − 0.988469I
5.95409 − 5.49753I 11.62281 + 4.60034I
u = 0.997643 + 0.681461I
a = 1.62326 + 1.33956I
b = −1.45118 + 0.32417I
5.95409 − 5.49753I 11.62281 + 4.60034I
u = 0.997643 − 0.681461I
a = −0.717630 + 0.372056I
b = 0.298652 + 0.988469I
5.95409 + 5.49753I 11.62281 − 4.60034I
u = 0.997643 − 0.681461I
a = 1.62326 − 1.33956I
b = −1.45118 − 0.32417I
5.95409 + 5.49753I 11.62281 − 4.60034I
u = 0.416995 + 0.648442I
a = −0.949384 + 0.075306I
b = −0.239105 + 0.481030I
3.20064 − 4.79464I 9.29089 + 5.61871I
u = 0.416995 + 0.648442I
a = −1.39815 + 0.63323I
b = 1.185950 − 0.159624I
3.20064 − 4.79464I 9.29089 + 5.61871I
u = 0.416995 − 0.648442I
a = −0.949384 − 0.075306I
b = −0.239105 − 0.481030I
3.20064 + 4.79464I 9.29089 − 5.61871I
u = 0.416995 − 0.648442I
a = −1.39815 − 0.63323I
b = 1.185950 + 0.159624I
3.20064 + 4.79464I 9.29089 − 5.61871I
u = −1.031610 + 0.673233I
a = 0.660128 + 0.130914I
b = 0.108465 − 1.062730I
1.02610 + 7.72193I 5.01562 − 5.32873I
u = −1.031610 + 0.673233I
a = −1.63202 + 1.29034I
b = 1.30307 + 0.59506I
1.02610 + 7.72193I 5.01562 − 5.32873I
16
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.031610 − 0.673233I
a = 0.660128 − 0.130914I
b = 0.108465 + 1.062730I
1.02610 − 7.72193I 5.01562 + 5.32873I
u = −1.031610 − 0.673233I
a = −1.63202 − 1.29034I
b = 1.30307 − 0.59506I
1.02610 − 7.72193I 5.01562 + 5.32873I
u = 1.036490 + 0.686644I
a = −0.859246 + 0.141872I
b = −0.088389 − 1.243250I
3.09358 − 12.88870I 8.12323 + 9.41526I
u = 1.036490 + 0.686644I
a = 1.66673 + 1.31376I
b = −1.42433 + 0.67775I
3.09358 − 12.88870I 8.12323 + 9.41526I
u = 1.036490 − 0.686644I
a = −0.859246 − 0.141872I
b = −0.088389 + 1.243250I
3.09358 + 12.88870I 8.12323 − 9.41526I
u = 1.036490 − 0.686644I
a = 1.66673 − 1.31376I
b = −1.42433 − 0.67775I
3.09358 + 12.88870I 8.12323 − 9.41526I
u = −0.730192 + 0.168194I
a = 1.037910 + 0.857867I
b = −1.158730 + 0.009887I
2.12065 + 0.19319I 2.79170 − 0.78328I
u = −0.730192 + 0.168194I
a = 2.40293 + 0.28821I
b = 0.655532 + 0.107869I
2.12065 + 0.19319I 2.79170 − 0.78328I
u = −0.730192 − 0.168194I
a = 1.037910 − 0.857867I
b = −1.158730 − 0.009887I
2.12065 − 0.19319I 2.79170 + 0.78328I
u = −0.730192 − 0.168194I
a = 2.40293 − 0.28821I
b = 0.655532 − 0.107869I
2.12065 − 0.19319I 2.79170 + 0.78328I
17
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.164238 + 0.469611I
a = −0.67492 + 1.27326I
b = −0.926457 + 0.376650I
4.93312 + 1.19641I 13.57525 − 0.85209I
u = 0.164238 + 0.469611I
a = −1.03530 + 1.51984I
b = 1.225220 + 0.155136I
4.93312 + 1.19641I 13.57525 − 0.85209I
u = 0.164238 − 0.469611I
a = −0.67492 − 1.27326I
b = −0.926457 − 0.376650I
4.93312 − 1.19641I 13.57525 + 0.85209I
u = 0.164238 − 0.469611I
a = −1.03530 − 1.51984I
b = 1.225220 − 0.155136I
4.93312 − 1.19641I 13.57525 + 0.85209I
18
III. I
u
3
= ⟨−u
3
+ b, u
3
+ u
2
+ 2a − 2u, u
4
− u
2
+ 1⟩
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
11
=
−
1
2
u
3
−
1
2
u
2
+ u
u
3
a
12
=
1
2
u
3
−
1
2
u
2
+ u
u
3
a
9
=
−
1
2
u
3
−
1
4
u
2
+ u +
1
4
1
2
u
2
+
1
2
u
a
1
=
u
3
0
a
6
=
−
1
2
u
3
+ u
2
+
1
2
u +
1
2
−1
a
4
=
−
1
2
u
3
−
1
2
u
2
−
1
4
u + 1
−
1
2
u
3
+
1
2
u
2
+
1
2
u −
1
2
a
5
=
−
1
2
u
3
+ u
2
+
1
2
u −
1
2
−1
a
10
=
−
3
2
u
3
−
3
4
u
2
+
3
2
u
1
2
u
3
+
1
2
u +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u
2
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
(u
2
− u + 1)
2
c
2
, c
4
, c
7
c
9
u
4
− u
2
+ 1
c
3
, c
8
16(16u
4
+ 16u
3
+ 8u
2
− 4u + 1)
c
5
, c
6
, c
11
c
12
(u
2
+ 1)
2
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
(y
2
+ y + 1)
2
c
2
, c
4
, c
7
c
9
(y
2
− y + 1)
2
c
3
, c
8
256(256y
4
+ 224y
2
+ 1)
c
5
, c
6
, c
11
c
12
(y + 1)
4
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.866025 + 0.500000I
a = 0.616025 − 0.433013I
b = 1.000000I
−1.64493 − 4.05977I 4.00000 + 6.92820I
u = 0.866025 − 0.500000I
a = 0.616025 + 0.433013I
b = − 1.000000I
−1.64493 + 4.05977I 4.00000 − 6.92820I
u = −0.866025 + 0.500000I
a = −1.116030 + 0.433013I
b = 1.000000I
−1.64493 + 4.05977I 4.00000 − 6.92820I
u = −0.866025 − 0.500000I
a = −1.116030 − 0.433013I
b = − 1.000000I
−1.64493 − 4.05977I 4.00000 + 6.92820I
22
IV. I
u
4
= ⟨b, a − 1, u
4
− u
3
+ 1⟩
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
11
=
1
0
a
12
=
1
0
a
9
=
u
2
+ 1
u
2
a
1
=
u
3
−1
a
6
=
1
0
a
4
=
−u
3
−u
3
+ u
a
5
=
−u
3
+ 1
1
a
10
=
u
3
+ u
2
u
3
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
23
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
+ u
3
+ 2u
2
+ 1
c
2
, c
4
, c
7
c
9
u
4
+ u
3
+ 1
c
3
u
4
− u
2
− 2u + 3
c
5
, c
12
(u − 1)
4
c
6
, c
11
u
4
c
8
, c
10
u
4
− u
3
+ 2u
2
+ 1
24
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
10
y
4
+ 3y
3
+ 6y
2
+ 4y + 1
c
2
, c
4
, c
7
c
9
y
4
− y
3
+ 2y
2
+ 1
c
3
y
4
− 2y
3
+ 7y
2
− 10y + 9
c
5
, c
12
(y −1)
4
c
6
, c
11
y
4
25
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.518913 + 0.666610I
a = 1.00000
b = 0
1.64493 6.00000
u = −0.518913 − 0.666610I
a = 1.00000
b = 0
1.64493 6.00000
u = 1.018910 + 0.602565I
a = 1.00000
b = 0
1.64493 6.00000
u = 1.018910 − 0.602565I
a = 1.00000
b = 0
1.64493 6.00000
26
V. I
u
5
= ⟨b + 1, −u
3
+ a − 1, u
4
− u
3
+ 1⟩
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
11
=
u
3
+ 1
−1
a
12
=
u
3
−1
a
9
=
−u
2
− u + 1
−u
3
+ u
2
+ u
a
1
=
u
3
−1
a
6
=
−u
3
1
a
4
=
u
3
− 2u − 1
−u
3
− u
2
+ u
a
5
=
−u
3
1
a
10
=
−u
3
+ u
2
+ 2
u
3
+ u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
27
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
+ u
3
+ 2u
2
+ 1
c
2
, c
4
, c
7
c
9
u
4
+ u
3
+ 1
c
3
, c
10
u
4
− u
3
+ 2u
2
+ 1
c
5
, c
12
u
4
c
6
, c
11
(u − 1)
4
c
8
u
4
− u
2
− 2u + 3
28
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
10
y
4
+ 3y
3
+ 6y
2
+ 4y + 1
c
2
, c
4
, c
7
c
9
y
4
− y
3
+ 2y
2
+ 1
c
5
, c
12
y
4
c
6
, c
11
(y −1)
4
c
8
y
4
− 2y
3
+ 7y
2
− 10y + 9
29
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.518913 + 0.666610I
a = 1.55204 + 0.24227I
b = −1.00000
1.64493 6.00000
u = −0.518913 − 0.666610I
a = 1.55204 − 0.24227I
b = −1.00000
1.64493 6.00000
u = 1.018910 + 0.602565I
a = 0.94796 + 1.65794I
b = −1.00000
1.64493 6.00000
u = 1.018910 − 0.602565I
a = 0.94796 − 1.65794I
b = −1.00000
1.64493 6.00000
30
VI. I
u
6
= ⟨b, a − 1, u + 1⟩
(i) Arc colorings
a
2
=
0
−1
a
7
=
1
0
a
8
=
1
1
a
3
=
1
0
a
11
=
1
0
a
12
=
1
0
a
9
=
2
1
a
1
=
−1
−1
a
6
=
1
0
a
4
=
1
0
a
5
=
2
1
a
10
=
3
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
31
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u + 1
c
2
, c
4
, c
5
c
7
, c
8
, c
9
c
10
, c
12
u − 1
c
3
, c
6
, c
11
u
32
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
c
9
, c
10
, c
12
y −1
c
3
, c
6
, c
11
y
33
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = 0
1.64493 6.00000
34
VII. I
u
7
= ⟨b + 1, a, u + 1⟩
(i) Arc colorings
a
2
=
0
−1
a
7
=
1
0
a
8
=
1
1
a
3
=
1
0
a
11
=
0
−1
a
12
=
−1
−1
a
9
=
1
1
a
1
=
−1
−1
a
6
=
1
1
a
4
=
0
−1
a
5
=
1
1
a
10
=
1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
35
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u + 1
c
2
, c
3
, c
4
c
6
, c
7
, c
9
c
10
, c
11
u − 1
c
5
, c
8
, c
12
u
36
(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
9
, c
10
, c
11
y −1
c
5
, c
8
, c
12
y
37
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = −1.00000
1.64493 6.00000
38
VIII. I
u
8
= ⟨−u
3
+ b, u
3
+ 2a − 2u − 1, u
4
− u
2
+ 1⟩
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
11
=
−
1
2
u
3
+ u +
1
2
u
3
a
12
=
1
2
u
3
+ u +
1
2
u
3
a
9
=
u
3
−
1
2
u +
1
2
1
2
u
3
+
1
2
u
2
−
1
2
u
a
1
=
u
3
0
a
6
=
1
2
u
3
+ u
2
+
1
2
−1
a
4
=
u
2
−
1
2
u −
1
2
−
1
2
u
3
−
1
2
u
2
+
1
2
u
a
5
=
1
2
u
3
+ u
2
−
1
2
−1
a
10
=
1
2
u
2
+
1
2
u
3
+
1
2
u
2
−
1
2
u −
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
39
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
(u
2
− u + 1)
2
c
2
, c
4
, c
7
c
9
u
4
− u
2
+ 1
c
3
, c
8
4(4u
4
+ 4u
3
+ 2u
2
+ 2u + 1)
c
5
, c
6
, c
11
c
12
(u
2
+ 1)
2
40
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
(y
2
+ y + 1)
2
c
2
, c
4
, c
7
c
9
(y
2
− y + 1)
2
c
3
, c
8
16(16y
4
− 4y
2
+ 1)
c
5
, c
6
, c
11
c
12
(y + 1)
4
41
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = 0.866025 + 0.500000I
a = 1.36603
b = 1.000000I
−1.64493 4.00000
u = 0.866025 − 0.500000I
a = 1.36603
b = − 1.000000I
−1.64493 4.00000
u = −0.866025 + 0.500000I
a = −0.366025
b = 1.000000I
−1.64493 4.00000
u = −0.866025 − 0.500000I
a = −0.366025
b = − 1.000000I
−1.64493 4.00000
42
IX. I
u
9
= ⟨b + 1, u
5
a − u
5
− u
3
a + 2u
3
+ au − u + 1⟩
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
11
=
a
−1
a
12
=
a − 1
−1
a
9
=
a
2
u
2
− 2u
2
a + u
2
+ a
−u
2
a + 2u
2
− 1
a
1
=
u
3
u
5
− u
3
+ u
a
6
=
−a + 1
1
a
4
=
−u
3
a
2
+ 2u
3
a + a
2
u − u
3
− 3au + u
u
3
a − 2u
3
− au + 3u
a
5
=
u
3
− a + 1
u
5
− u
3
+ u + 1
a
10
=
u
4
a
2
− 2u
4
a + u
4
+ u
2
a + a
−u
4
a + 2u
4
− u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
43
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
9
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
3.28987 12.0000
44
X. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
25(u + 1)
2
(u
2
− u + 1)
4
(u
4
+ u
3
+ 2u
2
+ 1)
2
· ((u
32
+ 11u
31
+ ··· + 2u + 1)
2
)(25u
40
+ 355u
39
+ ··· − 2766u + 169)
c
2
, c
7
5(u − 1)
2
(u
4
− u
2
+ 1)
2
(u
4
+ u
3
+ 1)
2
(u
32
+ u
31
+ ··· + 2u + 1)
2
· (5u
40
− 15u
39
+ ··· − 4u + 13)
c
3
, c
8
16384u(u − 1)(u
4
− u
2
− 2u + 3)(u
4
− u
3
+ 2u
2
+ 1)
· (4u
4
+ 4u
3
+ 2u
2
+ 2u + 1)(16u
4
+ 16u
3
+ 8u
2
− 4u + 1)
· (64u
40
+ 192u
39
+ ··· + 80u + 25)
· (4u
64
+ 56u
63
+ ··· + 96881428u + 10705749)
c
4
, c
9
5(u − 1)
2
(u
4
− u
2
+ 1)
2
(u
4
+ u
3
+ 1)
2
(u
32
+ u
31
+ ··· − u
2
+ 1)
2
· (5u
40
− 15u
39
+ ··· − 58u + 13)
c
5
, c
6
, c
11
c
12
u
5
(u − 1)
5
(u
2
+ 1)
4
(u
40
+ 4u
39
+ ··· + 36u + 4)
· (u
64
+ 4u
63
+ ··· + 9164u + 2061)
c
10
25(u − 1)
2
(u
2
− u + 1)
4
(u
4
− u
3
+ 2u
2
+ 1)
2
· ((u
32
− 15u
31
+ ··· − 2u + 1)
2
)(25u
40
− 455u
39
+ ··· + 718u + 169)
45
XI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
625(y −1)
2
(y
2
+ y + 1)
4
(y
4
+ 3y
3
+ 6y
2
+ 4y + 1)
2
· (y
32
+ 21y
31
+ ··· + 2y + 1)
2
· (625y
40
+ 14425y
39
+ ··· − 2166706y + 28561)
c
2
, c
7
25(y −1)
2
(y
2
− y + 1)
4
(y
4
− y
3
+ 2y
2
+ 1)
2
· ((y
32
− 11y
31
+ ··· − 2y + 1)
2
)(25y
40
− 355y
39
+ ··· + 2766y + 169)
c
3
, c
8
268435456(y)(y − 1)(y
4
− 2y
3
+ ··· − 10y + 9)(y
4
+ 3y
3
+ ··· + 4y + 1)
· (16y
4
− 4y
2
+ 1)(256y
4
+ 224y
2
+ 1)
· (4096y
40
− 53248y
39
+ ··· − 13450y + 625)
· (16y
64
− 592y
63
+ ··· − 2588329366713886y + 114613061651001)
c
4
, c
9
25(y −1)
2
(y
2
− y + 1)
4
(y
4
− y
3
+ 2y
2
+ 1)
2
· ((y
32
− 15y
31
+ ··· − 2y + 1)
2
)(25y
40
− 455y
39
+ ··· + 718y + 169)
c
5
, c
6
, c
11
c
12
y
5
(y −1)
5
(y + 1)
8
(y
40
− 12y
39
+ ··· − 696y + 16)
· (y
64
− 44y
63
+ ··· − 8369050y + 4247721)
c
10
625(y −1)
2
(y
2
+ y + 1)
4
(y
4
+ 3y
3
+ 6y
2
+ 4y + 1)
2
· (y
32
+ 5y
31
+ ··· + 2y + 1)
2
· (625y
40
+ 4425y
39
+ ··· − 2689202y + 28561)
46
