12n
0508
(K12n
0508
)
A knot diagram
1
Linearized knot diagam
3 6 12 7 2 9 11 5 3 12 8 9
Solving Sequence
2,6 3,9
7 10 1 5 4 8 12 11
c
2
c
6
c
9
c
1
c
5
c
4
c
8
c
12
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−11u
14
+ 3u
13
+ ··· + 36b + 14, −u
14
+ 3u
13
+ ··· + 12a − 8,
u
15
+ 2u
14
− 2u
13
− 5u
12
+ 4u
11
+ 9u
10
+ u
9
+ u
8
+ 3u
7
− 4u
6
+ 5u
5
+ 21u
4
+ 16u
3
+ 4u
2
− 2i
I
u
2
= h−7u
24
+ 2u
23
+ ··· + 32b + 140, −47u
25
− 117u
24
+ ··· + 368a + 2737, u
26
+ 2u
25
+ ··· − 46u − 23i
I
u
3
= hu
3
− u
2
+ 2b, −u
2
+ 2a + u, u
4
− u
2
+ 2i
I
u
4
= h4b
2
− 4b − 32u + 33, 2bu − 2b − u + 1, 2a + 1, u
2
− 2u + 1i
I
u
5
= h−u
3
+ u
2
+ 2b − u + 1, u
3
+ u
2
+ 2a − u + 1, u
4
+ 1i
I
u
6
= h−a
2
+ b − 2a, a
3
+ 2a
2
+ a + 1, u − 1i
I
u
7
= hba − a
2
+ a + 1, u − 1i
I
u
8
= hb − a − 1, u − 1i
I
v
1
= ha, b
3
− b − 1, v − 1i
I
v
2
= ha, b + 1, v − 1i
* 8 irreducible components of dim
C
= 0, with total 59 representations.
* 2 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
= h−11u
14
+ 3u
13
+ · · · + 36b + 14, −u
14
+ 3u
13
+ · · · + 12a − 8, u
15
+
2u
14
+ · · · + 4u
2
− 2i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
9
=
1
12
u
14
−
1
4
u
13
+ ··· +
1
6
u +
2
3
11
36
u
14
−
1
12
u
13
+ ··· −
8
9
u −
7
18
a
7
=
1
4
u
12
−
3
4
u
10
+ ··· +
1
2
u −
1
2
1
12
u
14
−
1
4
u
13
+ ··· +
2
3
u +
1
6
a
10
=
13
36
u
14
+
1
3
u
13
+ ··· +
11
9
u +
2
9
1
12
u
14
−
5
12
u
12
+ ··· −
1
3
u −
1
3
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
4
=
−
1
2
u
14
−
1
2
u
13
+ ··· − u +
3
2
−
1
2
u
14
−
1
2
u
13
+ ··· −
1
2
u + 1
a
8
=
−0.388889u
14
− 0.416667u
13
+ ··· − 0.277778u + 1.22222
−
1
6
u
14
−
1
4
u
13
+ ··· −
4
3
u +
1
6
a
12
=
1
2
u
6
−
1
2
u
4
+
1
2
u
3
+ 1
1
12
u
14
−
5
12
u
12
+ ··· −
1
3
u −
1
3
a
11
=
13
36
u
14
+
1
3
u
13
+ ··· +
11
9
u +
2
9
1
6
u
14
+
1
4
u
13
+ ··· −
1
6
u −
2
3
(ii) Obstruction class = −1
(iii) Cusp Shapes =
28
27
u
14
−
8
9
u
13
−
134
27
u
12
+
104
27
u
11
+ 10u
10
−
26
3
u
9
−
206
27
u
8
+
200
27
u
7
−
148
27
u
6
−
98
27
u
5
+
158
9
u
4
−
8
3
u
3
−
488
27
u
2
−
214
27
u −
160
27
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
15
+ 8u
14
+ ··· + 16u + 4
c
2
, c
5
, c
7
c
11
u
15
− 2u
14
+ ··· − 4u
2
+ 2
c
3
, c
12
2(2u
15
− 6u
14
+ ··· − 9u + 3)
c
4
, c
6
2(2u
15
+ 2u
14
+ ··· + 27u + 3)
c
8
3(3u
15
− 18u
14
+ ··· + 11u + 7)
c
9
3(3u
15
+ 18u
14
+ ··· − 17u + 7)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
15
+ 32y
13
+ ··· + 800y − 16
c
2
, c
5
, c
7
c
11
y
15
− 8y
14
+ ··· + 16y − 4
c
3
, c
12
4(4y
15
− 104y
14
+ ··· + 249y − 9)
c
4
, c
6
4(4y
15
+ 56y
14
+ ··· + 717y − 9)
c
8
9(9y
15
− 24y
14
+ ··· + 555y − 49)
c
9
9(9y
15
− 168y
14
+ ··· − 229y − 49)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.144120 + 1.054560I
a = −1.219520 + 0.251203I
b = −0.305685 − 0.517647I
−5.10750 + 3.19553I −5.98942 − 2.06971I
u = 0.144120 − 1.054560I
a = −1.219520 − 0.251203I
b = −0.305685 + 0.517647I
−5.10750 − 3.19553I −5.98942 + 2.06971I
u = −0.928567 + 0.624096I
a = −0.420023 + 0.318850I
b = 0.348565 + 0.966858I
2.54442 + 5.04477I −0.58192 − 5.69711I
u = −0.928567 − 0.624096I
a = −0.420023 − 0.318850I
b = 0.348565 − 0.966858I
2.54442 − 5.04477I −0.58192 + 5.69711I
u = 1.007800 + 0.747753I
a = 0.553989 + 0.660683I
b = 0.448740 + 1.074590I
−0.77508 − 5.73977I −11.31279 + 5.82144I
u = 1.007800 − 0.747753I
a = 0.553989 − 0.660683I
b = 0.448740 − 1.074590I
−0.77508 + 5.73977I −11.31279 − 5.82144I
u = −1.203300 + 0.403095I
a = −0.438882 − 1.115720I
b = −0.88076 − 2.14220I
−4.79918 + 9.00906I −10.02265 − 8.13965I
u = −1.203300 − 0.403095I
a = −0.438882 + 1.115720I
b = −0.88076 + 2.14220I
−4.79918 − 9.00906I −10.02265 + 8.13965I
u = −0.189655 + 0.562206I
a = 1.23134 + 1.07625I
b = 0.232219 − 0.118314I
1.46040 − 1.17055I 1.25283 + 2.21107I
u = −0.189655 − 0.562206I
a = 1.23134 − 1.07625I
b = 0.232219 + 0.118314I
1.46040 + 1.17055I 1.25283 − 2.21107I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.346970 + 0.412511I
a = 0.057661 + 0.976830I
b = −0.68130 + 1.78913I
−14.7969 + 6.7928I −12.44195 − 4.01641I
u = −1.346970 − 0.412511I
a = 0.057661 − 0.976830I
b = −0.68130 − 1.78913I
−14.7969 − 6.7928I −12.44195 + 4.01641I
u = 1.32413 + 0.56114I
a = 0.09611 − 1.50736I
b = 0.69422 − 2.43940I
−12.5822 − 14.8536I −10.17186 + 7.59627I
u = 1.32413 − 0.56114I
a = 0.09611 + 1.50736I
b = 0.69422 + 2.43940I
−12.5822 + 14.8536I −10.17186 − 7.59627I
u = 0.384887
a = 0.278645
b = −0.711991
−0.975030 −11.4640
6
II. I
u
2
= h−7u
24
+ 2u
23
+ · · · + 32b + 140, −47u
25
− 117u
24
+ · · · + 368a +
2737, u
26
+ 2u
25
+ · · · − 46u − 23i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
9
=
0.127717u
25
+ 0.317935u
24
+ ··· − 4.13587u − 7.43750
7
32
u
24
−
1
16
u
23
+ ··· −
23
32
u −
35
8
a
7
=
0.535326u
25
+ 0.726902u
24
+ ··· − 11.5476u − 17.8125
0.375000u
25
+ 0.531250u
24
+ ··· − 6.71875u − 12.3125
a
10
=
−0.122283u
25
+ 0.0366848u
24
+ ··· + 2.39538u − 1.62500
−0.187500u
25
− 0.0625000u
24
+ ··· + 3.59375u + 0.656250
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
4
=
−0.157609u
25
− 0.252717u
24
+ ··· + 0.442935u + 8.18750
−
1
8
u
25
−
3
16
u
24
+ ··· +
3
4
u +
81
16
a
8
=
0.377717u
25
+ 0.474185u
24
+ ··· − 8.38587u − 11.0313
0.250000u
25
+ 0.375000u
24
+ ··· − 4.96875u − 7.96875
a
12
=
0.347826u
25
+ 0.539402u
24
+ ··· − 7.51630u − 11.5938
0.156250u
25
+ 0.343750u
24
+ ··· − 3.59375u − 8
a
11
=
0.663043u
25
+ 0.951087u
24
+ ··· − 13.5272u − 21.9375
5
16
u
25
+
5
8
u
24
+ ··· −
125
16
u −
61
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
3
2
u
25
−
3
2
u
24
+ ··· +
147
4
u +
57
2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
26
+ 16u
25
+ ··· + 874u + 529
c
2
, c
5
, c
7
c
11
u
26
− 2u
25
+ ··· + 46u − 23
c
3
, c
12
2(2u
26
− 4u
25
+ ··· + 5854u + 13513)
c
4
, c
6
2(2u
26
+ 12u
25
+ ··· − 2212u − 4061)
c
8
(u
13
+ 2u
12
+ ··· − 3u + 1)
2
c
9
(u
13
− 2u
12
+ ··· + 9u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
26
− 12y
25
+ ··· − 3545358y + 279841
c
2
, c
5
, c
7
c
11
y
26
− 16y
25
+ ··· − 874y + 529
c
3
, c
12
4(4y
26
− 232y
25
+ ··· + 2.78070 × 10
8
y + 1.82601 × 10
8
)
c
4
, c
6
4(4y
26
+ 88y
25
+ ··· + 9.27579 × 10
7
y + 1.64917 × 10
7
)
c
8
(y
13
− 2y
12
+ ··· + 17y − 1)
2
c
9
(y
13
− 18y
12
+ ··· + 65y − 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.145808 + 0.983156I
a = 1.47860 − 0.04032I
b = 0.285975 + 0.539303I
−10.07100 − 1.92961I −9.33197 + 0.98070I
u = 0.145808 − 0.983156I
a = 1.47860 + 0.04032I
b = 0.285975 − 0.539303I
−10.07100 + 1.92961I −9.33197 − 0.98070I
u = −0.694794 + 0.669489I
a = 0.747191 + 0.070295I
b = −0.044639 − 0.692695I
3.24368 1.97600 + 0.I
u = −0.694794 − 0.669489I
a = 0.747191 − 0.070295I
b = −0.044639 + 0.692695I
3.24368 1.97600 + 0.I
u = 0.727228 + 0.765886I
a = −0.656960 − 0.521646I
b = −0.485985 − 0.701709I
0.0875120 −10.12424 + 0.I
u = 0.727228 − 0.765886I
a = −0.656960 + 0.521646I
b = −0.485985 + 0.701709I
0.0875120 −10.12424 + 0.I
u = 0.089788 + 1.062330I
a = 1.239930 − 0.503480I
b = 0.331705 + 0.510914I
−8.73749 + 9.07090I −7.83282 − 5.02365I
u = 0.089788 − 1.062330I
a = 1.239930 + 0.503480I
b = 0.331705 − 0.510914I
−8.73749 − 9.07090I −7.83282 + 5.02365I
u = 1.071990 + 0.329976I
a = 1.056560 + 0.393863I
b = 1.60771 + 0.71133I
−4.67191 + 1.36942I −12.56235 − 3.09698I
u = 1.071990 − 0.329976I
a = 1.056560 − 0.393863I
b = 1.60771 − 0.71133I
−4.67191 − 1.36942I −12.56235 + 3.09698I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.14128
a = −0.980723
b = −1.64813
−2.29729 −0.573860
u = −1.166120 + 0.256035I
a = −0.63202 − 1.51574I
b = −0.93587 − 2.03026I
−4.67191 + 1.36942I −12.56235 − 3.09698I
u = −1.166120 − 0.256035I
a = −0.63202 + 1.51574I
b = −0.93587 + 2.03026I
−4.67191 − 1.36942I −12.56235 + 3.09698I
u = −1.148780 + 0.376438I
a = 0.276926 + 1.188600I
b = 0.77110 + 1.96980I
−1.36409 + 4.88678I −5.58540 − 5.91732I
u = −1.148780 − 0.376438I
a = 0.276926 − 1.188600I
b = 0.77110 − 1.96980I
−1.36409 − 4.88678I −5.58540 + 5.91732I
u = −0.722662
a = −2.98213
b = −1.92811
−2.29729 −0.573860
u = −0.048328 + 0.709054I
a = −1.16615 − 1.24694I
b = −0.0850094 − 0.1052430I
−1.36409 − 4.88678I −5.58540 + 5.91732I
u = −0.048328 − 0.709054I
a = −1.16615 + 1.24694I
b = −0.0850094 + 0.1052430I
−1.36409 + 4.88678I −5.58540 − 5.91732I
u = 1.277190 + 0.579499I
a = −0.27185 − 1.43511I
b = 0.07667 − 2.38894I
−13.50590 − 3.70097I −11.32642 + 2.50956I
u = 1.277190 − 0.579499I
a = −0.27185 + 1.43511I
b = 0.07667 + 2.38894I
−13.50590 + 3.70097I −11.32642 − 2.50956I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.31273 + 0.58386I
a = 0.040670 + 1.389900I
b = −0.44938 + 2.27345I
−8.73749 − 9.07090I −7.83282 + 5.02365I
u = 1.31273 − 0.58386I
a = 0.040670 − 1.389900I
b = −0.44938 − 2.27345I
−8.73749 + 9.07090I −7.83282 − 5.02365I
u = −1.38277 + 0.40950I
a = −0.082363 − 0.796616I
b = 0.62637 − 1.37358I
−10.07100 + 1.92961I −9.33197 − 0.98070I
u = −1.38277 − 0.40950I
a = −0.082363 + 0.796616I
b = 0.62637 + 1.37358I
−10.07100 − 1.92961I −9.33197 + 0.98070I
u = −1.39325 + 0.44582I
a = −0.049113 + 0.700779I
b = −0.91053 + 1.15456I
−13.50590 − 3.70097I −11.32642 + 2.50956I
u = −1.39325 − 0.44582I
a = −0.049113 − 0.700779I
b = −0.91053 − 1.15456I
−13.50590 + 3.70097I −11.32642 − 2.50956I
12
III. I
u
3
= hu
3
− u
2
+ 2b, −u
2
+ 2a + u, u
4
− u
2
+ 2i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
9
=
1
2
u
2
−
1
2
u
−
1
2
u
3
+
1
2
u
2
a
7
=
1
2
u
3
−
1
2
u
2
−
1
2
u + 1
1
2
u
3
+ 1
a
10
=
1
2
u
2
−
1
2
u − 1
−
1
2
u
3
−
1
2
u
2
a
1
=
−u
2
+ 1
−u
2
+ 2
a
5
=
u
u
a
4
=
1
4
u
3
−
1
4
u
2
+ u +
1
2
3
4
u
3
+
1
4
u
2
+
1
2
u −
1
2
a
8
=
1
2
u
2
−
3
2
u
−
1
2
u
3
+
1
2
u
2
− u
a
12
=
−u
2
−
1
2
u + 1
−
1
2
u
2
+ 2
a
11
=
−
1
2
u
3
+
1
2
u
2
−
1
2
u + 1
−
1
2
u
3
+ u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 8
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
(u
2
− u + 2)
2
c
2
, c
5
, c
7
c
11
u
4
− u
2
+ 2
c
3
, c
12
2(2u
4
+ 2u
3
+ 5u
2
+ 4u + 1)
c
4
, c
6
2(2u
4
− 2u
3
+ 5u
2
− 4u + 1)
c
8
, c
9
(u + 1)
4
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
(y
2
+ 3y + 4)
2
c
2
, c
5
, c
7
c
11
(y
2
− y + 2)
2
c
3
, c
4
, c
6
c
12
4(4y
4
+ 16y
3
+ 13y
2
− 6y + 1)
c
8
, c
9
(y − 1)
4
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.978318 + 0.676097I
a = −0.239159 + 0.323389I
b = 0.452616 − 0.154683I
0.82247 − 5.33349I −6.00000 + 5.29150I
u = 0.978318 − 0.676097I
a = −0.239159 − 0.323389I
b = 0.452616 + 0.154683I
0.82247 + 5.33349I −6.00000 − 5.29150I
u = −0.978318 + 0.676097I
a = 0.739159 − 0.999486I
b = 0.04738 − 1.47756I
0.82247 + 5.33349I −6.00000 − 5.29150I
u = −0.978318 − 0.676097I
a = 0.739159 + 0.999486I
b = 0.04738 + 1.47756I
0.82247 − 5.33349I −6.00000 + 5.29150I
16
IV. I
u
4
= h4b
2
− 4b − 32u + 33, 2bu − 2b − u + 1, 2a + 1, u
2
− 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
2u − 1
a
9
=
−0.5
b
a
7
=
1
4
u
−
1
2
b +
3
4
u +
1
4
a
10
=
−b − u
−3u +
5
2
a
1
=
−2u + 2
−4u + 3
a
5
=
u
u
a
4
=
−
1
8
b +
11
8
u −
3
16
−
5
8
b +
33
8
u −
39
16
a
8
=
−b − 2u + 1
−2u +
3
2
a
12
=
−
7
2
u +
13
4
1
2
b −
5
2
u +
3
2
a
11
=
−b − 4u +
11
4
1
2
b − 4u +
5
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
10
(u − 1)
3
c
3
512(2u + 1)
3
c
4
512(2u − 1)
3
c
5
, c
8
, c
9
c
11
(u + 1)
3
c
6
64(2u − 1)
3
c
12
64(2u + 1)
3
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
8
, c
9
c
10
, c
11
(y − 1)
3
c
3
, c
4
262144(4y − 1)
3
c
6
, c
12
4096(4y − 1)
3
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.500000
b = 0.500000
−3.28987 −12.0000
u = 1.00000
a = −0.500000
b = 0.500000
−3.28987 −12.0000
u = 1.00000
a = −0.500000
b = 0.500000
−3.28987 −12.0000
20
V. I
u
5
= h−u
3
+ u
2
+ 2b − u + 1, u
3
+ u
2
+ 2a − u + 1, u
4
+ 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
9
=
−
1
2
u
3
−
1
2
u
2
+
1
2
u −
1
2
1
2
u
3
−
1
2
u
2
+
1
2
u −
1
2
a
7
=
1
2
u
3
− u
2
+
1
2
u
u
3
−
1
2
u
2
+ u +
1
2
a
10
=
−
1
2
u
3
−
1
2
u
2
+
1
2
u +
1
2
1
2
u
3
+
1
2
u
2
+
1
2
u −
1
2
a
1
=
−u
2
+ 1
1
a
5
=
u
u
a
4
=
−
1
2
u
3
+
1
2
u
2
+ u
1
2
u
2
+
1
2
u − 1
a
8
=
−
1
2
u
3
−
1
2
u
2
+
3
2
u −
1
2
1
2
u
3
−
1
2
u
2
+
3
2
u −
1
2
a
12
=
−
3
2
u
2
+
1
2
1
2
u
3
+
1
2
u + 1
a
11
=
−
1
2
u
3
− u
2
+
1
2
u − 1
−
1
2
u
2
+ u −
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
(u
2
+ 1)
2
c
2
, c
5
, c
7
c
11
u
4
+ 1
c
3
, c
4
, c
6
c
12
2(2u
4
+ 4u
3
+ 6u
2
+ 4u + 1)
c
8
, c
9
(u − 1)
4
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
(y + 1)
4
c
2
, c
5
, c
7
c
11
(y
2
+ 1)
2
c
3
, c
4
, c
6
c
12
4(4y
4
+ 8y
3
+ 8y
2
− 4y + 1)
c
8
, c
9
(y − 1)
4
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.707107 + 0.707107I
a = 0.207107 − 0.500000I
b = −0.500000 + 0.207107I
1.64493 −4.00000
u = 0.707107 − 0.707107I
a = 0.207107 + 0.500000I
b = −0.500000 − 0.207107I
1.64493 −4.00000
u = −0.707107 + 0.707107I
a = −1.207110 + 0.500000I
b = −0.500000 + 1.207110I
1.64493 −4.00000
u = −0.707107 − 0.707107I
a = −1.207110 − 0.500000I
b = −0.500000 − 1.207110I
1.64493 −4.00000
24
VI. I
u
6
= h−a
2
+ b − 2a, a
3
+ 2a
2
+ a + 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
9
=
a
a
2
+ 2a
a
7
=
a
2
−a
a
10
=
−a
2
a
a
1
=
0
−1
a
5
=
1
1
a
4
=
−a
2
−a
2
− a
a
8
=
−a
2
a
a
12
=
−a
2
a
a
11
=
−a
2
a
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
25
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u + 1)
3
c
3
, c
4
, c
8
c
9
u
3
− u − 1
c
6
u
3
− 2u
2
+ u − 1
c
7
, c
10
, c
11
u
3
c
12
u
3
+ 2u
2
+ u + 1
26
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y − 1)
3
c
3
, c
4
, c
8
c
9
y
3
− 2y
2
+ y − 1
c
6
, c
12
y
3
− 2y
2
− 3y − 1
c
7
, c
10
, c
11
y
3
27
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.122561 + 0.744862I
b = −0.78492 + 1.30714I
−1.64493 −6.00000
u = 1.00000
a = −0.122561 − 0.744862I
b = −0.78492 − 1.30714I
−1.64493 −6.00000
u = 1.00000
a = −1.75488
b = −0.430160
−1.64493 −6.00000
28
VII. I
u
7
= hba − a
2
+ a + 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
9
=
a
b
a
7
=
a
2
a
2
− a
a
10
=
−b + 2a
a
a
1
=
0
−1
a
5
=
1
1
a
4
=
−a
3
+ 1
−a
3
+ a
2
+ 1
a
8
=
−b + 2a
a
a
12
=
−a
2
−a
2
+ a
a
11
=
−a
2
− b + 2a
−a
2
+ 2a
(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.
29
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
−3.28987 −12.0000
30
VIII. I
u
8
= hb − a − 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
9
=
a
a + 1
a
7
=
a
2
a
2
+ a + 1
a
10
=
a − 1
a
a
1
=
0
−1
a
5
=
1
1
a
4
=
a
3
+ a
2
+ 1
a
3
+ 2a
2
+ 2a + 2
a
8
=
a − 1
a
a
12
=
−a
2
−a
2
− a − 1
a
11
=
−a
2
+ a − 1
−a
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.
31
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
−3.28987 −12.0000
32
IX. I
v
1
= ha, b
3
− b − 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
1
0
a
3
=
1
0
a
9
=
0
b
a
7
=
1
−b
2
a
10
=
−b
b
a
1
=
1
0
a
5
=
1
0
a
4
=
−b
2
+ 1
b
2
+ b
a
8
=
−b
b
a
12
=
1
−b
2
a
11
=
−b + 1
−b
2
+ b
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
33
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
3
c
3
u
3
+ 2u
2
+ u + 1
c
4
u
3
− 2u
2
+ u − 1
c
6
, c
8
, c
9
c
12
u
3
− u − 1
c
7
, c
10
, c
11
(u + 1)
3
34
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
3
c
3
, c
4
y
3
− 2y
2
− 3y − 1
c
6
, c
8
, c
9
c
12
y
3
− 2y
2
+ y − 1
c
7
, c
10
, c
11
(y − 1)
3
35
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −0.662359 + 0.562280I
−1.64493 −6.00000
v = 1.00000
a = 0
b = −0.662359 − 0.562280I
−1.64493 −6.00000
v = 1.00000
a = 0
b = 1.32472
−1.64493 −6.00000
36
X. I
v
2
= ha, b + 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
1
0
a
3
=
1
0
a
9
=
0
−1
a
7
=
1
−1
a
10
=
1
−1
a
1
=
1
0
a
5
=
1
0
a
4
=
0
1
a
8
=
1
−1
a
12
=
1
−1
a
11
=
1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
37
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
10
, c
11
u
c
3
, c
4
, c
6
c
12
u + 1
c
8
, c
9
u − 1
38
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
10
, c
11
y
c
3
, c
4
, c
6
c
8
, c
9
, c
12
y − 1
39
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
2
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
0 0
40
XI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
10
u
4
(u − 1)
3
(u + 1)
3
(u
2
+ 1)
2
(u
2
− u + 2)
2
(u
15
+ 8u
14
+ ··· + 16u + 4)
· (u
26
+ 16u
25
+ ··· + 874u + 529)
c
2
, c
7
u
4
(u − 1)
3
(u + 1)
3
(u
4
+ 1)(u
4
− u
2
+ 2)(u
15
− 2u
14
+ ··· − 4u
2
+ 2)
· (u
26
− 2u
25
+ ··· + 46u − 23)
c
3
8192(u + 1)(2u + 1)
3
(u
3
− u − 1)(u
3
+ 2u
2
+ u + 1)
· (2u
4
+ 2u
3
+ 5u
2
+ 4u + 1)(2u
4
+ 4u
3
+ 6u
2
+ 4u + 1)
· (2u
15
− 6u
14
+ ··· − 9u + 3)(2u
26
− 4u
25
+ ··· + 5854u + 13513)
c
4
8192(u + 1)(2u − 1)
3
(u
3
− u − 1)(u
3
− 2u
2
+ u − 1)
· (2u
4
− 2u
3
+ 5u
2
− 4u + 1)(2u
4
+ 4u
3
+ 6u
2
+ 4u + 1)
· (2u
15
+ 2u
14
+ ··· + 27u + 3)(2u
26
+ 12u
25
+ ··· − 2212u − 4061)
c
5
, c
11
u
4
(u + 1)
6
(u
4
+ 1)(u
4
− u
2
+ 2)(u
15
− 2u
14
+ ··· − 4u
2
+ 2)
· (u
26
− 2u
25
+ ··· + 46u − 23)
c
6
1024(u + 1)(2u − 1)
3
(u
3
− u − 1)(u
3
− 2u
2
+ u − 1)
· (2u
4
− 2u
3
+ 5u
2
− 4u + 1)(2u
4
+ 4u
3
+ 6u
2
+ 4u + 1)
· (2u
15
+ 2u
14
+ ··· + 27u + 3)(2u
26
+ 12u
25
+ ··· − 2212u − 4061)
c
8
3(u − 1)
5
(u + 1)
7
(u
3
− u − 1)
2
(u
13
+ 2u
12
+ ··· − 3u + 1)
2
· (3u
15
− 18u
14
+ ··· + 11u + 7)
c
9
3(u − 1)
5
(u + 1)
7
(u
3
− u − 1)
2
(u
13
− 2u
12
+ ··· + 9u + 1)
2
· (3u
15
+ 18u
14
+ ··· − 17u + 7)
c
12
1024(u + 1)(2u + 1)
3
(u
3
− u − 1)(u
3
+ 2u
2
+ u + 1)
· (2u
4
+ 2u
3
+ 5u
2
+ 4u + 1)(2u
4
+ 4u
3
+ 6u
2
+ 4u + 1)
· (2u
15
− 6u
14
+ ··· − 9u + 3)(2u
26
− 4u
25
+ ··· + 5854u + 13513)
41
XII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
4
(y − 1)
6
(y + 1)
4
(y
2
+ 3y + 4)
2
(y
15
+ 32y
13
+ ··· + 800y − 16)
· (y
26
− 12y
25
+ ··· − 3545358y + 279841)
c
2
, c
5
, c
7
c
11
y
4
(y − 1)
6
(y
2
+ 1)
2
(y
2
− y + 2)
2
(y
15
− 8y
14
+ ··· + 16y − 4)
· (y
26
− 16y
25
+ ··· − 874y + 529)
c
3
67108864(y − 1)(4y − 1)
3
(y
3
− 2y
2
− 3y − 1)(y
3
− 2y
2
+ y − 1)
· (4y
4
+ 8y
3
+ 8y
2
− 4y + 1)(4y
4
+ 16y
3
+ 13y
2
− 6y + 1)
· (4y
15
− 104y
14
+ ··· + 249y − 9)
· (4y
26
− 232y
25
+ ··· + 278070166y + 182601169)
c
4
67108864(y − 1)(4y − 1)
3
(y
3
− 2y
2
− 3y − 1)(y
3
− 2y
2
+ y − 1)
· (4y
4
+ 8y
3
+ 8y
2
− 4y + 1)(4y
4
+ 16y
3
+ 13y
2
− 6y + 1)
· (4y
15
+ 56y
14
+ ··· + 717y − 9)
· (4y
26
+ 88y
25
+ ··· + 92757862y + 16491721)
c
6
1048576(y − 1)(4y − 1)
3
(y
3
− 2y
2
− 3y − 1)(y
3
− 2y
2
+ y − 1)
· (4y
4
+ 8y
3
+ 8y
2
− 4y + 1)(4y
4
+ 16y
3
+ 13y
2
− 6y + 1)
· (4y
15
+ 56y
14
+ ··· + 717y − 9)
· (4y
26
+ 88y
25
+ ··· + 92757862y + 16491721)
c
8
9(y − 1)
12
(y
3
− 2y
2
+ y − 1)
2
(y
13
− 2y
12
+ ··· + 17y − 1)
2
· (9y
15
− 24y
14
+ ··· + 555y − 49)
c
9
9(y − 1)
12
(y
3
− 2y
2
+ y − 1)
2
(y
13
− 18y
12
+ ··· + 65y − 1)
2
· (9y
15
− 168y
14
+ ··· − 229y − 49)
c
12
1048576(y − 1)(4y − 1)
3
(y
3
− 2y
2
− 3y − 1)(y
3
− 2y
2
+ y − 1)
· (4y
4
+ 8y
3
+ 8y
2
− 4y + 1)(4y
4
+ 16y
3
+ 13y
2
− 6y + 1)
· (4y
15
− 104y
14
+ ··· + 249y − 9)
· (4y
26
− 232y
25
+ ··· + 278070166y + 182601169)
42
