11n
158
(K11n
158
)
1
Arc Sequences
9 8 1 10 2 3 1 6 4 5 8
Solving Sequence
1,4 3,6
7 8 9 10 2 5 11
c
3
c
6
c
7
c
8
c
9
c
2
c
5
c
10
c
1
, c
4
, c
11
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
16
+ b
15
+ ··· + 398b + 359,
20071843b
15
+ 5118248500u + ··· 9365383501b 11671609211,
822556148b
15
+ 4315217387b
14
+ ··· + 885456990500a + 1862768645227i
I
u
2
= hu
13
10u
12
+ 41u
11
83u
10
+ 64u
9
+ 62u
8
168u
7
+ 78u
6
+ 125u
5
192u
4
+ 109u
3
42u
2
+ 19u 5,
1132u
12
+ 15374u
11
+ ··· + 9505b 25570, 18468u
12
+ 176512u
11
+ ··· + 9505a 146541i
I
u
3
= hu
19
+ 9u
18
+ ··· + 52u + 16, 48933197u
18
364980751u
17
+ ··· + 35727512a 761474994,
78311833u
18
+ 588712101u
17
+ ··· + 17863756b + 816399608i
There are 3 irreducible components with 48 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
= hb
16
+ b
15
+ · · · + 398b + 359, 5.12 × 10
9
u 2.01 × 10
7
b
15
+ · · · 9.37 ×
10
9
b1.17×10
10
, 8.23×10
8
b
15
+4.32×10
9
b
14
+· · ·+8.85×10
11
a+1.86×10
12
i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
0.00392162b
15
0.00280910b
14
+ ··· + 1.82980b + 2.28039
a
3
=
0.00392162b
15
0.00280910b
14
+ ··· + 1.82980b + 2.28039
0.00392162b
15
0.00280910b
14
+ ··· + 1.82980b + 2.28039
a
6
=
0.000928962b
15
0.00487344b
14
+ ··· + 1.69090b 2.10374
b
a
7
=
0.00665112b
15
0.000529657b
14
+ ··· 0.617531b 1.01179
0.00758008b
15
+ 0.00434378b
14
+ ··· 1.30843b + 1.09194
a
8
=
0.000928962b
15
0.00487344b
14
+ ··· + 0.690903b 2.10374
0.00758008b
15
+ 0.00434378b
14
+ ··· 1.30843b + 1.09194
a
9
=
0.00642415b
15
+ 0.00150914b
14
+ ··· + 0.347537b + 1.73467
0.00450462b
15
+ 0.00723276b
14
+ ··· 2.85118b + 0.335106
a
10
=
0.00642415b
15
+ 0.00150914b
14
+ ··· + 0.347537b + 1.73467
0.0104803b
15
+ 0.0114048b
14
+ ··· 4.62021b + 0.326292
a
2
=
0.00269531b
15
+ 0.00382665b
14
+ ··· 2.76705b 0.857186
0.00933016b
15
+ 0.0150005b
14
+ ··· 3.53292b 3.37503
a
5
=
0.00140001b
15
+ 0.00153322b
14
+ ··· 1.08895b 1.35105
0.00945258b
15
+ 0.0116315b
14
+ ··· 4.37471b 2.56086
a
11
=
0.00140001b
15
0.00153322b
14
+ ··· + 1.08895b + 1.35105
0.000444714b
15
0.0102686b
14
+ ··· + 4.43548b + 4.83243
a
11
=
0.00140001b
15
0.00153322b
14
+ ··· + 1.08895b + 1.35105
0.000444714b
15
0.0102686b
14
+ ··· + 4.43548b + 4.83243
(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 = 1.89923 + 0.40053I
a = 1.31435 2.27830I
b = 2.60224 2.30506I
14.02845 1.41510I 13.8267 + 4.9087I
u = 1.89923 0.40053I
a = 1.31435 + 2.27830I
b = 2.60224 + 2.30506I
14.02845 + 1.41510I 13.8267 4.9087I
u = 1.89923 + 0.40053I
a = 0.311766 0.387658I
b = 1.012250 0.233757I
6.13277 1.41510I 13.8267 + 4.9087I
u = 1.89923 0.40053I
a = 0.311766 + 0.387658I
b = 1.012250 + 0.233757I
6.13277 + 1.41510I 13.8267 4.9087I
u = 0.399232 + 0.325640I
a = 1.37761 1.40200I
b = 0.951916 0.937444I
7.02670 3.16396I 10.17326 + 2.56480I
u = 0.399232 0.325640I
a = 1.37761 + 1.40200I
b = 0.951916 + 0.937444I
7.02670 + 3.16396I 10.17326 2.56480I
u = 0.399232 + 0.325640I
a = 1.54970 + 1.59621I
b = 0.754672 0.417445I
0.86898 3.16396I 10.17326 + 2.56480I
u = 0.399232 0.325640I
a = 1.54970 1.59621I
b = 0.754672 + 0.417445I
0.86898 + 3.16396I 10.17326 2.56480I
u = 0.399232 0.325640I
a = 0.196524 0.524037I
b = 0.481163 1.014985I
0.86898 + 3.16396I 10.17326 2.56480I
u = 0.399232 + 0.325640I
a = 0.196524 + 0.524037I
b = 0.481163 + 1.014985I
0.86898 3.16396I 10.17326 + 2.56480I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.89923 + 0.40053I
a = 0.198750 0.009956I
b = 1.077164 0.328644I
14.02845 1.41510I 13.8267 + 4.9087I
u = 1.89923 0.40053I
a = 0.198750 + 0.009956I
b = 1.077164 + 0.328644I
14.02845 + 1.41510I 13.8267 4.9087I
u = 1.89923 0.40053I
a = 0.73789 1.26169I
b = 1.59478 1.23974I
6.13277 + 1.41510I 13.8267 4.9087I
u = 1.89923 + 0.40053I
a = 0.73789 + 1.26169I
b = 1.59478 + 1.23974I
6.13277 1.41510I 13.8267 + 4.9087I
u = 0.399232 + 0.325640I
a = 2.16504 4.14887I
b = 1.66797 0.62694I
7.02670 3.16396I 10.17326 + 2.56480I
u = 0.399232 0.325640I
a = 2.16504 + 4.14887I
b = 1.66797 + 0.62694I
7.02670 + 3.16396I 10.17326 2.56480I
4
II. I
u
2
= hu
13
10u
12
+ · · · + 19u 5, 1132u
12
+ 15374u
11
+ · · · + 9505b
25570, 1.85 × 10
4
u
12
+ 1.77 × 10
5
u
11
+ · · · + 9505a 1.47 × 10
5
i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
3
=
u
u
a
6
=
1.94298u
12
18.5704u
11
+ ··· 36.2904u + 15.4173
0.119095u
12
1.61746u
11
+ ··· 6.61494u + 2.69016
a
7
=
3.10973u
12
28.6447u
11
+ ··· 51.6021u + 21.8465
1.28585u
12
11.6917u
11
+ ··· 21.9267u + 9.11941
a
8
=
1.82388u
12
16.9530u
11
+ ··· 29.6754u + 12.7271
1.28585u
12
11.6917u
11
+ ··· 21.9267u + 9.11941
a
9
=
1.14887u
12
+ 10.7218u
11
+ ··· + 17.9187u 7.26270
0.775802u
12
6.89963u
11
+ ··· 12.3623u + 5.52762
a
10
=
1.14887u
12
+ 10.7218u
11
+ ··· + 17.9187u 7.26270
0.430615u
12
3.29742u
11
+ ··· 3.53635u + 1.69332
a
2
=
0.582009u
12
+ 5.39842u
11
+ ··· + 11.7529u 4.31836
0.421673u
12
+ 3.72120u
11
+ ··· + 7.73982u 2.91005
a
5
=
1.77359u
12
16.8679u
11
+ ··· 32.6267u + 14.7106
0.329406u
12
+ 2.03240u
11
+ ··· 0.885534u + 1.43030
a
11
=
0.348869u
12
3.72183u
11
+ ··· 11.3187u + 7.06270
0.233140u
12
+ 1.67659u
11
+ ··· 0.565807u + 1.74435
a
11
=
0.348869u
12
3.72183u
11
+ ··· 11.3187u + 7.06270
0.233140u
12
+ 1.67659u
11
+ ··· 0.565807u + 1.74435
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.090577 0.377410I
a = 0.672667 0.573571I
b = 0.700644 1.107340I
1.52980 + 2.82140I 9.56946 2.96660I
u = 1.090577 + 0.377410I
a = 0.672667 + 0.573571I
b = 0.700644 + 1.107340I
1.52980 2.82140I 9.56946 + 2.96660I
u = 0.108344 0.413697I
a = 2.07131 0.18744I
b = 0.537731 + 0.725306I
1.59640 + 3.38566I 2.41525 6.49353I
u = 0.108344 + 0.413697I
a = 2.07131 + 0.18744I
b = 0.537731 0.725306I
1.59640 3.38566I 2.41525 + 6.49353I
u = 0.617802 0.087807I
a = 0.59816 + 2.02242I
b = 0.144427 + 0.388914I
2.09623 + 2.92285I 9.7624 12.7438I
u = 0.617802 + 0.087807I
a = 0.59816 2.02242I
b = 0.144427 0.388914I
2.09623 2.92285I 9.7624 + 12.7438I
u = 0.928070 1.060784I
a = 1.219006 0.510829I
b = 0.826774 0.115283I
4.89940 + 5.64504I 5.13604 4.51836I
u = 0.928070 + 1.060784I
a = 1.219006 + 0.510829I
b = 0.826774 + 0.115283I
4.89940 5.64504I 5.13604 + 4.51836I
u = 1.78018 0.29183I
a = 0.331620 0.602768I
b = 1.178782 0.503266I
5.39720 + 0.76633I 5.46823 + 1.57837I
u = 1.78018 + 0.29183I
a = 0.331620 + 0.602768I
b = 1.178782 + 0.503266I
5.39720 0.76633I 5.46823 1.57837I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.90523 0.76417I
a = 0.503298 + 1.075759I
b = 1.017339 + 0.875093I
8.02879 + 2.87155I 15.3113 3.5915I
u = 1.90523 + 0.76417I
a = 0.503298 1.075759I
b = 1.017339 0.875093I
8.02879 2.87155I 15.3113 + 3.5915I
u = 1.93528
a = 0.901050
b = 2.01341
13.7330 11.3356
7
III.
I
u
3
= hu
19
+9u
18
+· · · +52u + 16, 4.89 × 10
7
u
18
3.65 × 10
8
u
17
+· · · +3.57 ×
10
7
a 7.61 ×10
8
, 7.83×10
7
u
18
+5.89× 10
8
u
17
+· · ·+ 1.79×10
7
b +8.16 ×10
8
i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
3
=
u
u
a
6
=
1.36962u
18
+ 10.2157u
17
+ ··· + 28.5407u + 21.3134
4.38384u
18
32.9557u
17
+ ··· 123.489u 45.7015
a
7
=
14.3633u
18
+ 107.666u
17
+ ··· + 384.195u + 159.070
8.60980u
18
+ 64.4946u
17
+ ··· + 232.165u + 92.0554
a
8
=
5.75346u
18
+ 43.1714u
17
+ ··· + 152.030u + 67.0149
8.60980u
18
+ 64.4946u
17
+ ··· + 232.165u + 92.0554
a
9
=
5.81896u
18
+ 44.4494u
17
+ ··· + 170.182u + 83.5347
19.0221u
18
139.043u
17
+ ··· 481.129u 176.164
a
10
=
5.81896u
18
+ 44.4494u
17
+ ··· + 170.182u + 83.5347
7.38363u
18
51.7054u
17
+ ··· 162.331u 49.4251
a
2
=
14.8183u
18
108.826u
17
+ ··· 379.568u 150.370
24.5390u
18
179.672u
17
+ ··· 619.182u 237.093
a
5
=
9.15412u
18
67.3226u
17
+ ··· 241.693u 77.9317
0.240422u
18
+ 0.0766104u
17
+ ··· 16.7004u 3.93744
a
11
=
20.7405u
18
151.106u
17
+ ··· 516.459u 182.478
35.5588u
18
259.931u
17
+ ··· 897.028u 331.848
a
11
=
20.7405u
18
151.106u
17
+ ··· 516.459u 182.478
35.5588u
18
259.931u
17
+ ··· 897.028u 331.848
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.95261 0.29538I
a = 0.757150 1.076471I
b = 1.88854 1.21307I
16.5060 12.7254I 9.92028 + 5.57941I
u = 1.95261 + 0.29538I
a = 0.757150 + 1.076471I
b = 1.88854 + 1.21307I
16.5060 + 12.7254I 9.92028 5.57941I
u = 1.80267 0.31950I
a = 0.484309 + 0.899825I
b = 1.61268 + 0.78549I
8.63276 7.82560I 8.78089 + 5.39108I
u = 1.80267 + 0.31950I
a = 0.484309 0.899825I
b = 1.61268 0.78549I
8.63276 + 7.82560I 8.78089 5.39108I
u = 1.67630 0.02437I
a = 0.190831 + 0.017576I
b = 1.44827 0.39201I
15.6607 + 4.6824I 10.38940 2.40952I
u = 1.67630 + 0.02437I
a = 0.190831 0.017576I
b = 1.44827 + 0.39201I
15.6607 4.6824I 10.38940 + 2.40952I
u = 1.66571 0.20021I
a = 0.222503 0.548270I
b = 1.44086 0.25116I
8.13053 0.97689I 10.22917 + 0.27205I
u = 1.66571 + 0.20021I
a = 0.222503 + 0.548270I
b = 1.44086 + 0.25116I
8.13053 + 0.97689I 10.22917 0.27205I
u = 0.489753 0.127157I
a = 0.07795 2.53440I
b = 0.012261 0.532740I
2.26122 + 2.63337I 2.15678 + 7.21953I
u = 0.489753 + 0.127157I
a = 0.07795 + 2.53440I
b = 0.012261 + 0.532740I
2.26122 2.63337I 2.15678 7.21953I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.431648
a = 1.74062
b = 0.342901
1.52055 7.33434
u = 0.442879 0.073324I
a = 0.416327 + 0.055605I
b = 0.528236 0.452678I
0.928589 + 0.684123I 8.21629 4.45735I
u = 0.442879 + 0.073324I
a = 0.416327 0.055605I
b = 0.528236 + 0.452678I
0.928589 0.684123I 8.21629 + 4.45735I
u = 0.578683 0.929258I
a = 0.960951 + 0.354045I
b = 0.919683 0.105737I
0.22723 + 2.57483I 7.79227 4.00370I
u = 0.578683 + 0.929258I
a = 0.960951 0.354045I
b = 0.919683 + 0.105737I
0.22723 2.57483I 7.79227 + 4.00370I
u = 1.098147 0.704965I
a = 0.338482 + 0.860780I
b = 0.724744 + 0.848916I
6.47985 + 3.00781I 9.19206 3.40012I
u = 1.098147 + 0.704965I
a = 0.338482 0.860780I
b = 0.724744 0.848916I
6.47985 3.00781I 9.19206 + 3.40012I
u = 1.18316 1.41586I
a = 1.227909 0.096806I
b = 1.225449 + 0.160603I
5.79784 + 5.72974I 14.4693 5.0563I
u = 1.18316 + 1.41586I
a = 1.227909 + 0.096806I
b = 1.225449 0.160603I
5.79784 5.72974I 14.4693 + 5.0563I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
13
+ u
12
u
11
3u
10
2u
9
+ u
8
2u
6
+ u
5
4u
4
2u
3
2u
2
1)
(u
16
+ 3u
15
+ ··· 14u + 41)(u
19
+ u
18
+ ··· u 1)
c
2
(u
13
3u
11
2u
10
+ 3u
9
+ u
8
9u
7
11u
6
+ u
5
2u
4
10u
3
3u
2
1)
(u
16
+ u
15
+ ··· + 398u + 359)(u
19
10u
17
+ ··· + 21u 6)
c
3
(1 2u + u
2
+ 3u
3
+ u
4
)
4
(u
13
+ 10u
12
+ ··· + 19u + 5)
(u
19
9u
18
+ ··· + 52u 16)
c
4
(u
2
u 1)
8
(u
13
7u
11
+ ··· + 2u + 1)(u
19
+ 9u
18
+ ··· + 4u
2
16)
c
6
, c
11
(u
13
u
12
+ ··· + u 1)(u
16
+ u
15
+ ··· + 314u + 59)
(u
19
u
18
+ ··· + 2u 1)
c
7
(u
13
+ u
12
+ ··· + u + 1)(u
16
+ u
15
+ ··· + 314u + 59)
(u
19
u
18
+ ··· + 2u 1)
c
8
(u
4
u
3
+ u
2
+ 1)
4
(u
13
6u
12
+ ··· 5u
2
+ 1)
(u
19
+ 9u
18
+ ··· + 34u + 4)
c
9
, c
10
(u
2
u 1)
8
(u
13
7u
11
+ ··· + 2u 1)(u
19
+ 9u
18
+ ··· + 4u
2
16)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
13
3y
12
+ ··· 4y 1)(y
16
y
15
+ ··· + 6528y + 1681)
(y
19
+ 15y
18
+ ··· 47y 1)
c
2
(y
13
6y
12
+ ··· 6y 1)(y
16
13y
15
+ ··· 519558y + 128881)
(y
19
20y
18
+ ··· + 273y 36)
c
3
(1 2y + 15y
2
7y
3
+ y
4
)
4
(y
13
18y
12
+ ··· 59y 25)
(y
19
25y
18
+ ··· + 6064y 256)
c
4
, c
9
, c
10
(y
2
3y + 1)
8
(y
13
14y
12
+ ··· + 12y 1)
(y
19
17y
18
+ ··· + 128y 256)
c
6
, c
7
, c
11
(y
13
7y
12
+ ··· 9y 1)(y
16
21y
15
+ ··· 63668y + 3481)
(y
19
33y
18
+ ··· + 12y 1)
c
8
(1 + 2y + 3y
2
+ y
3
+ y
4
)
4
(y
13
2y
12
+ ··· + 10y 1)
(y
19
y
18
+ ··· 116y 16)
12