11a
350
(K11a
350
)
1
Arc Sequences
6 5 1 11 9 10 4 3 2 7 8
Solving Sequence
4,11 5,8
1 3 9 2 7 10 6
c
4
c
11
c
3
c
8
c
2
c
7
c
10
c
6
c
1
, c
5
, c
9
Representation Ideals
I =
5
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, a 1, b 1i
I
u
2
= hu
4
+ u
3
u
2
u 1, u
2
+ b 1, u
3
+ a + 2ui
I
u
3
= hu
20
+ 12u
18
+ ··· + 2u + 1,
1004384011943253u
19
+ 596388417531502u
18
+ ··· + 1506447924749558a + 145454116419295,
1924336217084222u
19
761995836313501u
18
+ ··· + 1506447924749558b 1534485576412749i
I
u
4
= hu
24
+ 5u
23
+ ··· + 5u + 1, 1.77351 × 10
21
u
23
+ 2.33204 × 10
22
u
22
+ ··· + 1.84096 × 10
22
a + 9.52568 × 10
22
,
3.19673 × 10
22
u
23
1.29822 × 10
23
u
22
+ ··· + 1.84096 × 10
22
b + 1.69641 × 10
22
i
I
u
5
= hu
90
+ 13u
88
+ ··· 2331u 697,
3.32974 × 10
395
u
89
+ 3.91857 × 10
394
u
88
+ ··· + 7.51178 × 10
396
b + 1.55103 × 10
397
,
8.98383 × 10
397
u
89
+ 4.40075 × 10
397
u
88
+ ··· + 5.23571 × 10
399
a 2.59039 × 10
400
i
There are 5 irreducible components with 140 representations.
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= hu
2
+ u + 1, a 1, b 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
1
1
a
5
=
0
1
a
8
=
0
u
a
1
=
1
u
a
3
=
u + 1
u + 1
a
9
=
u 1
1
a
2
=
u + 1
0
a
7
=
u
u
a
10
=
1
1
a
6
=
u
u
a
6
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 1.00000
b = 1.00000
3.28987 6.00000
u = 0.500000 + 0.866025I
a = 1.00000
b = 1.00000
3.28987 6.00000
3
II. I
u
2
= hu
4
+ u
3
u
2
u 1, u
2
+ b 1, u
3
+ a + 2ui
(i) Arc colorings
a
4
=
1
0
a
11
=
u
3
2u
u
2
+ 1
a
5
=
u
3
+ 2u
u
3
+ u
2
u 2
a
8
=
0
u
a
1
=
u
3
2u
u
2
a
3
=
0
u
3
u
2
u 1
a
9
=
0
u
a
2
=
1
u
3
2u 1
a
7
=
u
u
a
10
=
u
u
3
u
2
+ u + 1
a
6
=
u
3
+ 2u
u
3
+ u
2
u 1
a
6
=
u
3
+ 2u
u
3
+ u
2
u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.51288
a = 0.436911
b = 1.28879
0.459232 9.15137
u = 0.332924 0.670769I
a = 1.07833 + 1.42030I
b = 1.339093 0.446630I
5.36351 2.52742I 0.91810 + 4.26254I
u = 0.332924 + 0.670769I
a = 1.07833 1.42030I
b = 1.339093 + 0.446630I
5.36351 + 2.52742I 0.91810 4.26254I
u = 1.17872
a = 0.719740
b = 0.389391
3.68806 12.6848
5
III. I
u
3
=
hu
20
+12u
18
+· · ·+2u+1, 1.00×10
15
u
19
+5.96×10
14
u
18
+· · ·+1.51×10
15
a+
1.45 × 10
14
, 1.92 × 10
15
u
19
7.62 × 10
14
u
18
+ · · · + 1.51 × 10
15
b 1.53 × 10
15
i
(i) Arc colorings
a
4
=
1
0
a
11
=
0.666723u
19
0.395890u
18
+ ··· 7.55730u 0.0965544
1.27740u
19
+ 0.505823u
18
+ ··· 1.30161u + 1.01861
a
5
=
0.215007u
19
+ 1.19786u
18
+ ··· + 4.53728u + 2.00382
0.133507u
19
+ 1.00513u
18
+ ··· + 1.31497u + 0.994920
a
8
=
0
u
a
1
=
0.666723u
19
0.395890u
18
+ ··· 7.55730u 0.0965544
0.921777u
19
+ 0.630606u
18
+ ··· + 0.156893u + 1.41450
a
3
=
1.53607u
19
+ 0.379409u
18
+ ··· + 5.81323u + 2.41870
0.487299u
19
0.278951u
18
+ ··· 3.16063u 0.582346
a
9
=
1.43324u
19
+ 0.178004u
18
+ ··· + 0.521690u + 0.403000
0.232443u
19
0.776398u
18
+ ··· 5.22717u 2.21910
a
2
=
0.230371u
19
0.484579u
18
+ ··· 0.838000u 0.497919
0.856046u
19
1.03523u
18
+ ··· 5.12814u 1.45889
a
7
=
u
u
a
10
=
0.781758u
19
+ 0.119275u
18
+ ··· 6.36455u + 0.805159
1.39243u
19
+ 1.02099u
18
+ ··· 0.108861u + 1.92033
a
6
=
1.23049u
19
+ 1.32381u
18
+ ··· + 0.907807u 1.35726
0.574971u
19
0.264639u
18
+ ··· + 0.334506u 1.23993
a
6
=
1.23049u
19
+ 1.32381u
18
+ ··· + 0.907807u 1.35726
0.574971u
19
0.264639u
18
+ ··· + 0.334506u 1.23993
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.95246 1.17976I
a = 0.480385 + 0 .108335I
b = 0.930555 0.833096I
4.88933 3.47667I 0.85068 1.18559I
u = 0.95246 + 1 .17976I
a = 0.480385 0.108335I
b = 0.930555 + 0 .833096I
4.88933 + 3.47667I 0.85068 + 1.18559I
u = 0.82142 1.38988I
a = 0.831644 + 0.388470I
b = 1.08938 + 1.17378I
3.05353 9.22667I 1.88985 + 9.76468I
u = 0.82142 + 1 .38988I
a = 0.831644 0.388470I
b = 1.08938 1.17378I
3.05353 + 9.22667I 1.88985 9.76468I
u = 0.344777 0.171189I
a = 3.04159 + 2 .84285I
b = 0.512164 + 0 .027302I
0.28461 4.56548I 2.10704 + 11.49589I
u = 0.344777 + 0 .171189I
a = 3.04159 2.84285I
b = 0.512164 0.027302I
0.28461 + 4.56548I 2.10704 11.49589I
u = 0.247229 0.712054I
a = 1.53262 0.86551I
b = 2.01723 + 0.53024I
5.18058 1.49359
u = 0.247229 + 0 .712054I
a = 1.53262 + 0.86551I
b = 2.01723 0.53024I
5.18058 1.49359
u = 0.083868 0.715519I
a = 0.203218 + 1.313127I
b = 0.359282 + 1 .032369I
2.05119 + 4.76907I 4.19559 4.84396I
u = 0.083868 + 0 .715519I
a = 0.203218 1.313127I
b = 0.359282 1.032369I
2.05119 4.76907I 4.19559 + 4.84396I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.021497 0.702286I
a = 0.85059 + 1 .57411I
b = 0.263480 0.599946I
4.88933 3.47667I 0.85068 1.18559I
u = 0.021497 + 0 .702286I
a = 0.85059 1.57411I
b = 0.263480 + 0 .599946I
4.88933 + 3.47667I 0.85068 + 1 .18559I
u = 0.12157 3 .08967I
a = 0.0935213 0.0468344I
b = 0.081756 0.345839I
0.0546371 11.3948
u = 0.12157 + 3 .08967I
a = 0.0935213 + 0 .0468344I
b = 0.081756 + 0 .345839I
0.0546371 11.3948
u = 0.758707 0.684264I
a = 1.55289 + 0.00082I
b = 0.349179 0.763759I
0.28461 + 4.56548I 2.10704 11.49589I
u = 0.758707 + 0 .684264I
a = 1.55289 0.00082I
b = 0.349179 + 0 .763759I
0.28461 4.56548I 2.10704 + 11.49589I
u = 0.769326 0.148206I
a = 0.294260 0.614060I
b = 1.85833 + 0 .06011I
3.05353 + 9.22667I 1.88985 9.76468I
u = 0.769326 + 0 .148206I
a = 0.294260 + 0 .614060I
b = 1.85833 0.06011I
3.05353 9.22667I 1.88985 + 9.76468I
u = 0.778657 0.810317I
a = 1.139968 0.137068I
b = 1.049781 0.529863I
2.05119 + 4.76907I 4.19559 4.84396I
u = 0.778657 + 0 .810317I
a = 1.139968 + 0.137068I
b = 1.049781 + 0 .529863I
2.05119 4.76907I 4.19559 + 4 .84396I
8
IV. I
u
4
=
hu
24
+5u
23
+· · ·+5u+1, 1.77×10
21
u
23
+2.33×10
22
u
22
+· · ·+1.84×10
22
a+
9.53 × 10
22
, 3.20× 10
22
u
23
1.30 × 10
23
u
22
+ · · · + 1.84 × 10
22
b + 1.70 × 10
22
i
(i) Arc colorings
a
4
=
1
0
a
11
=
0.0963361u
23
1.26676u
22
+ ··· 9.53253u 5.17431
1.73645u
23
+ 7.05188u
22
+ ··· + 0.294313u 0.921482
a
5
=
4.62275u
23
+ 19.0713u
22
+ ··· + 5.20858u + 0.334058
2.98775u
23
+ 9.96658u
22
+ ··· 17.2892u 5.52017
a
8
=
0
u
a
1
=
0.0963361u
23
1.26676u
22
+ ··· 9.53253u 5.17431
2.58570u
23
+ 11.7047u
22
+ ··· + 8.94016u + 0.826954
a
3
=
4.70063u
23
+ 22.6495u
22
+ ··· + 30.9144u + 6.96158
2.30023u
23
+ 9.00070u
22
+ ··· 2.13202u 0.624051
a
9
=
1.03260u
23
+ 6.73243u
22
+ ··· + 12.8715u + 5.54934
1.74443u
23
+ 8.33782u
22
+ ··· + 5.56945u + 0.493106
a
2
=
0.493106u
23
0.721096u
22
+ ··· + 15.8450u + 3.10392
0.310299u
23
+ 5.64788u
22
+ ··· + 35.9680u + 9.64576
a
7
=
u
u
a
10
=
0.711836u
23
+ 1.60538u
22
+ ··· 7.30204u 5.05623
2.35195u
23
+ 9.92402u
22
+ ··· + 2.52481u 0.803405
a
6
=
1.11471u
23
6.93641u
22
+ ··· 20.5579u 3.38095
2.40004u
23
+ 10.0654u
22
+ ··· 1.66561u 2.20692
a
6
=
1.11471u
23
6.93641u
22
+ ··· 20.5579u 3.38095
2.40004u
23
+ 10.0654u
22
+ ··· 1.66561u 2.20692
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 4.60596
a = 0.0381895
b = 0.261896
0.0136435 185.449
u = 1.80800
a = 0.228116
b = 0.889432
0.0704593 10.7549
u = 0.913544 0.934068I
a = 1.233681 0.115558I
b = 0.785793 1.019607I
2.20266 13.23266I 2.28061 + 10.36959I
u = 0.913544 + 0 .934068I
a = 1.233681 + 0 .115558I
b = 0.785793 + 1 .019607I
2.20266 + 13.23266I 2.28061 10.36959I
u = 0.897247 0.630246I
a = 0.933814 0.602137I
b = 0.585130 + 0.498475I
1.59590 4.68195I 5.09129 + 10 .67393I
u = 0.897247 + 0 .630246I
a = 0.933814 + 0.602137I
b = 0.585130 0.498475I
1.59590 + 4 .68195I 5.09129 10.67393I
u = 0.397178 0.844071I
a = 0.625486 + 0 .273992I
b = 1.04842 1.16694I
5.31187 3.98651I 5.66668 + 8.11376I
u = 0.397178 + 0 .844071I
a = 0.625486 0.273992I
b = 1.04842 + 1 .16694I
5.31187 + 3 .98651I 5.66668 8.11376I
u = 0.315740 0.142853I
a = 3.12574 0.79725I
b = 0.321002 0.655022I
0.03065 2.94504I 0.62404 + 2 .53282I
u = 0.315740 + 0 .142853I
a = 3.12574 + 0.79725I
b = 0.321002 + 0 .655022I
0.03065 + 2 .94504I 0.62404 2.53282I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.284422 0.660333I
a = 1.60718 0.88125I
b = 1.94093 + 0 .79249I
5.07438 1.59386I 2.13582 + 1.10254I
u = 0.284422 + 0 .660333I
a = 1.60718 + 0.88125I
b = 1.94093 0.79249I
5.07438 + 1.59386I 2.13582 1.10254I
u = 0.275157 0.682551I
a = 1.23917 1.99924I
b = 1.165374 + 0 .199995I
5.80334 2.74033I 15.5158 + 11.8363I
u = 0.275157 + 0 .682551I
a = 1.23917 + 1.99924I
b = 1.165374 0.199995I
5.80334 + 2.74033I 15.5158 11.8363I
u = 0.219244 0.655965I
a = 0.797642 0.055736I
b = 0.263742 + 0 .556060I
0.112664 + 1 .342819I 0.96375 5.78549I
u = 0.219244 + 0 .655965I
a = 0.797642 + 0 .055736I
b = 0.263742 0.556060I
0.112664 1.342819I 0.96375 + 5.78549I
u = 0.669369 0.399651I
a = 2.20939 0.64260I
b = 0.153633 + 0 .560178I
0.94261 + 4.71787I 11.6555 14.8007I
u = 0.669369 + 0 .399651I
a = 2.20939 + 0 .64260I
b = 0.153633 0.560178I
0.94261 4.71787I 11.6555 + 14 .8007I
u = 0.948562 0.118544I
a = 0.581109 0.319272I
b = 0.471369 0.789803I
2.99609 1.94907I 10.80827 + 2.58369I
u = 0.948562 + 0 .118544I
a = 0.581109 + 0.319272I
b = 0.471369 + 0 .789803I
2.99609 + 1.94907I 10.80827 2.58369I
11
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.95916 1. 28491I
a = 0.945428 + 0 .223381I
b = 1.27764 + 1 .17372I
3.9513 + 19 .3243I 0.92200 10.02171I
u = 0.95916 + 1. 28491I
a = 0.945428 0.223381I
b = 1.27764 1.17372I
3.9513 19.3243I 0.92200 + 10.02171I
u = 0.99393 1. 39275I
a = 0.691469 0.002103I
b = 0.969764 0.856772I
7.26188 + 10.39509I 4.97109 9.26898I
u = 0.99393 + 1. 39275I
a = 0.691469 + 0.002103I
b = 0.969764 + 0.856772I
7.26188 10.39509I 4.97109 + 9.26898I
12
V. I
u
5
= hu
90
+ 13u
88
+ · · · 2331u 697, 3.33 × 10
395
u
89
+ 3.92 ×
10
394
u
88
+ · · · + 7.51 × 10
396
b + 1.55 × 10
397
, 8.98 × 10
397
u
89
+ 4.40 ×
10
397
u
88
+ · · · + 5.24 × 10
399
a 2.59 × 10
400
i
(i) Arc colorings
a
4
=
1
0
a
11
=
0.0171588u
89
0.00840525u
88
+ ··· + 12.1863u + 4.94754
0.0443270u
89
0.00521657u
88
+ ··· + 109.925u 2.06479
a
5
=
0.0148842u
89
0.00924215u
88
+ ··· + 56.6153u + 20.9406
0.00893073u
89
+ 0.0387779u
88
+ ··· 286.423u 68.4010
a
8
=
0
u
a
1
=
0.0171588u
89
0.00840525u
88
+ ··· + 12.1863u + 4.94754
0.0351084u
89
0.00121783u
88
+ ··· + 78.3728u 7.92325
a
3
=
0.00168374u
89
+ 0.00862368u
88
+ ··· 3.83748u 13.2749
0.00556089u
89
+ 0.0372899u
88
+ ··· 278.664u 68.8320
a
9
=
0.0344163u
89
0.0153682u
88
+ ··· + 216.569u + 42.9654
0.0678970u
89
0.0149324u
88
+ ··· + 132.480u + 3.08375
a
2
=
0.0400693u
89
+ 0.00355555u
88
+ ··· + 65.6161u 13.1441
0.0633408u
89
+ 0.0000414706u
88
+ ··· 10.8900u + 22.1890
a
7
=
u
u
a
10
=
0.0191629u
89
0.00721730u
88
+ ··· + 23.6897u + 2.72503
0.0463311u
89
0.00402862u
88
+ ··· + 121.428u 4.28730
a
6
=
0.0394519u
89
0.0139733u
88
+ ··· + 228.214u + 41.0694
0.0476243u
89
0.0103409u
88
+ ··· + 100.448u + 1.61779
a
6
=
0.0394519u
89
0.0139733u
88
+ ··· + 228.214u + 41.0694
0.0476243u
89
0.0103409u
88
+ ··· + 100.448u + 1.61779
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
13
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.86135
a = 0.476317
b = 0.144622
1.80906 10.9203
u = 1.53129 0.40027I
a = 0.376675 0.236887I
b = 0.439194 0.308476I
2.94180 + 3.09675I 0.67036 8.19799I
u = 1.53129 + 0 .40027I
a = 0.376675 + 0 .236887I
b = 0.439194 + 0.308476I
2.94180 3.09675I 0.67036 + 8.19799I
u = 1.12939 1.35997I
a = 0.418028 0.108447I
b = 0.794756 + 0.815720I
5.20109 3.69707I 17.8159 + 11.7922I
u = 1.12939 + 1 .35997I
a = 0.418028 + 0.108447I
b = 0.794756 0.815720I
5.20109 + 3.69707I 17.8159 11.7922I
u = 1.01755 1.10772I
a = 0.107355 0.395530I
b = 0.276482 0.830194I
1.98063 + 6.23670I 4.70829 11.04703I
u = 1.01755 + 1 .10772I
a = 0.107355 + 0 .395530I
b = 0.276482 + 0.830194I
1.98063 6.23670I 4.70829 + 11.04703I
u = 0.96765 1.13256I
a = 0.723353 0.041982I
b = 1.33220 0.77413I
6.00576 5.14134I 4.61013 + 6.12801I
u = 0.96765 + 1 .13256I
a = 0.723353 + 0 .041982I
b = 1.33220 + 0 .77413I
6.00576 + 5.14134I 4.61013 6.12801I
u = 0.943877 0.072123I
a = 1.127525 0.396586I
b = 0.162898 0.301281I
0.30613 2.93570I 3.28195 + 5.62494I
14
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.943877 + 0 .072123I
a = 1.127525 + 0.396586I
b = 0.162898 + 0 .301281I
0.30613 + 2.93570I 3.28195 5.62494I
u = 0.91660 1.45408I
a = 0.707584 + 0.366299I
b = 1.00411 + 1.12746I
2.29876 9.05121I 7.57611 + 8 .61795I
u = 0.91660 + 1 .45408I
a = 0.707584 0.366299I
b = 1.00411 1.12746I
2.29876 + 9.05121I 7.57611 8.61795I
u = 0.89064 1.22962I
a = 0.974495 0.194580I
b = 1.26392 1.22083I
5.23865 10.99060I 4.17703 + 8.81757I
u = 0.89064 + 1 .22962I
a = 0.974495 + 0 .194580I
b = 1.26392 + 1 .22083I
5.23865 + 10.99060I 4.17703 8.81757I
u = 0.805949 0.810035I
a = 1.369133 + 0.135187I
b = 0.774856 + 0.614237I
1.98063 6.23670I 4.70829 + 11.04703I
u = 0.805949 + 0 .810035I
a = 1.369133 0.135187I
b = 0.774856 0.614237I
1.98063 + 6.23670I 4.70829 11.04703I
u = 0.71086 1.28822I
a = 1.013324 + 0.443464I
b = 1.30629 + 1.27421I
3.90289 8.93089I 8.55811 + 8.07031I
u = 0.71086 + 1 .28822I
a = 1.013324 0.443464I
b = 1.30629 1.27421I
3.90289 + 8.93089I 8.55811 8.07031I
u = 0.698365 0.048277I
a = 1.230849 + 0.164102I
b = 0.595096 + 0.752202I
3.31934 + 1.40971I 9.63872 4.90184I
15
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.698365 + 0 .048277I
a = 1.230849 0.164102I
b = 0.595096 0.752202I
3.31934 1.40971I 9.63872 + 4.90184I
u = 0.619879 1.143885I
a = 0.847945 0.061859I
b = 1.21360 1.06865I
5.61745 4.44792I 7.47292 + 6.68943I
u = 0.619879 + 1 .143885I
a = 0.847945 + 0 .061859I
b = 1.21360 + 1 .06865I
5.61745 + 4.44792I 7.47292 6.68943I
u = 0.543413 0.608268I
a = 1.82562 0.73399I
b = 0.383906 0.772337I
2.54435 + 3.08820I 4.24618 + 0.69599I
u = 0.543413 + 0 .608268I
a = 1.82562 + 0 .73399I
b = 0.383906 + 0 .772337I
2.54435 3.08820I 4.24618 0.69599I
u = 0.535466 0.312111I
a = 1.089138 0.344682I
b = 0.83964 1.37237I
2.70611 5.73656I 9.46948 + 10.34279I
u = 0.535466 + 0 .312111I
a = 1.089138 + 0 .344682I
b = 0.83964 + 1 .37237I
2.70611 + 5.73656I 9.46948 10.34279I
u = 0.507872 0.568783I
a = 1.85572 + 0.35781I
b = 0.849469 + 0.617285I
2.49917 2.89475I 6.29512 + 4.04782I
u = 0.507872 + 0 .568783I
a = 1.85572 0.35781I
b = 0.849469 0.617285I
2.49917 + 2.89475I 6.29512 4.04782I
u = 0.479257 0.791392I
a = 0.205058 1.279989I
b = 0.656940 + 0.143421I
4.85797 6.40077
16
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.479257 + 0 .791392I
a = 0.205058 + 1 .279989I
b = 0.656940 0.143421I
4.85797 6.40077
u = 0.314482 1.066295I
a = 0.270382 + 0.209277I
b = 0.78646 + 1.43056I
3.71252 2.09738I 4.75808 + 1.45431I
u = 0.314482 + 1 .066295I
a = 0.270382 0.209277I
b = 0.78646 1.43056I
3.71252 + 2.09738I 4.75808 1.45431I
u = 0.278683 0.741266I
a = 1.271719 + 0 .384129I
b = 1.98266 0.86162I
6.04090 + 0.51555I 9.73034 2.24226I
u = 0.278683 + 0 .741266I
a = 1.271719 0.384129I
b = 1.98266 + 0 .86162I
6.04090 0.51555I 9.73034 + 2.24226I
u = 0.23191 1.52682I
a = 0.162650 0.667592I
b = 0.13101 1.43940I
0.07742 6.14163I 2.62442 + 6.27682I
u = 0.23191 + 1 .52682I
a = 0.162650 + 0 .667592I
b = 0.13101 + 1 .43940I
0.07742 + 6.14163I 2.62442 6.27682I
u = 0.220217 0.884440I
a = 1.41425 + 1 .00894I
b = 1.52010 0.23280I
6.04090 0.51555I 9.73034 + 2.24226I
u = 0.220217 + 0 .884440I
a = 1.41425 1.00894I
b = 1.52010 + 0 .23280I
6.04090 + 0.51555I 9.73034 2.24226I
u = 0.202911 0.802221I
a = 0.239026 + 0.129561I
b = 0.07172 1.54625I
5.61745 4.44792I 7.47292 + 6.68943I
17
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.202911 + 0 .802221I
a = 0.239026 0.129561I
b = 0.07172 + 1 .54625I
5.61745 + 4 .44792I 7.47292 6.68943I
u = 0.072897 0.360777I
a = 1.69941 + 1.82660I
b = 0.56031 1.30023I
0.30613 2.93570I 3.28195 + 5.62494I
u = 0.072897 + 0 .360777I
a = 1.69941 1.82660I
b = 0.56031 + 1 .30023I
0.30613 + 2 .93570I 3.28195 5.62494I
u = 0.002400 0.566715I
a = 1.30349 2.14817I
b = 0.999244 + 0 .434820I
3.71252 + 2 .09738I 4.75808 1.45431I
u = 0.002400 + 0 .566715I
a = 1.30349 + 2.14817I
b = 0.999244 0.434820I
3.71252 2.09738I 4.75808 + 1 .45431I
u = 0.019987 0.705680I
a = 1.14359 2.12828I
b = 0.458617 + 0 .275074I
5.20109 + 3 .69707I 17.8159 11.7922I
u = 0.019987 + 0 .705680I
a = 1.14359 + 2 .12828I
b = 0.458617 0.275074I
5.20109 3.69707I 17.8159 + 11 .7922I
u = 0.080321 1.117672I
a = 0.328051 + 1.003899I
b = 0.435614 0.257393I
6.00576 5.14134I 4.61013 + 6 .12801I
u = 0.080321 + 1 .117672I
a = 0.328051 1.003899I
b = 0.435614 + 0 .257393I
6.00576 + 5 .14134I 4.61013 6.12801I
u = 0.115651 0.545467I
a = 1.21803 0.73893I
b = 2.06751 + 1 .42472I
3.90289 8.93089I 8.55811 + 8 .07031I
18
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.115651 + 0 .545467I
a = 1.21803 + 0 .73893I
b = 2.06751 1.42472I
3.90289 + 8 .93089I 8.55811 8.07031I
u = 0.16533 1. 80552I
a = 0.210076 + 0 .290744I
b = 0.171940 + 0 .840992I
0.581066 + 0.443460I 0.61628 4.85512I
u = 0.16533 + 1. 80552I
a = 0.210076 0.290744I
b = 0.171940 0.840992I
0.581066 0.443460I 0.61628 + 4.85512I
u = 0.167555 0 .950964I
a = 1.07819 0.97727I
b = 0.852676 + 0 .346090I
2.94180 + 3 .09675I 0.67036 8.19799I
u = 0.167555 + 0 .950964I
a = 1.07819 + 0 .97727I
b = 0.852676 0.346090I
2.94180 3.09675I 0.67036 + 8 .19799I
u = 0.207589 0 .469922I
a = 3.26674 + 0.95831I
b = 0.617172 0.025267I
0.06231 4.05987I 8.44249 0.74405I
u = 0.207589 + 0 .469922I
a = 3.26674 0.95831I
b = 0.617172 + 0.025267I
0.06231 + 4 .05987I 8.44249 + 0.74405I
u = 0.332333 0 .878248I
a = 1.11689 + 1.36705I
b = 1.133728 0.356923I
5.23865 + 10.99060I 4.17703 8.81757I
u = 0.332333 + 0 .878248I
a = 1.11689 1.36705I
b = 1.133728 + 0.356923I
5.23865 10.99060I 4.17703 + 8.81757I
u = 0.408621 0 .900284I
a = 1.090236 0.006918I
b = 0.615500 + 0 .539952I
0.40566 + 1 .48274I 2.55230 3.28537I
19
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.408621 + 0 .900284I
a = 1.090236 + 0. 006918I
b = 0.615500 0.539952I
0.40566 1.48274I 2.55230 + 3.28537I
u = 0.495841 1.030741I
a = 1.19640 0.76143I
b = 0.778973 0.793692I
0.29490 + 6.42715I 2.19076 11.02163I
u = 0.495841 + 1 .030741I
a = 1.19640 + 0.76143I
b = 0.778973 + 0.793692I
0.29490 6.42715I 2.19076 + 11 .02163I
u = 0.512233 0.648024I
a = 1.096099 0.115186I
b = 1.65035 0.28520I
0.29490 + 6.42715I 2.19076 11.02163I
u = 0.512233 + 0 .648024I
a = 1.096099 + 0. 115186I
b = 1.65035 + 0 .28520I
0.29490 6.42715I 2.19076 + 11 .02163I
u = 0.650238 0.770616I
a = 0.061475 + 0. 152433I
b = 0.241049 + 0.794859I
0.40566 + 1.48274I 2.55230 3.28537I
u = 0.650238 + 0 .770616I
a = 0.061475 0.152433I
b = 0.241049 0.794859I
0.40566 1.48274I 2.55230 + 3.28537I
u = 0.771121 0.933795I
a = 0.512320 + 0. 628800I
b = 0.127098 + 0.767595I
2.49917 + 2.89475I 6.29512 4.04782I
u = 0.771121 + 0 .933795I
a = 0.512320 0.628800I
b = 0.127098 0.767595I
2.49917 2.89475I 6.29512 + 4.04782I
u = 0.808260 0.907026I
a = 1.128763 + 0. 174468I
b = 0.99330 + 1 .13322I
2.54435 + 3.08820I 4.24618 + 0.69599I
20
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.808260 + 0 .907026I
a = 1.128763 0.174468I
b = 0.99330 1 .13322I
2.54435 3.08820I 4.24618 0.69599I
u = 0.822790 0.362700I
a = 0.582795 + 0 .205483I
b = 1.86613 + 1 .09512I
2.29876 + 9.05121I 7.57611 8.61795I
u = 0.822790 + 0 .362700I
a = 0.582795 0.205483I
b = 1.86613 1 .09512I
2.29876 9.05121I 7.57611 + 8 .61795I
u = 0.844655 0.809311I
a = 1.277548 + 0 .019305I
b = 0.474629 + 0 .965798I
0.06231 + 4.05987I 8.44249 + 0.74405I
u = 0.844655 + 0 .809311I
a = 1.277548 0.019305I
b = 0.474629 0.965798I
0.06231 4.05987I 8.44249 0.74405I
u = 0.87421 1 .20478I
a = 0.188426 0.565768I
b = 0.491008 + 0 .034057I
0.581066 + 0 .443460I 0.61628 4.85512I
u = 0.87421 + 1 .20478I
a = 0.188426 + 0 .565768I
b = 0.491008 0.034057I
0.581066 0.443460I 0.61628 + 4 .85512I
u = 0.879902 0.854615I
a = 1.039705 0.014239I
b = 1.59546 0.81234I
0.07742 + 6.14163I 2.62442 6.27682I
u = 0.879902 + 0 .854615I
a = 1.039705 + 0.014239I
b = 1.59546 + 0 .81234I
0.07742 6.14163I 2.62442 + 6 .27682I
u = 0.937741 0.997672I
a = 0.908424 0.268173I
b = 0.667750 0.681982I
2.70611 + 5.73656I 9.46948 10.34279I
21
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.937741 + 0 .997672I
a = 0.908424 + 0.268173I
b = 0.667750 + 0.681982I
2.70611 5.73656I 9.46948 + 10.34279I
u = 1.01061 1. 30844I
a = 0.863217 0.194805I
b = 1.16053 0.93960I
1.78184 + 10.59477I 2.34787 10.61099I
u = 1.01061 + 1. 30844I
a = 0.863217 + 0.194805I
b = 1.16053 + 0.93960I
1.78184 10.59477I 2.34787 + 10.61099I
u = 1.03662 1. 44950I
a = 0.606342 + 0 .083738I
b = 1.016690 + 0 .657524I
7.04800 4.85291
u = 1.03662 + 1. 44950I
a = 0.606342 0.083738I
b = 1.016690 0.657524I
7.04800 4.85291
u = 1.043389 0 .776682I
a = 0.440999 0.250176I
b = 0.494667 0.713171I
3.31934 + 1.40971I 9.63872 4.90184I
u = 1.043389 + 0 .776682I
a = 0.440999 + 0.250176I
b = 0.494667 + 0.713171I
3.31934 1.40971I 9.63872 + 4 .90184I
u = 1.20673
a = 0.594183
b = 1.55197
1.80906 10.9203
u = 1.75907 0. 63951I
a = 0.288929 + 0 .315662I
b = 0.417651 + 0.265533I
1.78184 10.59477I 2.34787 + 10.61099I
u = 1.75907 + 0. 63951I
a = 0.288929 0.315662I
b = 0.417651 0.265533I
1.78184 + 10.59477I 2.34787 10.61099I
22
VI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
u
24
(u + 1)
2
(u
4
+ u
3
+ ··· 2u + 1)(u
20
u
19
+ ··· + 5u + 1)
(u
90
+ 4u
89
+ ··· + 679u 263)
c
2
u
24
(u
2
+ u + 1)(u
4
3u
3
+ ··· + u + 1)(u
20
3u
19
+ ··· + 2u + 1)
(u
90
+ u
89
+ ··· + 675u + 103)
c
3
u
24
(u
2
+ u + 1)(u
4
+ 3u
3
+ ··· u + 1)(u
20
+ 3u
19
+ ··· 2u + 1)
(u
90
+ u
89
+ ··· + 675u + 103)
c
4
(u + 1)
2
(u
4
u
3
+ ··· + 2u + 1)(u
20
+ u
19
+ ··· 5u + 1)
(u
24
5u
23
+ ··· 8u + 1)(u
90
+ 4u
89
+ ··· + 679u 263)
c
5
u
24
(u
2
+ u + 1)(u
4
+ u
3
+ ··· u 1)(u
20
+ u
18
+ ··· u + 1)
(u
90
8u
88
+ ··· + 22u 1)
c
6
u
26
(u
4
+ u
3
2u
2
2u + 1)
(u
10
3u
8
+ 2u
7
8u
5
+ 9u
4
+ 9u
3
5u
2
4u 2)
2
(u
90
40u
88
+ ··· 40824u + 11664)
c
7
(u
2
+ u + 1)(u
4
u
3
+ ··· + u 1)(u
20
+ 12u
18
+ ··· 2u + 1)
(u
24
+ 5u
23
+ ··· + 5u + 1)(u
90
+ 13u
88
+ ··· 2331u 697)
c
8
u
4
(u + 1)
2
(u
20
+ 13u
18
+ ··· + 366u
2
+ 113)
(u
24
+ u
23
+ ··· 320u + 64)(u
90
+ 4u
89
+ ··· 810u + 81)
c
9
u
24
(u
2
+ u + 1)(u
4
+ u
3
+ ··· u 1)(u
20
+ 12u
18
+ ··· + 2u + 1)
(u
90
+ 13u
88
+ ··· 2331u 697)
c
10
u
2
(u
4
u
3
2u
2
+ 2u + 1)
(u
10
3u
8
2u
7
+ 8u
5
+ 9u
4
9u
3
5u
2
+ 4u 2)
2
(u
24
+ 2u
23
+ ··· + 106u + 36)(u
90
40u
88
+ ··· 40824u + 11664)
c
11
(u
2
+ u + 1)(u
4
u
3
+ ··· + u 1)(u
20
+ u
18
+ ··· + u + 1)
(u
24
u
23
+ ··· 7u + 1)(u
90
8u
88
+ ··· + 22u 1)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
y
24
(y 1)
2
(y
4
5y
3
+ ··· 8y + 1)(y
20
13y
19
+ ··· + 11y + 1)
(y
90
20y
89
+ ··· 4190907y + 69169)
c
2
y
24
(y
2
+ y + 1)(y
4
7y
3
+ ··· + y + 1)(y
20
5y
19
+ ··· 6y + 1)
(y
90
7y
89
+ ··· + 296069y + 10609)
c
3
y
24
(y
2
+ y + 1)(y
4
7y
3
+ ··· + y + 1)(y
20
5y
19
+ ··· 6y + 1)
(y
90
7y
89
+ ··· + 296069y + 10609)
c
4
(y 1)
2
(y
4
5y
3
+ ··· 8y + 1)(y
20
13y
19
+ ··· + 11y + 1)
(y
24
7y
23
+ ··· 6y + 1)(y
90
20y
89
+ ··· 4190907y + 69169)
c
5
y
24
(y
2
+ y + 1)(y
4
+ y
3
+ ··· 3y + 1)(y
20
+ 2y
19
+ ··· + 19y + 1)
(y
90
16y
89
+ ··· 90y + 1)
c
6
y
26
(y
4
5y
3
+ 10y
2
8y + 1)
(4 + 4y + 61y
2
235y
3
+ 253y
4
74y
5
32y
6
+ 14y
7
+ 9y
8
6y
9
+ y
10
)
2
(y
90
80y
89
+ ··· + 844636896y + 136048896)
c
7
(y
2
+ y + 1)(y
4
3y
3
+ ··· + y + 1)(y
20
+ 24y
19
+ ··· + 4y
2
+ 1)
(y
24
21y
23
+ ··· 5y + 1)
(y
90
+ 26y
89
+ ··· + 21978055y + 485809)
c
8
y
4
(y 1)
2
(113 + 366y + 507y
2
+ 607y
3
+ 499y
4
+ 367y
5
+ 197y
6
+ 105y
7
+ 29y
8
+ 13y
9
+ y
10
)
2
(y
24
23y
23
+ ··· + 30720y + 4096)
(y
90
+ 40y
89
+ ··· 465102y + 6561)
c
9
y
24
(y
2
+ y + 1)(y
4
3y
3
+ ··· + y + 1)(y
20
+ 24y
19
+ ··· + 4y
2
+ 1)
(y
90
+ 26y
89
+ ··· + 21978055y + 485809)
c
10
y
2
(y
4
5y
3
+ 10y
2
8y + 1)
(4 + 4y + 61y
2
235y
3
+ 253y
4
74y
5
32y
6
+ 14y
7
+ 9y
8
6y
9
+ y
10
)
2
(y
24
12y
23
+ ··· + 9860y + 1296)
(y
90
80y
89
+ ··· + 844636896y + 136048896)
c
11
(y
2
+ y + 1)(y
4
+ y
3
+ ··· 3y + 1)(y
20
+ 2y
19
+ ··· + 19y + 1)
(y
24
+ 7y
23
+ ··· 29y + 1)(y
90
16y
89
+ ··· 90y + 1)
24