9
34
(K9a
28
)
1
Arc Sequences
7 9 8 2 3 4 1 6 5
Solving Sequence
1,7 2,4
5 8 3 6 9
c
1
c
4
c
7
c
3
c
6
c
9
c
2
, c
5
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
5
+ 4u
3
u
2
+ 6u + 1, u
4
3u
3
13u
2
+ 19b 25, 12u
4
+ 2u
3
42u
2
+ 19a + 19u 53i
I
u
2
= hu
22
u
21
+ ··· 380u + 31, 1.54528 × 10
34
u
21
1.00011 × 10
35
u
20
+ ··· + 8.70880 × 10
36
b 3.68171 × 10
37
,
1.06728 × 10
38
u
21
+ 6.15849 × 10
37
u
20
+ ··· + 2.69973 × 10
38
a + 9.47724 × 10
39
i
I
u
3
= hu
17
+ 2u
16
+ ··· u 1, 2.21511 × 10
15
u
16
9.02960 × 10
14
u
15
+ ··· + 9.87751 × 10
16
b + 7.97715 × 10
16
,
3.86652 × 10
16
u
16
1.00971 × 10
17
u
15
+ ··· + 9.87751 × 10
16
a + 2.45682 × 10
17
i
There are 3 irreducible components with 44 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
5
+ 4u
3
u
2
+ 6u + 1, u
4
3u
3
13u
2
+ 19b 25, 12u
4
+
2u
3
42u
2
+ 19a + 19u 53i
(i) Arc colorings
a
1
=
1
0
a
7
=
12
19
u
4
2
19
u
3
+ ··· u +
53
19
0.0526316u
4
+ 0.157895u
3
+ 0.684211u
2
+ 1.31579
a
2
=
12
19
u
4
+
2
19
u
3
+ ··· + u
53
19
5
19
u
4
+
4
19
u
3
+ ··· + u
30
19
a
4
=
0
u
a
5
=
4
19
u
4
7
19
u
3
+ ··· u +
24
19
0.0526316u
4
0.157895u
3
+ 0.315789u
2
+ 0.684211
a
8
=
13
19
u
4
+
1
19
u
3
+ ··· u +
78
19
0.0526316u
4
+ 0.157895u
3
+ 0.684211u
2
+ 1.31579
a
3
=
12
19
u
4
+
2
19
u
3
+ ··· + u
53
19
6
19
u
4
+
1
19
u
3
+ ··· + u
17
19
a
6
=
12
19
u
4
2
19
u
3
+ ··· u +
53
19
0.368421u
4
+ 0.105263u
3
+ 1.78947u
2
+ 1.21053
a
9
=
0
2
19
u
4
6
19
u
3
+ ··· u
12
19
a
9
=
0
2
19
u
4
6
19
u
3
+ ··· u
12
19
(ii) Obstruction class = 1
(iii) Cusp Shapes =
44
19
u
4
+
20
19
u
3
173
19
u
2
+ 9u
302
19
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.48913 1.70661I
a = 0.315141 0.634725I
b = 0.372466 + 1.263916I
3.01018 + 5.17259I 1.67537 5.94701I
u = 0.48913 + 1.70661I
a = 0.315141 + 0.634725I
b = 0.372466 1.263916I
3.01018 5.17259I 1.67537 + 5.94701I
u = 0.159685
a = 3.00636
b = 1.33263
2.14584 17.5699
u = 0.568970 1.289650I
a = 0.688042 + 0.434689I
b = 0.038780 0.656277I
0.29233 + 3.70382I 0.53969 6.40947I
u = 0.568970 + 1.289650I
a = 0.688042 0.434689I
b = 0.038780 + 0.656277I
0.29233 3.70382I 0.53969 + 6.40947I
3
II. I
u
2
=
hu
22
u
21
+· · ·380u+31, 1.55×10
34
u
21
1.00×10
35
u
20
+· · ·+8.71×10
36
b
3.68× 10
37
, 1.07×10
38
u
21
+6.16 ×10
37
u
20
+· · · +2.70 ×10
38
a + 9.48× 10
39
i
(i) Arc colorings
a
1
=
1
0
a
7
=
0.395330u
21
0.228115u
20
+ ··· + 317.522u 35.1044
0.00177439u
21
+ 0.0114840u
20
+ ··· 25.3251u + 4.22758
a
2
=
0.611634u
21
+ 0.383149u
20
+ ··· 480.444u + 51.1271
0.0142107u
21
0.0237106u
20
+ ··· + 35.7588u 5.52493
a
4
=
0
u
a
5
=
0.186635u
21
0.114700u
20
+ ··· + 137.977u 13.4913
0.311095u
21
+ 0.198692u
20
+ ··· 245.048u + 25.9814
a
8
=
0.393556u
21
0.216631u
20
+ ··· + 292.197u 30.8768
0.00177439u
21
+ 0.0114840u
20
+ ··· 25.3251u + 4.22758
a
3
=
0.832403u
21
0.511437u
20
+ ··· + 611.450u 61.6271
0.237985u
21
0.142936u
20
+ ··· + 183.419u 18.5201
a
6
=
0.395330u
21
0.228115u
20
+ ··· + 317.522u 35.1044
0.0789277u
21
+ 0.0550670u
20
+ ··· 76.6117u + 9.41125
a
9
=
0.238038u
21
+ 0.116179u
20
+ ··· 121.039u + 11.2955
0.311769u
21
0.218329u
20
+ ··· + 269.380u 28.5070
a
9
=
0.238038u
21
+ 0.116179u
20
+ ··· 121.039u + 11.2955
0.311769u
21
0.218329u
20
+ ··· + 269.380u 28.5070
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0.899579u
21
0.361023u
20
+ ··· + 303.383u 6.98934
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.09449 1.39486I
a = 0.074289 + 0.679601I
b = 0.234018 0.605151I
0.80290 2.70441I 3.46762 0.08333I
u = 1.09449 + 1.39486I
a = 0.074289 0.679601I
b = 0.234018 + 0.605151I
0.80290 + 2.70441I 3.46762 + 0.08333I
u = 0.93463 1.96170I
a = 0.115334 0.803335I
b = 0.290349 + 1.272232I
4.63073 + 5.00074I 7.84059 6.22751I
u = 0.93463 + 1.96170I
a = 0.115334 + 0.803335I
b = 0.290349 1.272232I
4.63073 5.00074I 7.84059 + 6.22751I
u = 0.77437 1.80213I
a = 0.324756 + 0.268135I
b = 0.290349 1.272232I
4.63073 5.00074I 7.84059 + 6.22751I
u = 0.77437 + 1.80213I
a = 0.324756 0.268135I
b = 0.290349 + 1.272232I
4.63073 + 5.00074I 7.84059 6.22751I
u = 0.539408 1.137214I
a = 0.333150 0.877548I
b = 0.57044 + 1.34258I
1.76023 + 5.92443I 3.17045 10.02355I
u = 0.539408 + 1.137214I
a = 0.333150 + 0.877548I
b = 0.57044 1.34258I
1.76023 5.92443I 3.17045 + 10.02355I
u = 0.114441 1.174872I
a = 0.167178 + 0.800914I
b = 0.572881 0.536287I
0.42400 2.24779I 3.63582 + 5.06360I
u = 0.114441 + 1.174872I
a = 0.167178 0.800914I
b = 0.572881 + 0.536287I
0.42400 + 2.24779I 3.63582 5.06360I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.308557 0.368076I
a = 1.19849 2.03397I
b = 0.234018 0.605151I
0.80290 2.70441I 3.46762 0.08333I
u = 0.308557 + 0.368076I
a = 1.19849 + 2.03397I
b = 0.234018 + 0.605151I
0.80290 + 2.70441I 3.46762 + 0.08333I
u = 0.338231 0.031077I
a = 0.547725 0.798746I
b = 1.10821
2.37876 12.2613
u = 0.338231 + 0.031077I
a = 0.547725 + 0.798746I
b = 1.10821
2.37876 12.2613
u = 0.420994 0.244996I
a = 0.05369 1.45636I
b = 0.572881 0.536287I
0.42400 2.24779I 3.63582 + 5.06360I
u = 0.420994 + 0.244996I
a = 0.05369 + 1.45636I
b = 0.572881 + 0.536287I
0.42400 + 2.24779I 3.63582 5.06360I
u = 0.49755 1.99148I
a = 0.121501 + 0.568413I
b = 0.57044 1.34258I
1.76023 5.92443I 3.17045 + 10.02355I
u = 0.49755 + 1.99148I
a = 0.121501 0.568413I
b = 0.57044 + 1.34258I
1.76023 + 5.92443I 3.17045 10.02355I
u = 1.00148 2.34859I
a = 0.107251 0.294522I
b = 0.253759 + 0.946686I
0.13765 5.21629I 0.43603 + 9.01278I
u = 1.00148 + 2.34859I
a = 0.107251 + 0.294522I
b = 0.253759 0.946686I
0.13765 + 5.21629I 0.43603 9.01278I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.161649 0.261604I
a = 0.08199 + 1.64659I
b = 0.253759 0.946686I
0.13765 + 5.21629I 0.43603 9.01278I
u = 1.161649 + 0.261604I
a = 0.08199 1.64659I
b = 0.253759 + 0.946686I
0.13765 5.21629I 0.43603 + 9.01278I
6
III.
I
u
3
= hu
17
+2u
16
+· · ·u1, 2.22×10
15
u
16
9.03×10
14
u
15
+· · ·+9.88×10
16
b+
7.98× 10
16
, 3.87×10
16
u
16
1.01 ×10
17
u
15
+· · · +9.88 ×10
16
a + 2.46× 10
17
i
(i) Arc colorings
a
1
=
1
0
a
7
=
0.391446u
16
+ 1.02223u
15
+ ··· 13.0404u 2.48729
0.0224258u
16
+ 0.00914157u
15
+ ··· + 1.22876u 0.807607
a
2
=
0.391446u
16
+ 1.02223u
15
+ ··· 13.0404u 2.48729
0.173298u
16
0.322219u
15
+ ··· + 1.04983u 0.988218
a
4
=
0
u
a
5
=
0.0974109u
16
0.476617u
15
+ ··· + 5.09590u + 4.82732
0.293327u
16
+ 0.571923u
15
+ ··· 2.92587u + 0.369021
a
8
=
0.369021u
16
+ 1.03137u
15
+ ··· 11.8116u 3.29489
0.0224258u
16
+ 0.00914157u
15
+ ··· + 1.22876u 0.807607
a
3
=
0.405634u
16
0.475431u
15
+ ··· + 2.25729u 5.43691
0.263711u
16
0.505477u
15
+ ··· + 5.25736u 0.218148
a
6
=
0.391446u
16
+ 1.02223u
15
+ ··· 13.0404u 2.48729
0.00967049u
16
+ 0.0857508u
15
+ ··· + 0.597981u 1.04694
a
9
=
0.998996u
16
1.79308u
15
+ ··· + 11.7594u 3.30602
0.335837u
16
0.690061u
15
+ ··· + 5.84255u + 0.405634
a
9
=
0.998996u
16
1.79308u
15
+ ··· + 11.7594u 3.30602
0.335837u
16
0.690061u
15
+ ··· + 5.84255u + 0.405634
(ii) Obstruction class = 1
(iii) Cusp Shapes =
199586521049607439
98775143384339033
u
16
+
400105814902270941
98775143384339033
u
15
+ ···
1727727253549674884
98775143384339033
u +
6651220862961528
98775143384339033
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.70241 1.65095I
a = 0.495612 0.476418I
b = 0.048681 + 1.008066I
3.35770 + 0.12402I 5.97884 + 0.38118I
u = 0.70241 + 1.65095I
a = 0.495612 + 0.476418I
b = 0.048681 1.008066I
3.35770 0.12402I 5.97884 0.38118I
u = 0.65766 2.21038I
a = 0.215737 + 0.697632I
b = 0.59542 1.30831I
3.3114 + 14.3446I 1.18187 8.40363I
u = 0.65766 + 2.21038I
a = 0.215737 0.697632I
b = 0.59542 + 1.30831I
3.3114 14.3446I 1.18187 + 8.40363I
u = 0.49993 1.35736I
a = 0.338995 0.795720I
b = 0.546851 + 1.063672I
3.79067 2.00597I 6.21078 + 1.26630I
u = 0.49993 + 1.35736I
a = 0.338995 + 0.795720I
b = 0.546851 1.063672I
3.79067 + 2.00597I 6.21078 1.26630I
u = 0.408215 0.182807I
a = 2.28584 + 0.49755I
b = 0.582313 0.090917I
1.32135 1.62186I 2.58195 + 4.11393I
u = 0.408215 + 0.182807I
a = 2.28584 0.49755I
b = 0.582313 + 0.090917I
1.32135 + 1.62186I 2.58195 4.11393I
u = 0.005851 0.246041I
a = 2.72594 + 4.02166I
b = 1.115484 0.170377I
0.27750 8.29795I 1.06571 + 6.88359I
u = 0.005851 + 0.246041I
a = 2.72594 4.02166I
b = 1.115484 + 0.170377I
0.27750 + 8.29795I 1.06571 6.88359I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.136079 0.806342I
a = 0.543654 + 0.293993I
b = 0.423210 0.769632I
0.28619 1.83578I 2.59246 + 3.36751I
u = 0.136079 + 0.806342I
a = 0.543654 0.293993I
b = 0.423210 + 0.769632I
0.28619 + 1.83578I 2.59246 3.36751I
u = 0.338391
a = 0.457584
b = 1.18539
2.39123 15.5891
u = 0.45293 2.22807I
a = 0.341175 0.700130I
b = 0.437546 + 1.154220I
4.31147 5.61068I 7.96642 + 8.06049I
u = 0.45293 + 2.22807I
a = 0.341175 + 0.700130I
b = 0.437546 1.154220I
4.31147 + 5.61068I 7.96642 8.06049I
u = 0.504163 1.309703I
a = 0.276141 + 0.557698I
b = 0.28698 1.44004I
5.40594 3.12036I 5.53287 + 3.71986I
u = 0.504163 + 1.309703I
a = 0.276141 0.557698I
b = 0.28698 + 1.44004I
5.40594 + 3.12036I 5.53287 3.71986I
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
5
2u
4
+ 3u
3
3u
2
+ u 1)
(1 + 2u + 5u
2
+ 9u
3
+ 15u
4
+ 18u
5
+ 20u
6
+ 18u
7
+ 13u
8
+ 8u
9
+ 3u
10
+ u
11
)
2
(u
17
5u
16
+ ··· 21u + 4)
c
2
, c
8
(u
5
u
4
+ ··· + u
2
1)(u
17
+ u
16
+ ··· u 1)(u
22
+ 3u
21
+ ··· + 6u + 1)
c
3
, c
9
(u
5
u
3
+ u
2
+ u 1)(u
17
+ 5u
13
+ ··· + 4u 1)
(u
22
+ u
21
+ ··· 10u + 1)
c
4
, c
6
(u
5
+ 2u
4
+ ··· + 3u + 1)(u
17
+ 2u
16
+ ··· + 8u 1)
(u
22
u
21
+ ··· 4u + 1)
c
5
(u
5
3u
4
+ 5u
3
4u
2
+ 3u 1)
(u
11
5u
10
+ 12u
9
15u
8
+ 8u
7
+ 4u
6
8u
5
+ 3u
4
+ 3u
3
3u
2
+ 1)
2
(u
17
+ 10u
16
+ ··· + 5u + 2)
c
7
(u
5
+ 2u
4
+ 3u
3
+ 3u
2
+ u + 1)
(1 + 2u + 5u
2
+ 9u
3
+ 15u
4
+ 18u
5
+ 20u
6
+ 18u
7
+ 13u
8
+ 8u
9
+ 3u
10
+ u
11
)
2
(u
17
5u
16
+ ··· 21u + 4)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
(y
5
+ 2y
4
y
3
7y
2
5y 1)
(1 6y 19y
2
37y
3
55y
4
56y
5
24y
6
+ 20y
7
+ 35y
8
+ 22y
9
+ 7y
10
+ y
11
)
2
(y
17
+ 9y
16
+ ··· + 17y 16)
c
2
, c
8
(y
5
3y
4
+ ··· + 2y 1)(y
17
+ 7y
16
+ ··· 19y 1)
(y
22
5y
21
+ ··· + 72y
2
+ 1)
c
3
, c
9
(y
5
2y
4
+ ··· + 3y 1)(y
17
+ 10y
15
+ ··· + 10y 1)
(y
22
y
21
+ ··· 8y + 1)
c
4
, c
6
(y
5
+ 2y
4
+ ··· + 3y 1)(y
17
12y
16
+ ··· + 46y 1)
(y
22
+ 3y
21
+ ··· + 8y + 1)
c
5
(y
5
+ y
4
+ 7y
3
+ 8y
2
+ y 1)
(1 + 6y 15y
2
+ 19y
3
3y
4
+ 8y
5
12y
6
+ 28y
7
9y
8
+ 10y
9
y
10
+ y
11
)
2
(y
17
+ 16y
15
+ ··· 11y 4)
11