12a
0348
(K12a
0348
)
A knot diagram
1
Linearized knot diagam
3 6 9 7 10 2 12 4 1 5 8 11
Solving Sequence
4,8 9,11
12 1 3 7 5 10 6 2
c
8
c
11
c
12
c
3
c
7
c
4
c
10
c
5
c
2
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= ⟨2.44733 × 10
21
u
19
− 1.13296 × 10
22
u
18
+ ··· + 5.42213 × 10
21
b − 2.42529 × 10
22
,
− 1.95629 × 10
21
u
19
+ 5.54886 × 10
22
u
18
+ ··· + 2.41285 × 10
23
a + 6.92349 × 10
23
,
9u
20
− 45u
19
+ ··· − 430u + 89⟩
I
u
2
= ⟨−2.31722 × 10
21
u
33
− 5.93285 × 10
21
u
32
+ ··· − 2.13415 × 10
21
a + 2.34088 × 10
21
,
8.20059 × 10
21
au
33
+ 9.19800 × 10
21
u
33
+ ··· − 7.14710 × 10
21
a − 3.57339 × 10
22
, 3u
34
+ 6u
33
+ ··· + u
2
+ 1⟩
I
u
3
= ⟨b, a − 1, u
4
− u
3
+ 2u
2
− 2u + 1⟩
I
u
4
= ⟨−u
3
+ b − u + 1, −u
3
+ u
2
+ a − 2u, u
4
− u
3
+ 2u
2
− 2u + 1⟩
I
u
5
= ⟨b, a − 1, u
2
+ u + 1⟩
I
u
6
= ⟨b − u, a, u
2
+ u + 1⟩
I
u
7
= ⟨−au + b + a + u − 1, 2a
2
− au − 3a + 2u + 1, u
2
+ 1⟩
I
u
8
= ⟨b
2
+ b + 1, −u
5
a
2
+ 2u
5
a + ··· − 2b − 1, −u
3
a + u
3
+ bu − au + b + u,
u
6
a
2
− 2u
6
a + 2u
4
a
2
+ u
6
− 3u
4
a + a
2
u
2
+ u
3
a + u
4
− u
2
a − u
3
+ au + u
2
+ u + 1⟩
I
u
9
= ⟨−u
5
a
2
+ 2u
5
a − 2u
3
a
2
− u
5
+ 3u
3
a − a
2
u + u
2
a − u
3
+ au − u
2
+ b + a − u,
u
6
a
2
− 2u
6
a + 2u
4
a
2
+ u
6
− 3u
4
a + a
2
u
2
− 2u
3
a + u
4
− u
2
a + 2u
3
− 2au + u
2
+ u + 1⟩
* 7 irreducible components of dim
C
= 0, with total 104 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).
1
I. I
u
1
= ⟨2.45 × 10
21
u
19
− 1.13 × 10
22
u
18
+ · · · + 5.42 × 10
21
b − 2.43 ×
10
22
, −1.96 × 10
21
u
19
+ 5.55 × 10
22
u
18
+ · · · + 2.41 × 10
23
a + 6.92 ×
10
23
, 9u
20
− 45u
19
+ · · · − 430u + 89⟩
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
11
=
0.00810779u
19
− 0.229972u
18
+ ··· + 10.0312u − 2.86943
−0.451360u
19
+ 2.08951u
18
+ ··· − 19.3998u + 4.47296
a
12
=
−0.443253u
19
+ 1.85954u
18
+ ··· − 9.36858u + 1.60353
−0.451360u
19
+ 2.08951u
18
+ ··· − 19.3998u + 4.47296
a
1
=
−0.211003u
19
+ 0.993904u
18
+ ··· − 3.69980u + 1.00079
0.0403786u
19
− 0.205972u
18
+ ··· + 0.306071u − 0.245024
a
3
=
u
u
3
+ u
a
7
=
0.0858884u
19
− 0.602361u
18
+ ··· + 12.6293u − 2.96434
−0.149037u
19
+ 0.448835u
18
+ ··· − 0.0305593u − 0.376609
a
5
=
0.225218u
19
− 1.17037u
18
+ ··· + 15.3833u − 4.17263
−0.277055u
19
+ 0.981406u
18
+ ··· + 0.712270u − 0.967405
a
10
=
0.142110u
19
− 0.738313u
18
+ ··· + 10.1501u − 1.73455
−0.0701601u
19
+ 0.526991u
18
+ ··· − 9.31954u + 2.50170
a
6
=
0.262889u
19
− 1.03833u
18
+ ··· + 8.32215u − 2.29128
−0.0555259u
19
+ 0.0487443u
18
+ ··· + 8.11030u − 2.81062
a
2
=
0.0853475u
19
− 0.306104u
18
+ ··· + 3.79747u − 0.473020
0.251374u
19
− 1.26858u
18
+ ··· + 13.5561u − 3.51608
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
18263454827549106375
169441495364258615324
u
19
−
59899476849528023793
42360373841064653831
u
18
+ ··· +
14114427005478277957347
169441495364258615324
u −
4369480590430740839387
169441495364258615324
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
12
u
20
+ 12u
19
+ ··· + 27u + 4
c
2
, c
6
, c
7
c
11
u
20
+ 2u
19
+ ··· + u + 2
c
3
, c
8
9(9u
20
+ 45u
19
+ ··· + 430u + 89)
c
4
, c
9
4(4u
20
− 8u
19
+ ··· + 33u
2
+ 9)
c
5
, c
10
9(9u
20
+ 45u
19
+ ··· + 268u + 89)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
20
− 8y
19
+ ··· + 655y + 16
c
2
, c
6
, c
7
c
11
y
20
+ 12y
19
+ ··· + 27y + 4
c
3
, c
8
81(81y
20
+ 1161y
19
+ ··· + 40982y + 7921)
c
4
, c
9
16(16y
20
+ 80y
19
+ ··· + 594y + 81)
c
5
, c
10
81(81y
20
+ 837y
19
+ ··· + 66838y + 7921)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.092200 + 1.047180I
a = 0.004677 + 0.790560I
b = −0.09546 − 1.64917I
−8.80426 − 0.45536I 1.9853 + 17.0975I
u = 0.092200 − 1.047180I
a = 0.004677 − 0.790560I
b = −0.09546 + 1.64917I
−8.80426 + 0.45536I 1.9853 − 17.0975I
u = 0.848738 + 0.341219I
a = −0.396390 + 0.042942I
b = 0.693866 + 0.427483I
0.08901 − 2.93076I 1.95121 + 2.50627I
u = 0.848738 − 0.341219I
a = −0.396390 − 0.042942I
b = 0.693866 − 0.427483I
0.08901 + 2.93076I 1.95121 − 2.50627I
u = 0.613570 + 0.532135I
a = −1.169550 − 0.185382I
b = 0.168318 − 1.217040I
−9.43070 − 2.05540I −10.55085 + 3.27009I
u = 0.613570 − 0.532135I
a = −1.169550 + 0.185382I
b = 0.168318 + 1.217040I
−9.43070 + 2.05540I −10.55085 − 3.27009I
u = −0.479857 + 0.602074I
a = 0.353859 − 1.186080I
b = 0.026282 − 0.889205I
−2.87875 + 1.45206I −8.03079 − 4.11530I
u = −0.479857 − 0.602074I
a = 0.353859 + 1.186080I
b = 0.026282 + 0.889205I
−2.87875 − 1.45206I −8.03079 + 4.11530I
u = 1.256360 + 0.179722I
a = −0.906871 − 0.475961I
b = 0.603304 − 1.077240I
−3.61567 − 12.98850I −3.22409 + 10.41992I
u = 1.256360 − 0.179722I
a = −0.906871 + 0.475961I
b = 0.603304 + 1.077240I
−3.61567 + 12.98850I −3.22409 − 10.41992I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.202083 + 0.692929I
a = 0.668174 + 0.227261I
b = −0.254612 + 0.364368I
0.359490 − 1.106510I 4.17160 + 6.47143I
u = 0.202083 − 0.692929I
a = 0.668174 − 0.227261I
b = −0.254612 − 0.364368I
0.359490 + 1.106510I 4.17160 − 6.47143I
u = 0.37431 + 1.41465I
a = 1.021310 + 0.740873I
b = −0.944071 − 0.466625I
5.54214 − 7.36961I 4.38642 + 3.37286I
u = 0.37431 − 1.41465I
a = 1.021310 − 0.740873I
b = −0.944071 + 0.466625I
5.54214 + 7.36961I 4.38642 − 3.37286I
u = −0.33594 + 1.49255I
a = −1.032280 + 0.658846I
b = 0.816010 − 0.603612I
8.32420 + 1.49101I 7.14130 + 1.02494I
u = −0.33594 − 1.49255I
a = −1.032280 − 0.658846I
b = 0.816010 + 0.603612I
8.32420 − 1.49101I 7.14130 − 1.02494I
u = 0.54073 + 1.45381I
a = 1.82309 + 0.12465I
b = −0.682856 + 1.134390I
1.4774 − 19.2240I −0.44918 + 10.82584I
u = 0.54073 − 1.45381I
a = 1.82309 − 0.12465I
b = −0.682856 − 1.134390I
1.4774 + 19.2240I −0.44918 − 10.82584I
u = −0.61219 + 1.52578I
a = −1.62445 + 0.10961I
b = 0.669219 + 1.040740I
5.64729 + 12.61550I 2.61904 − 9.47385I
u = −0.61219 − 1.52578I
a = −1.62445 − 0.10961I
b = 0.669219 − 1.040740I
5.64729 − 12.61550I 2.61904 + 9.47385I
6
II. I
u
2
= ⟨−2.32 × 10
21
u
33
− 5.93 × 10
21
u
32
+ · · · − 2.13 × 10
21
a + 2.34 ×
10
21
, 8.20 × 10
21
au
33
+ 9.20 × 10
21
u
33
+ · · · − 7.15 × 10
21
a − 3.57 ×
10
22
, 3u
34
+ 6u
33
+ · · · + u
2
+ 1⟩
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
11
=
a
1.08578u
33
+ 2.77996u
32
+ ··· + a − 1.09687
a
12
=
1.08578u
33
+ 2.77996u
32
+ ··· + 2a − 1.09687
1.08578u
33
+ 2.77996u
32
+ ··· + a − 1.09687
a
1
=
1.97436au
33
− 16.8142u
33
+ ··· − 3.43039a + 13.1329
0.681005au
33
− 1.25817u
33
+ ··· − 0.357162a + 0.186404
a
3
=
u
u
3
+ u
a
7
=
−0.589433au
33
− 1.32608u
33
+ ··· + 0.705146a + 13.5864
0.514636au
33
− 1.30333u
33
+ ··· − 0.245900a + 4.59327
a
5
=
−7.17034au
33
+ 27.2048u
33
+ ··· + 5.86324a − 7.07975
−2.39818au
33
+ 3.29061u
33
+ ··· + 0.291662a + 2.38603
a
10
=
12.4359u
33
+ 3au
32
+ ··· + a − 33.6250
−0.874985au
33
+ 5.42874u
33
+ ··· + 1.39011a − 9.26373
a
6
=
−3au
33
+ 1.73278u
33
+ ··· + 42.6932u + 6.26647
−0.608399u
33
− 1.49941u
32
+ ··· + 1.09687u + 0.361927
a
2
=
1.04587au
33
− 17.2962u
33
+ ··· − 3.60194a + 13.7910
0.263401au
33
− 1.07806u
33
+ ··· − 0.456131a + 0.659305
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
110192114528878828278
33346026295216206221
u
33
−
312161458732301950125
133384105180864824884
u
32
+ ··· −
118687147920568685837
66692052590432412442
u −
611488195859711746311
133384105180864824884
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
12
u
68
+ 28u
67
+ ··· + 3540540u + 405769
c
2
, c
6
, c
7
c
11
u
68
+ 4u
67
+ ··· + 2840u + 637
c
3
, c
8
9(3u
34
− 6u
33
+ ··· + u
2
+ 1)
2
c
4
, c
9
64(64u
68
+ 64u
67
+ ··· + 770022u + 2211093)
c
5
, c
10
9(3u
34
− 6u
33
+ ··· − 4u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
68
+ 28y
67
+ ··· + 3856761155056y + 164648481361
c
2
, c
6
, c
7
c
11
y
68
+ 28y
67
+ ··· + 3540540y + 405769
c
3
, c
8
81(9y
34
+ 240y
33
+ ··· + 2y + 1)
2
c
4
, c
9
4096
· (4096y
68
+ 106496y
67
+ ··· + 7083972171144y + 4888932254649)
c
5
, c
10
81(9y
34
+ 168y
33
+ ··· + 2y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.928923 + 0.129584I
a = −0.887196 − 0.818849I
b = 0.120625 − 1.127660I
−6.72431 + 5.79611I −7.79847 − 4.21340I
u = 0.928923 + 0.129584I
a = −0.487954 + 0.335298I
b = 0.579811 + 1.099130I
−6.72431 + 5.79611I −7.79847 − 4.21340I
u = 0.928923 − 0.129584I
a = −0.887196 + 0.818849I
b = 0.120625 + 1.127660I
−6.72431 − 5.79611I −7.79847 + 4.21340I
u = 0.928923 − 0.129584I
a = −0.487954 − 0.335298I
b = 0.579811 − 1.099130I
−6.72431 − 5.79611I −7.79847 + 4.21340I
u = −1.059230 + 0.243669I
a = −0.847568 + 0.653323I
b = 0.592845 + 1.069630I
−1.74401 + 7.89373I −0.77647 − 6.38566I
u = −1.059230 + 0.243669I
a = −0.610778 − 0.013981I
b = 0.736784 − 0.428981I
−1.74401 + 7.89373I −0.77647 − 6.38566I
u = −1.059230 − 0.243669I
a = −0.847568 − 0.653323I
b = 0.592845 − 1.069630I
−1.74401 − 7.89373I −0.77647 + 6.38566I
u = −1.059230 − 0.243669I
a = −0.610778 + 0.013981I
b = 0.736784 + 0.428981I
−1.74401 − 7.89373I −0.77647 + 6.38566I
u = 0.196215 + 1.099790I
a = 0.344789 + 0.498969I
b = −0.236178 + 1.094890I
1.71677 − 1.07665I 3.45366 + 0.91009I
u = 0.196215 + 1.099790I
a = −1.19817 − 1.18268I
b = 0.753486 + 0.716576I
1.71677 − 1.07665I 3.45366 + 0.91009I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.196215 − 1.099790I
a = 0.344789 − 0.498969I
b = −0.236178 − 1.094890I
1.71677 + 1.07665I 3.45366 − 0.91009I
u = 0.196215 − 1.099790I
a = −1.19817 + 1.18268I
b = 0.753486 − 0.716576I
1.71677 + 1.07665I 3.45366 − 0.91009I
u = −0.411561 + 1.108320I
a = 0.173904 − 0.612229I
b = −0.143739 − 1.063970I
0.82810 + 6.47536I 0.07862 − 6.80334I
u = −0.411561 + 1.108320I
a = −2.12606 + 0.12473I
b = 0.664230 + 0.990769I
0.82810 + 6.47536I 0.07862 − 6.80334I
u = −0.411561 − 1.108320I
a = 0.173904 + 0.612229I
b = −0.143739 + 1.063970I
0.82810 − 6.47536I 0.07862 + 6.80334I
u = −0.411561 − 1.108320I
a = −2.12606 − 0.12473I
b = 0.664230 − 0.990769I
0.82810 − 6.47536I 0.07862 + 6.80334I
u = 0.061990 + 1.217400I
a = 1.271910 − 0.079555I
b = −0.594585 + 1.258760I
2.60150 − 1.59411I 4.61171 + 4.12369I
u = 0.061990 + 1.217400I
a = −1.55260 − 0.58417I
b = 0.832284 + 1.073760I
2.60150 − 1.59411I 4.61171 + 4.12369I
u = 0.061990 − 1.217400I
a = 1.271910 + 0.079555I
b = −0.594585 − 1.258760I
2.60150 + 1.59411I 4.61171 − 4.12369I
u = 0.061990 − 1.217400I
a = −1.55260 + 0.58417I
b = 0.832284 − 1.073760I
2.60150 + 1.59411I 4.61171 − 4.12369I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.309763 + 1.205810I
a = 1.19397 − 0.75885I
b = −1.018550 + 0.481300I
−1.13644 + 4.47248I −0.55538 − 5.09131I
u = −0.309763 + 1.205810I
a = −0.267392 − 0.127975I
b = −0.024069 + 1.347120I
−1.13644 + 4.47248I −0.55538 − 5.09131I
u = −0.309763 − 1.205810I
a = 1.19397 + 0.75885I
b = −1.018550 − 0.481300I
−1.13644 − 4.47248I −0.55538 + 5.09131I
u = −0.309763 − 1.205810I
a = −0.267392 + 0.127975I
b = −0.024069 − 1.347120I
−1.13644 − 4.47248I −0.55538 + 5.09131I
u = 0.119470 + 1.302700I
a = 0.899516 − 0.373320I
b = −0.849347 + 0.300420I
5.51314 − 3.80458I 6.50154 + 2.43385I
u = 0.119470 + 1.302700I
a = −1.75555 + 0.34611I
b = 0.739450 − 1.017200I
5.51314 − 3.80458I 6.50154 + 2.43385I
u = 0.119470 − 1.302700I
a = 0.899516 + 0.373320I
b = −0.849347 − 0.300420I
5.51314 + 3.80458I 6.50154 − 2.43385I
u = 0.119470 − 1.302700I
a = −1.75555 − 0.34611I
b = 0.739450 + 1.017200I
5.51314 + 3.80458I 6.50154 − 2.43385I
u = −0.191943 + 1.294500I
a = −1.40999 − 0.47598I
b = 0.980900 + 0.597909I
4.02612 + 8.09590I 3.41300 − 8.32326I
u = −0.191943 + 1.294500I
a = 1.55696 + 0.03633I
b = −0.660088 − 1.203800I
4.02612 + 8.09590I 3.41300 − 8.32326I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.191943 − 1.294500I
a = −1.40999 + 0.47598I
b = 0.980900 − 0.597909I
4.02612 − 8.09590I 3.41300 + 8.32326I
u = −0.191943 − 1.294500I
a = 1.55696 − 0.03633I
b = −0.660088 + 1.203800I
4.02612 − 8.09590I 3.41300 + 8.32326I
u = −0.678482 + 0.094917I
a = −0.801065 + 0.534099I
b = 0.747677 + 0.320842I
−4.52580 − 0.82511I −5.73325 − 0.12383I
u = −0.678482 + 0.094917I
a = −1.24497 + 1.03985I
b = 0.168560 + 1.112030I
−4.52580 − 0.82511I −5.73325 − 0.12383I
u = −0.678482 − 0.094917I
a = −0.801065 − 0.534099I
b = 0.747677 − 0.320842I
−4.52580 + 0.82511I −5.73325 + 0.12383I
u = −0.678482 − 0.094917I
a = −1.24497 − 1.03985I
b = 0.168560 − 1.112030I
−4.52580 + 0.82511I −5.73325 + 0.12383I
u = 0.364940 + 0.575344I
a = 0.926102 − 0.464353I
b = 0.488575 + 0.554913I
0.78238 − 1.45136I 2.79629 + 5.22795I
u = 0.364940 + 0.575344I
a = −0.0415032 + 0.1092230I
b = −0.528462 + 0.608685I
0.78238 − 1.45136I 2.79629 + 5.22795I
u = 0.364940 − 0.575344I
a = 0.926102 + 0.464353I
b = 0.488575 − 0.554913I
0.78238 + 1.45136I 2.79629 − 5.22795I
u = 0.364940 − 0.575344I
a = −0.0415032 − 0.1092230I
b = −0.528462 − 0.608685I
0.78238 + 1.45136I 2.79629 − 5.22795I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.446114 + 1.243030I
a = −0.368802 + 0.014090I
b = 0.003968 − 1.305630I
−3.22322 − 10.68950I −3.43174 + 7.82810I
u = 0.446114 + 1.243030I
a = 1.83566 − 0.03089I
b = −0.709112 + 1.154840I
−3.22322 − 10.68950I −3.43174 + 7.82810I
u = 0.446114 − 1.243030I
a = −0.368802 − 0.014090I
b = 0.003968 + 1.305630I
−3.22322 + 10.68950I −3.43174 − 7.82810I
u = 0.446114 − 1.243030I
a = 1.83566 + 0.03089I
b = −0.709112 − 1.154840I
−3.22322 + 10.68950I −3.43174 − 7.82810I
u = 0.030836 + 1.337180I
a = 1.022660 + 0.469683I
b = −0.932328 − 0.336335I
6.63609 − 2.25268I 7.17939 + 3.46008I
u = 0.030836 + 1.337180I
a = −1.34321 + 0.62136I
b = 0.917345 − 0.650827I
6.63609 − 2.25268I 7.17939 + 3.46008I
u = 0.030836 − 1.337180I
a = 1.022660 − 0.469683I
b = −0.932328 + 0.336335I
6.63609 + 2.25268I 7.17939 − 3.46008I
u = 0.030836 − 1.337180I
a = −1.34321 − 0.62136I
b = 0.917345 + 0.650827I
6.63609 + 2.25268I 7.17939 − 3.46008I
u = −0.237485 + 0.550986I
a = 0.95003 + 1.63364I
b = −0.558906 + 0.935813I
−0.10373 − 2.96497I −1.46555 − 1.43016I
u = −0.237485 + 0.550986I
a = 2.75289 + 0.07158I
b = 0.389516 − 0.711934I
−0.10373 − 2.96497I −1.46555 − 1.43016I
14
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.237485 − 0.550986I
a = 0.95003 − 1.63364I
b = −0.558906 − 0.935813I
−0.10373 + 2.96497I −1.46555 + 1.43016I
u = −0.237485 − 0.550986I
a = 2.75289 − 0.07158I
b = 0.389516 + 0.711934I
−0.10373 + 2.96497I −1.46555 + 1.43016I
u = −0.461359 + 0.273852I
a = −0.696005 + 0.649290I
b = −0.663647 − 0.671451I
−0.67894 + 5.72667I −2.62369 − 8.41817I
u = −0.461359 + 0.273852I
a = −0.01371 + 2.29902I
b = 0.529575 + 1.033290I
−0.67894 + 5.72667I −2.62369 − 8.41817I
u = −0.461359 − 0.273852I
a = −0.696005 − 0.649290I
b = −0.663647 + 0.671451I
−0.67894 − 5.72667I −2.62369 + 8.41817I
u = −0.461359 − 0.273852I
a = −0.01371 − 2.29902I
b = 0.529575 − 1.033290I
−0.67894 − 5.72667I −2.62369 + 8.41817I
u = −0.46554 + 1.43082I
a = 0.994812 − 0.782444I
b = −0.933512 + 0.483430I
3.47191 + 13.30640I 0. − 7.02472I
u = −0.46554 + 1.43082I
a = 1.78917 − 0.09812I
b = −0.681591 − 1.143150I
3.47191 + 13.30640I 0. − 7.02472I
u = −0.46554 − 1.43082I
a = 0.994812 + 0.782444I
b = −0.933512 − 0.483430I
3.47191 − 13.30640I 0. + 7.02472I
u = −0.46554 − 1.43082I
a = 1.78917 + 0.09812I
b = −0.681591 + 1.143150I
3.47191 − 13.30640I 0. + 7.02472I
15
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.47159 + 1.50485I
a = −0.959034 − 0.635623I
b = 0.801025 + 0.589395I
7.00842 − 7.08959I 0
u = 0.47159 + 1.50485I
a = −1.66282 − 0.03292I
b = 0.679056 − 1.037650I
7.00842 − 7.08959I 0
u = 0.47159 − 1.50485I
a = −0.959034 + 0.635623I
b = 0.801025 − 0.589395I
7.00842 + 7.08959I 0
u = 0.47159 − 1.50485I
a = −1.66282 + 0.03292I
b = 0.679056 + 1.037650I
7.00842 + 7.08959I 0
u = 0.195287 + 0.174782I
a = 1.98483 − 4.05832I
b = −0.653211 − 0.893273I
−1.28851 − 0.65000I −6.53552 + 1.99005I
u = 0.195287 + 0.174782I
a = 3.07716 − 4.26636I
b = 0.461611 − 1.028320I
−1.28851 − 0.65000I −6.53552 + 1.99005I
u = 0.195287 − 0.174782I
a = 1.98483 + 4.05832I
b = −0.653211 + 0.893273I
−1.28851 + 0.65000I −6.53552 − 1.99005I
u = 0.195287 − 0.174782I
a = 3.07716 + 4.26636I
b = 0.461611 + 1.028320I
−1.28851 + 0.65000I −6.53552 − 1.99005I
16
III. I
u
3
= ⟨b, a − 1, u
4
− u
3
+ 2u
2
− 2u + 1⟩
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
11
=
1
0
a
12
=
1
0
a
1
=
1
0
a
3
=
u
u
3
+ u
a
7
=
1
0
a
5
=
−u
u
a
10
=
−u
2
+ 1
u
2
a
6
=
−u
3
u
3
+ u
a
2
=
u
3
− u
2
+ 2u
u
3
+ u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
+ 4u − 2
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
+ u + 1)
2
c
2
, c
6
(u
2
− u + 1)
2
c
3
, c
4
, c
5
c
8
, c
10
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
7
, c
11
, c
12
u
4
c
9
u
4
+ 3u
3
+ 2u
2
+ 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
(y
2
+ y + 1)
2
c
3
, c
4
, c
5
c
8
, c
10
y
4
+ 3y
3
+ 2y
2
+ 1
c
7
, c
11
, c
12
y
4
c
9
y
4
− 5y
3
+ 6y
2
+ 4y + 1
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.621744 + 0.440597I
a = 1.00000
b = 0
− 2.02988I 0. + 3.46410I
u = 0.621744 − 0.440597I
a = 1.00000
b = 0
2.02988I 0. − 3.46410I
u = −0.121744 + 1.306620I
a = 1.00000
b = 0
2.02988I 0. − 3.46410I
u = −0.121744 − 1.306620I
a = 1.00000
b = 0
− 2.02988I 0. + 3.46410I
20
IV. I
u
4
= ⟨−u
3
+ b − u + 1, −u
3
+ u
2
+ a − 2u, u
4
− u
3
+ 2u
2
− 2u + 1⟩
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
11
=
u
3
− u
2
+ 2u
u
3
+ u − 1
a
12
=
2u
3
− u
2
+ 3u − 1
u
3
+ u − 1
a
1
=
u
u
3
+ u
a
3
=
u
u
3
+ u
a
7
=
−u
−u
3
− u
a
5
=
−u
3
1
a
10
=
u
2
+ 1
u
3
+ 2u − 1
a
6
=
u
u
3
+ u
a
2
=
u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
+ 4u − 2
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
4
c
3
, c
5
, c
8
c
9
, c
10
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
4
u
4
+ 3u
3
+ 2u
2
+ 1
c
7
, c
11
(u
2
− u + 1)
2
c
12
(u
2
+ u + 1)
2
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y
4
c
3
, c
5
, c
8
c
9
, c
10
y
4
+ 3y
3
+ 2y
2
+ 1
c
4
y
4
− 5y
3
+ 6y
2
+ 4y + 1
c
7
, c
11
, c
12
(y
2
+ y + 1)
2
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.621744 + 0.440597I
a = 0.929304 + 0.758745I
b = −0.500000 + 0.866025I
− 2.02988I 0. + 3.46410I
u = 0.621744 − 0.440597I
a = 0.929304 − 0.758745I
b = −0.500000 − 0.866025I
2.02988I 0. − 3.46410I
u = −0.121744 + 1.306620I
a = 2.07070 + 0.75874I
b = −0.500000 − 0.866025I
2.02988I 0. − 3.46410I
u = −0.121744 − 1.306620I
a = 2.07070 − 0.75874I
b = −0.500000 + 0.866025I
− 2.02988I 0. + 3.46410I
24
V. I
u
5
= ⟨b, a − 1, u
2
+ u + 1⟩
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
−u − 1
a
11
=
1
0
a
12
=
1
0
a
1
=
1
0
a
3
=
u
u + 1
a
7
=
1
0
a
5
=
−u
u
a
10
=
u + 2
−u − 1
a
6
=
−1
u + 1
a
2
=
0
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u + 2
25
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
9
u
2
+ u + 1
c
2
, c
3
, c
4
c
5
, c
6
, c
8
c
10
u
2
− u + 1
c
7
, c
11
, c
12
u
2
26
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
9
, c
10
y
2
+ y + 1
c
7
, c
11
, c
12
y
2
27
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 1.00000
b = 0
− 2.02988I 0. + 3.46410I
u = −0.500000 − 0.866025I
a = 1.00000
b = 0
2.02988I 0. − 3.46410I
28
VI. I
u
6
= ⟨b − u, a, u
2
+ u + 1⟩
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
−u − 1
a
11
=
0
u
a
12
=
u
u
a
1
=
u
u + 1
a
3
=
u
u + 1
a
7
=
−u
−u − 1
a
5
=
−1
2u
a
10
=
−u
−u − 2
a
6
=
u
u + 1
a
2
=
u
u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u + 2
29
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
2
c
3
, c
5
, c
7
c
8
, c
9
, c
10
c
11
u
2
− u + 1
c
4
, c
12
u
2
+ u + 1
30
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y
2
c
3
, c
4
, c
5
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y
2
+ y + 1
31
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0
b = −0.500000 + 0.866025I
− 2.02988I 0. + 3.46410I
u = −0.500000 − 0.866025I
a = 0
b = −0.500000 − 0.866025I
2.02988I 0. − 3.46410I
32
VII. I
u
7
= ⟨−au + b + a + u − 1, 2a
2
− au − 3a + 2u + 1, u
2
+ 1⟩
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
−1
a
11
=
a
au − a − u + 1
a
12
=
au − u + 1
au − a − u + 1
a
1
=
2au − a −
5
2
u +
3
2
−au + a + 2u − 1
a
3
=
u
0
a
7
=
au −
1
2
u +
1
2
au + a − u − 2
a
5
=
1
2
au +
1
2
a −
1
2
u +
1
2
−a + u + 1
a
10
=
1
2
au +
1
2
a +
1
2
u +
1
2
−a
a
6
=
au
−au − a + u + 1
a
2
=
au −
1
2
u +
1
2
−au + a + 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
33
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
− 3u + 1)
2
c
2
, c
6
, c
7
c
11
u
4
+ 3u
2
+ 1
c
3
, c
5
, c
8
c
10
(u
2
+ 1)
2
c
4
, c
9
4(4u
4
− 4u
3
+ 2u
2
+ 2u + 1)
c
12
(u
2
+ 3u + 1)
2
34
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
(y
2
− 7y + 1)
2
c
2
, c
6
, c
7
c
11
(y
2
+ 3y + 1)
2
c
3
, c
5
, c
8
c
10
(y + 1)
4
c
4
, c
9
16(16y
4
+ 28y
2
+ 1)
35
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 0.190983 + 0.809017I
b = − 1.61803I
−8.88264 −4.00000
u = 1.000000I
a = 1.309020 − 0.309017I
b = 0.618034I
−0.986960 −4.00000
u = − 1.000000I
a = 0.190983 − 0.809017I
b = 1.61803I
−8.88264 −4.00000
u = − 1.000000I
a = 1.309020 + 0.309017I
b = − 0.618034I
−0.986960 −4.00000
36
VIII. I
u
8
= ⟨b
2
+ b + 1, −u
5
a
2
+ 2u
5
a + · · · − 2b − 1, −u
3
a + u
3
+ bu − au +
b + u, u
6
a
2
− 2u
6
a + · · · + u + 1⟩
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
11
=
a
b
a
12
=
b + a
b
a
1
=
−
1
2
u
5
a
2
+ u
5
a + ··· − u −
1
2
b + 1
a
3
=
u
u
3
+ u
a
7
=
1
2
u
5
a
2
− u
5
a + ··· + u +
1
2
−b − 1
a
5
=
−
1
2
u
5
a
3
+
3
2
u
5
a
2
+ ··· −
1
2
a +
1
2
−au + 2u
a
10
=
1
2
u
5
a
3
−
3
2
u
5
a
2
+ ··· +
3
2
a −
1
2
−u
2
a + 2u
2
+ b
a
6
=
−a + u + 1
u
3
− b + u
a
2
=
−
1
2
u
5
a
2
+ u
5
a + ··· − b −
1
2
u
5
a − u
4
a − u
5
+ 3u
3
a + u
4
− u
2
a − 2u
3
+ 2au + u
2
− b − u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8b + 4
(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.
37
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
− 4.05977I − 6.92820I
38
IX. I
u
9
= ⟨−u
5
a
2
+ 2u
5
a + · · · + b + a, u
6
a
2
− 2u
6
a + · · · + u + 1⟩
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
11
=
a
u
5
a
2
− 2u
5
a + 2u
3
a
2
+ u
5
− 3u
3
a + a
2
u − u
2
a + u
3
− au + u
2
− a + u
a
12
=
u
5
a
2
− 2u
5
a + 2u
3
a
2
+ u
5
− 3u
3
a + a
2
u − u
2
a + u
3
− au + u
2
+ u
u
5
a
2
− 2u
5
a + 2u
3
a
2
+ u
5
− 3u
3
a + a
2
u − u
2
a + u
3
− au + u
2
− a + u
a
1
=
−u
5
a
3
+ 3u
5
a
2
+ ··· + a
2
− a
u
5
a
2
− 2u
5
a + ··· − a + 1
a
3
=
u
u
3
+ u
a
7
=
u
5
a
3
− 3u
5
a
2
+ ··· − a
2
+ a
−u
5
a
2
+ 2u
5
a + ··· + a − 1
a
5
=
a
3
u
3
− 3u
3
a
2
+ a
3
u + 3u
3
a − 2a
2
u − u
3
− a
2
+ au + 2a − 1
−au + 2u
a
10
=
u
4
a
3
− 3u
4
a
2
+ a
3
u
2
+ 3u
4
a − 2a
2
u
2
− u
4
− a
2
u + u
2
a + 2au + a − u
u
5
a
2
− 2u
5
a + 2u
3
a
2
+ u
5
− 3u
3
a + a
2
u − 2u
2
a + u
3
− au + 3u
2
− a + u
a
6
=
u
5
a
3
− 3u
5
a
2
+ ··· + 2a − 1
u
3
+ u − 1
a
2
=
−u
5
a
3
+ 3u
5
a
2
+ ··· − a + 1
u
5
a
2
− 3u
5
a + ··· − a + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
(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.
39
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
9
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
40
X. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
6
(u
2
− 3u + 1)
2
(u
2
+ u + 1)
3
(u
20
+ 12u
19
+ ··· + 27u + 4)
· (u
68
+ 28u
67
+ ··· + 3540540u + 405769)
c
2
, c
6
, c
7
c
11
u
6
(u
2
− u + 1)
3
(u
4
+ 3u
2
+ 1)(u
20
+ 2u
19
+ ··· + u + 2)
· (u
68
+ 4u
67
+ ··· + 2840u + 637)
c
3
, c
8
81(u
2
+ 1)
2
(u
2
− u + 1)
2
(u
4
+ u
3
+ 2u
2
+ 2u + 1)
2
· (9u
20
+ 45u
19
+ ··· + 430u + 89)(3u
34
− 6u
33
+ ··· + u
2
+ 1)
2
c
4
, c
9
1024(u
2
− u + 1)(u
2
+ u + 1)(u
4
+ u
3
+ ··· + 2u + 1)(u
4
+ 3u
3
+ 2u
2
+ 1)
· (4u
4
− 4u
3
+ 2u
2
+ 2u + 1)(4u
20
− 8u
19
+ ··· + 33u
2
+ 9)
· (64u
68
+ 64u
67
+ ··· + 770022u + 2211093)
c
5
, c
10
81(u
2
+ 1)
2
(u
2
− u + 1)
2
(u
4
+ u
3
+ 2u
2
+ 2u + 1)
2
· (9u
20
+ 45u
19
+ ··· + 268u + 89)(3u
34
− 6u
33
+ ··· − 4u + 1)
2
c
12
u
6
(u
2
+ u + 1)
3
(u
2
+ 3u + 1)
2
(u
20
+ 12u
19
+ ··· + 27u + 4)
· (u
68
+ 28u
67
+ ··· + 3540540u + 405769)
41
XI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
6
(y
2
− 7y + 1)
2
(y
2
+ y + 1)
3
(y
20
− 8y
19
+ ··· + 655y + 16)
· (y
68
+ 28y
67
+ ··· + 3856761155056y + 164648481361)
c
2
, c
6
, c
7
c
11
y
6
(y
2
+ y + 1)
3
(y
2
+ 3y + 1)
2
(y
20
+ 12y
19
+ ··· + 27y + 4)
· (y
68
+ 28y
67
+ ··· + 3540540y + 405769)
c
3
, c
8
6561(y + 1)
4
(y
2
+ y + 1)
2
(y
4
+ 3y
3
+ 2y
2
+ 1)
2
· (81y
20
+ 1161y
19
+ ··· + 40982y + 7921)
· (9y
34
+ 240y
33
+ ··· + 2y + 1)
2
c
4
, c
9
1048576(y
2
+ y + 1)
2
(y
4
− 5y
3
+ ··· + 4y + 1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (16y
4
+ 28y
2
+ 1)(16y
20
+ 80y
19
+ ··· + 594y + 81)
· (4096y
68
+ 106496y
67
+ ··· + 7083972171144y + 4888932254649)
c
5
, c
10
6561(y + 1)
4
(y
2
+ y + 1)
2
(y
4
+ 3y
3
+ 2y
2
+ 1)
2
· (81y
20
+ 837y
19
+ ··· + 66838y + 7921)
· (9y
34
+ 168y
33
+ ··· + 2y + 1)
2
42
