10
104
(K10a
118
)
1
Arc Sequences
8 6 1 9 10 3 4 2 5 7
Solving Sequence
2,8 4,9
5 1 3 7 6 10
c
8
c
4
c
1
c
3
c
7
c
6
c
10
c
2
, c
5
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
7
3u
6
+ 8u
5
15u
4
+ 20u
3
21u
2
+ 10u 1,
14u
6
38u
5
+ 95u
4
189u
3
+ 226u
2
+ 43b 254u + 92,
19u
6
70u
5
+ 175u
4
364u
3
+ 491u
2
+ 43a 529u + 303i
I
u
2
= hu
15
+ u
14
+ ··· + 13u + 1,
2836311062u
14
4161649487u
13
+ ··· + 1085175397a 37334618369,
6343880351u
14
4549484601u
13
+ ··· + 1085175397b 24261485287i
I
u
3
= hu
30
7u
29
+ ··· + 310u 25,
2.83886 × 10
60
u
29
+ 2.00494 × 10
61
u
28
+ ··· + 9.06522 × 10
61
b 4.93675 × 10
62
,
1.17639 × 10
62
u
29
8.00708 × 10
62
u
28
+ ··· + 4.53261 × 10
62
a + 7.64064 × 10
63
i
There are 3 irreducible components with 52 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
7
3u
6
+· · ·+10u1, 14u
6
38u
5
+· · ·+43b+92, 19u
6
70u
5
+· · ·+43a+303i
(i) Arc colorings
a
2
=
1
0
a
8
=
0.441860u
6
+ 1.62791u
5
+ ··· + 12.3023u 7.04651
0.325581u
6
+ 0.883721u
5
+ ··· + 5.90698u 2.13953
a
4
=
0
u
a
9
=
0.116279u
6
+ 0.744186u
5
+ ··· + 6.39535u 4.90698
0.325581u
6
+ 0.883721u
5
+ ··· + 5.90698u 2.13953
a
5
=
0.697674u
6
+ 1.46512u
5
+ ··· + 6.37209u + 1.55814
0.255814u
6
0.837209u
5
+ ··· 4.06977u + 1.39535
a
1
=
1.95349u
6
+ 5.30233u
5
+ ··· + 32.4419u 10.8372
0.558140u
6
+ 1.37209u
5
+ ··· + 7.69767u 2.95349
a
3
=
0.441860u
6
1.62791u
5
+ ··· 12.3023u + 8.04651
0.558140u
6
1.37209u
5
+ ··· 7.69767u + 2.95349
a
7
=
0.441860u
6
+ 1.62791u
5
+ ··· + 12.3023u 7.04651
0.0465116u
6
+ 0.302326u
5
+ ··· + 2.44186u 1.83721
a
6
=
1.83721u
6
4.55814u
5
+ ··· 26.0465u + 6.93023
0.325581u
6
0.883721u
5
+ ··· 5.90698u + 2.13953
a
10
=
1.83721u
6
+ 4.55814u
5
+ ··· + 26.0465u 5.93023
0.255814u
6
+ 0.837209u
5
+ ··· + 4.06977u 1.39535
(ii) Obstruction class = 1
(iii) Cusp Shapes =
166
43
u
6
+
383
43
u
5
1065
43
u
4
+
1854
43
u
3
2219
43
u
2
+
2422
43
u
624
43
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.21748 1.74792I
a = 0.649452 + 0.131055I
b = 0.859004 0.613195I
1.45010 + 4.82255I 1.50641 5.81707I
u = 0.21748 + 1.74792I
a = 0.649452 0.131055I
b = 0.859004 + 0.613195I
1.45010 4.82255I 1.50641 + 5.81707I
u = 0.132774
a = 5.59576
b = 1.43826
5.57629 7.84921
u = 0.59726 1.44367I
a = 0.343727 + 0.399701I
b = 0.765267 + 0.295664I
2.11479 + 2.13385I 6.73578 5.40456I
u = 0.59726 + 1.44367I
a = 0.343727 0.399701I
b = 0.765267 0.295664I
2.11479 2.13385I 6.73578 + 5.40456I
u = 0.713207
a = 1.82040
b = 0.543068
6.50483 9.77983
u = 1.39446
a = 0.804715
b = 1.70772
2.88904 5.52812
3
II.
I
u
2
= hu
15
+ u
14
+ · · · + 13u + 1, 2.84 × 10
9
u
14
4.16 × 10
9
u
13
+ · · · + 1.09 ×
10
9
a3.73×10
10
, 6.34×10
9
u
14
4.55×10
9
u
13
+· · ·+1.09×10
9
b2.43×10
10
i
(i) Arc colorings
a
2
=
1
0
a
8
=
2.61369u
14
+ 3.83500u
13
+ ··· + 234.689u + 34.4042
5.84595u
14
+ 4.19240u
13
+ ··· + 202.195u + 22.3572
a
4
=
0
u
a
9
=
3.23226u
14
0.357394u
13
+ ··· + 32.4941u + 12.0470
5.84595u
14
+ 4.19240u
13
+ ··· + 202.195u + 22.3572
a
5
=
29.7456u
14
20.1784u
13
+ ··· 927.104u 97.0905
0.106446u
14
0.405615u
13
+ ··· 24.8709u 3.99398
a
1
=
7.67162u
14
6.70125u
13
+ ··· 332.186u 37.6152
3.67764u
14
2.60083u
13
+ ··· 121.162u 12.5643
a
3
=
3.92232u
14
1.04663u
13
+ ··· 16.0282u + 4.11295
3.67764u
14
+ 2.60083u
13
+ ··· + 121.162u + 12.5643
a
7
=
2.61369u
14
+ 3.83500u
13
+ ··· + 234.689u + 34.4042
4.99844u
14
+ 3.70527u
13
+ ··· + 183.704u + 21.1359
a
6
=
5.00115u
14
0.776007u
13
+ ··· + 34.9424u + 15.4817
5.84595u
14
+ 4.19240u
13
+ ··· + 202.195u + 22.3572
a
10
=
22.9379u
14
+ 15.0216u
13
+ ··· + 681.905u + 71.1410
0.106446u
14
+ 0.405615u
13
+ ··· + 24.8709u + 3.99398
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
51770215051
1085175397
u
14
36321666149
1085175397
u
13
+ ···
1691614369554
1085175397
u
172793741600
1085175397
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.14450 1.53934I
a = 0.349830 0.104240I
b = 0.955976 0.550854I
2.89422 + 6.37595I 2.35312 7.90831I
u = 1.14450 + 1.53934I
a = 0.349830 + 0.104240I
b = 0.955976 + 0.550854I
2.89422 6.37595I 2.35312 + 7.90831I
u = 0.379744 0.426871I
a = 1.92696 + 0.97656I
b = 0.104299 0.883953I
7.67422 + 2.17377I 7.19312 2.21789I
u = 0.379744 + 0.426871I
a = 1.92696 0.97656I
b = 0.104299 + 0.883953I
7.67422 2.17377I 7.19312 + 2.21789I
u = 0.363500
a = 4.03505
b = 0.810873
5.97579 5.93617
u = 0.265387 1.302841I
a = 0.395076 0.839348I
b = 1.144624 0.283738I
4.34026 3.41455I 3.26031 + 4.30453I
u = 0.265387 + 1.302841I
a = 0.395076 + 0.839348I
b = 1.144624 + 0.283738I
4.34026 + 3.41455I 3.26031 4.30453I
u = 0.246839 0.030877I
a = 2.15485 2.00132I
b = 0.198076 0.530950I
1.036954 0.848562I 5.31510 + 2.72513I
u = 0.246839 + 0.030877I
a = 2.15485 + 2.00132I
b = 0.198076 + 0.530950I
1.036954 + 0.848562I 5.31510 2.72513I
u = 0.108197 1.248190I
a = 0.082518 + 0.325057I
b = 0.748915 + 0.301809I
1.32635 + 1.58430I 2.05695 3.17357I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.108197 + 1.248190I
a = 0.082518 0.325057I
b = 0.748915 0.301809I
1.32635 1.58430I 2.05695 + 3.17357I
u = 0.47742 1.71628I
a = 0.728532 + 0.408764I
b = 1.299487 + 0.476281I
5.82203 9.38410I 3.97952 + 7.17475I
u = 0.47742 + 1.71628I
a = 0.728532 0.408764I
b = 1.299487 0.476281I
5.82203 + 9.38410I 3.97952 7.17475I
u = 1.13261 1.70886I
a = 0.811136 0.155800I
b = 1.36114 0.66333I
0.18522 14.10713I 0.21037 + 7.77333I
u = 1.13261 + 1.70886I
a = 0.811136 + 0.155800I
b = 1.36114 + 0.66333I
0.18522 + 14.10713I 0.21037 7.77333I
6
III. I
u
3
=
hu
30
7u
29
+· · ·+310u25, 2.84×10
60
u
29
+2.00×10
61
u
28
+· · ·+9.07×10
61
b
4.94 × 10
62
, 1.18 × 10
62
u
29
8.01 × 10
62
u
28
+ · · · + 4.53 × 10
62
a + 7.64 × 10
63
i
(i) Arc colorings
a
2
=
1
0
a
8
=
0.259540u
29
+ 1.76655u
28
+ ··· + 177.704u 16.8570
0.0313159u
29
0.221169u
28
+ ··· 46.6273u + 5.44582
a
4
=
0
u
a
9
=
0.290856u
29
+ 1.98772u
28
+ ··· + 224.332u 22.3029
0.0313159u
29
0.221169u
28
+ ··· 46.6273u + 5.44582
a
5
=
0.393277u
29
2.71248u
28
+ ··· 394.244u + 40.0163
0.0960661u
29
+ 0.656937u
28
+ ··· + 64.4414u 5.93801
a
1
=
0.153667u
29
1.00376u
28
+ ··· 56.1761u + 6.84379
0.0838534u
29
+ 0.562817u
28
+ ··· + 53.4761u 4.34607
a
3
=
0.221034u
29
1.41819u
28
+ ··· + 9.86065u 4.46191
0.00963692u
29
+ 0.0503878u
28
+ ··· + 8.72117u 0.372363
a
7
=
0.259540u
29
+ 1.76655u
28
+ ··· + 177.704u 16.8570
0.0349057u
29
0.242593u
28
+ ··· 37.5450u + 4.19011
a
6
=
0.00413707u
29
+ 0.0444402u
28
+ ··· + 113.823u 13.9150
0.0181896u
29
0.113895u
28
+ ··· + 14.0709u 0.692588
a
10
=
0.223946u
29
1.53030u
28
+ ··· 249.493u + 26.2220
0.115776u
29
+ 0.791112u
28
+ ··· + 67.0119u 4.59285
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0.331444u
29
2.31943u
28
+ ··· 261.816u + 20.0577
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.48326 1.70876I
a = 0.809450 + 0.084039I
b = 1.293622 + 0.494060I
3.96189 + 7.22895I 2.25407 6.47803I
u = 1.48326 + 1.70876I
a = 0.809450 0.084039I
b = 1.293622 0.494060I
3.96189 7.22895I 2.25407 + 6.47803I
u = 0.582023
a = 0.153159
b = 1.93909
3.64718 7.53837
u = 0.54889 1.72674I
a = 0.769650 0.282582I
b = 1.110708 0.353649I
1.58157 + 4.35870I 1.97513 7.41010I
u = 0.54889 + 1.72674I
a = 0.769650 + 0.282582I
b = 1.110708 + 0.353649I
1.58157 4.35870I 1.97513 + 7.41010I
u = 0.41170 1.41665I
a = 0.751948 0.066896I
b = 0.982304 0.859538I
0.49041 + 2.82812I 1.00910 2.97945I
u = 0.41170 + 1.41665I
a = 0.751948 + 0.066896I
b = 0.982304 + 0.859538I
0.49041 2.82812I 1.00910 + 2.97945I
u = 0.10143 1.90226I
a = 0.920871 0.566634I
b = 0.996102 + 0.183217I
0.17569 + 4.40083I 4.27520 3.49859I
u = 0.10143 + 1.90226I
a = 0.920871 + 0.566634I
b = 0.996102 0.183217I
0.17569 4.40083I 4.27520 + 3.49859I
u = 0.129260 0.273797I
a = 1.31410 + 2.67479I
b = 0.108683 + 0.973790I
1.58157 + 4.35870I 1.97513 7.41010I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.129260 + 0.273797I
a = 1.31410 2.67479I
b = 0.108683 0.973790I
1.58157 4.35870I 1.97513 + 7.41010I
u = 0.145272 1.011817I
a = 0.851144 0.237339I
b = 1.42674 0.46099I
5.71916 1.53058I 8.50440 + 4.43065I
u = 0.145272 + 1.011817I
a = 0.851144 + 0.237339I
b = 1.42674 + 0.46099I
5.71916 + 1.53058I 8.50440 4.43065I
u = 0.218320 1.108685I
a = 0.409780 + 0.523142I
b = 0.940257 + 0.348069I
1.58157 + 1.29754I 1.97513 + 1.45120I
u = 0.218320 + 1.108685I
a = 0.409780 0.523142I
b = 0.940257 0.348069I
1.58157 1.29754I 1.97513 1.45120I
u = 0.218527 0.470543I
a = 1.71214 0.95448I
b = 0.177135 + 1.347849I
3.96189 7.22895I 2.25407 + 6.47803I
u = 0.218527 + 0.470543I
a = 1.71214 + 0.95448I
b = 0.177135 1.347849I
3.96189 + 7.22895I 2.25407 6.47803I
u = 0.264664 0.140760I
a = 3.23675 2.16615I
b = 0.598654 + 0.701784I
3.96189 + 1.57271I 2.25407 0.51914I
u = 0.264664 + 0.140760I
a = 3.23675 + 2.16615I
b = 0.598654 0.701784I
3.96189 1.57271I 2.25407 + 0.51914I
u = 0.335807 1.278966I
a = 0.822986 0.128575I
b = 1.25578 + 0.86711I
0.17569 4.40083I 4.27520 + 3.49859I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.335807 + 1.278966I
a = 0.822986 + 0.128575I
b = 1.25578 0.86711I
0.17569 + 4.40083I 4.27520 3.49859I
u = 0.67455 1.32965I
a = 0.998626 + 0.359869I
b = 1.233505 + 0.007630I
5.71916 + 1.53058I 8.50440 4.43065I
u = 0.67455 + 1.32965I
a = 0.998626 0.359869I
b = 1.233505 0.007630I
5.71916 1.53058I 8.50440 + 4.43065I
u = 0.820174 0.651930I
a = 0.350518 0.308500I
b = 0.207639 0.081681I
1.58157 + 1.29754I 1.97513 + 1.45120I
u = 0.820174 + 0.651930I
a = 0.350518 + 0.308500I
b = 0.207639 + 0.081681I
1.58157 1.29754I 1.97513 1.45120I
u = 0.96834 1.96626I
a = 0.571909 + 0.355619I
b = 0.817375 0.142812I
0.49041 + 2.82812I 1.00910 2.97945I
u = 0.96834 + 1.96626I
a = 0.571909 0.355619I
b = 0.817375 + 0.142812I
0.49041 2.82812I 1.00910 + 2.97945I
u = 1.33013 0.90847I
a = 0.264031 0.020166I
b = 1.068125 + 0.364252I
3.96189 1.57271I 2.25407 + 0.51914I
u = 1.33013 + 0.90847I
a = 0.264031 + 0.020166I
b = 1.068125 0.364252I
3.96189 + 1.57271I 2.25407 0.51914I
u = 2.05678
a = 0.910060
b = 1.50212
3.64718 7.53837
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
7
u
6
+ ··· u + 1)(u
15
+ u
14
+ ··· 3u
3
1)
(u
30
u
29
+ ··· 64u 7)
c
2
, c
8
(u
7
+ u
6
+ ··· u 1)(u
15
+ u
14
+ ··· 3u
3
1)
(u
30
u
29
+ ··· 64u 7)
c
3
(u
3
u
2
+ 1)
10
(u
7
+ 3u
6
+ 3u
5
u
4
4u
3
2u
2
+ 1)
(u
15
+ 12u
14
+ ··· + 240u + 32)
c
4
, c
5
(u
5
u
4
2u
3
+ u
2
+ u + 1)
6
(u
7
4u
5
u
4
+ 4u
3
+ 2u
2
1)
(u
15
+ 7u
14
+ ··· 6u
2
+ 8)
c
7
, c
10
(u
7
+ u
4
2u
3
1)(u
15
3u
13
+ ··· + u 1)(u
30
+ 3u
29
+ ··· + 14u 1)
c
9
(u
5
u
4
2u
3
+ u
2
+ u + 1)
6
(u
7
4u
5
+ u
4
+ 4u
3
2u
2
+ 1)
(u
15
+ 7u
14
+ ··· 6u
2
+ 8)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
6
c
8
(y
7
7y
6
+ 19y
5
29y
4
+ 30y
3
19y
2
+ 7y 1)
(y
15
9y
14
+ ··· + 4y
2
1)(y
30
21y
29
+ ··· 1884y + 49)
c
3
(y
3
y
2
+ 2y 1)
10
(y
7
3y
6
+ 7y
5
13y
4
+ 6y
3
2y
2
+ 4y 1)
(y
15
4y
14
+ ··· 1280y 1024)
c
4
, c
5
, c
9
(y
5
5y
4
+ 8y
3
3y
2
y 1)
6
(y
7
8y
6
+ 24y
5
33y
4
+ 20y
3
6y
2
+ 4y 1)
(y
15
13y
14
+ ··· + 96y 64)
c
7
, c
10
(y
7
4y
5
+ ··· + 2y
2
1)(y
15
6y
14
+ ··· + 13y 1)
(y
30
+ 7y
29
+ ··· 116y + 1)
12