10
66
(K10a
40
)
1
Arc Sequences
6 7 10 8 3 2 9 5 1 4
Solving Sequence
7,9 3,8
2 6 1 10 5 4
c
7
c
2
c
6
c
1
c
9
c
5
c
4
c
3
, c
8
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
2
2, b 1, 2a + ui
I
u
2
= ha
24
a
23
+ ··· + 3a + 1, 9.20781 × 10
33
u 1.27023 × 10
32
a
23
+ ··· 4.24103 × 10
33
a + 1.21758 × 10
34
,
9.20781 × 10
33
b + 8.38185 × 10
32
a
23
+ ··· + 3.25059 × 10
34
a + 1.65649 × 10
34
i
I
u
3
= hu
16
3u
15
+ ··· + 2u 2,
u
15
+ u
14
+ 7u
13
4u
12
21u
11
+ 32u
9
+ 22u
8
18u
7
37u
6
11u
5
+ 14u
4
+ 15u
3
+ 8u
2
+ 2a 2,
2u
15
+ 3u
14
+ ··· + b 3i
I
v
1
= hv + 1, b + 1, ai
There are 4 irreducible components with 43 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
2, b 1, 2a + ui
(i) Arc colorings
a
7
=
0
u
a
9
=
1
2
u
1
a
3
=
1
0
a
8
=
1
2
u
u + 1
a
2
=
1
2
a
6
=
u
u
a
1
=
1
0
a
10
=
1
2
u 1
1
a
5
=
0
u
a
4
=
1
2
u
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 20
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.41421
a = 0.707107
b = 1.00000
8.22467 20.0000
u = 1.41421
a = 0.707107
b = 1.00000
8.22467 20.0000
3
II.
I
u
2
= ha
24
a
23
+· · ·+3a+1, 9.21×10
33
u1.27×10
32
a
23
+· · ·4.24×10
33
a+
1.22 × 10
34
, 9.21 × 10
33
b + 8.38 × 10
32
a
23
+ · · · + 3.25 × 10
34
a + 1.66 × 10
34
i
(i) Arc colorings
a
7
=
0
0.0137952a
23
0.0110554a
22
+ ··· + 0.460591a 1.32234
a
9
=
a
0.0910298a
23
+ 0.0864552a
22
+ ··· 3.53026a 1.79900
a
3
=
1
0
a
8
=
0.00894321a
23
0.0173226a
22
+ ··· + 0.0220145a + 0.00273977
0.0140329a
23
0.0422675a
22
+ ··· 1.64306a 1.21603
a
2
=
1
0.0378419a
23
0.0479410a
22
+ ··· + 1.16892a 1.75386
a
6
=
0.0137952a
23
0.0110554a
22
+ ··· + 0.460591a 1.32234
0.0592934a
23
+ 0.0703203a
22
+ ··· 1.90953a + 0.979417
a
1
=
0.0378419a
23
0.0479410a
22
+ ··· + 1.16892a 0.753855
0.0771691a
23
+ 0.113697a
22
+ ··· 1.93993a 0.509640
a
10
=
0.0781747a
23
+ 0.148387a
22
+ ··· + 0.309080a + 1.01093
0.0777965a
23
0.129702a
22
+ ··· 0.321513a 1.03329
a
5
=
0.0454982a
23
+ 0.0592650a
22
+ ··· 1.44894a 0.342919
0.0592934a
23
+ 0.0703203a
22
+ ··· 1.90953a + 0.979417
a
4
=
0.0808769a
23
+ 0.0931088a
22
+ ··· 1.39866a 0.358199
0.0000539024a
23
+ 0.0309495a
22
+ ··· + 1.26422a + 1.40296
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0.160789a
23
0.136052a
22
+ ··· + 5.30200a 13.4949
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.227035 0.729376I
a = 1.74721 0.33031I
b = 1.54253 0.63371I
1.84911 + 3.88480I 7.19439 4.17140I
u = 0.227035 + 0.729376I
a = 1.74721 + 0.33031I
b = 1.54253 + 0.63371I
1.84911 3.88480I 7.19439 + 4.17140I
u = 1.383156 0.208829I
a = 1.048381 0.638733I
b = 0.460849 0.796651I
8.42885 + 3.88480I 16.8056 4.1714I
u = 1.383156 + 0.208829I
a = 1.048381 + 0.638733I
b = 0.460849 + 0.796651I
8.42885 3.88480I 16.8056 + 4.1714I
u = 1.383156 + 0.208829I
a = 0.740319 0.608118I
b = 1.53179 + 2.23329I
8.42885 3.88480I 16.8056 + 4.1714I
u = 1.383156 0.208829I
a = 0.740319 + 0.608118I
b = 1.53179 2.23329I
8.42885 + 3.88480I 16.8056 4.1714I
u = 1.39026 0.29206I
a = 0.562829 0.247525I
b = 1.52512 + 0.54241I
3.28987 7.58818I 12.00000 + 5.13539I
u = 1.39026 + 0.29206I
a = 0.562829 + 0.247525I
b = 1.52512 0.54241I
3.28987 + 7.58818I 12.00000 5.13539I
u = 0.851576 + 0.246566I
a = 0.135015 0.800436I
b = 0.228521 + 0.294293I
0.174773 + 0.093609I 10.00912 + 0.76204I
u = 0.851576 0.246566I
a = 0.135015 + 0.800436I
b = 0.228521 0.294293I
0.174773 0.093609I 10.00912 0.76204I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.851576 0.246566I
a = 0.027684 1.049247I
b = 0.853628 + 0.103990I
0.174773 0.093609I 10.00912 0.76204I
u = 0.851576 + 0.246566I
a = 0.027684 + 1.049247I
b = 0.853628 0.103990I
0.174773 + 0.093609I 10.00912 + 0.76204I
u = 1.343201 0.063939I
a = 0.018260 0.156234I
b = 1.63366 + 0.54208I
6.40496 0.09361I 13.99088 0.76204I
u = 1.343201 + 0.063939I
a = 0.018260 + 0.156234I
b = 1.63366 0.54208I
6.40496 + 0.09361I 13.99088 + 0.76204I
u = 0.228302 0.503204I
a = 0.36401 2.59009I
b = 0.291931 0.532652I
3.28987 1.20211I 12.00000 + 5.63740I
u = 0.228302 + 0.503204I
a = 0.36401 + 2.59009I
b = 0.291931 + 0.532652I
3.28987 + 1.20211I 12.00000 5.63740I
u = 1.39026 + 0.29206I
a = 0.785377 0.784885I
b = 1.65407 + 1.38887I
3.28987 + 7.58818I 12.00000 5.13539I
u = 1.39026 0.29206I
a = 0.785377 + 0.784885I
b = 1.65407 1.38887I
3.28987 7.58818I 12.00000 + 5.13539I
u = 0.227035 + 0.729376I
a = 0.974378 0.391964I
b = 0.867048 + 0.072234I
1.84911 3.88480I 7.19439 + 4.17140I
u = 0.227035 0.729376I
a = 0.974378 + 0.391964I
b = 0.867048 0.072234I
1.84911 + 3.88480I 7.19439 4.17140I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.343201 0.063939I
a = 0.995513 0.202926I
b = 0.377496 0.258035I
6.40496 0.09361I 13.99088 0.76204I
u = 1.343201 + 0.063939I
a = 0.995513 + 0.202926I
b = 0.377496 + 0.258035I
6.40496 + 0.09361I 13.99088 + 0.76204I
u = 0.228302 0.503204I
a = 1.66043 0.86559I
b = 1.50510 1.09585I
3.28987 1.20211I 12.00000 + 5.63740I
u = 0.228302 + 0.503204I
a = 1.66043 + 0.86559I
b = 1.50510 + 1.09585I
3.28987 + 1.20211I 12.00000 5.63740I
7
III.
I
u
3
= hu
16
3u
15
+· · ·+2u2, u
15
+u
14
+· · ·+2a2, 2u
15
+3u
14
+· · ·+b3i
(i) Arc colorings
a
7
=
0
u
a
9
=
1
2
u
15
1
2
u
14
+ ··· 4u
2
+ 1
2u
15
3u
14
+ ··· 7u
2
+ 3
a
3
=
1
0
a
8
=
1
2
u
15
+
1
2
u
14
+ ··· + 2u
2
1
u
15
+ u
14
+ ··· + u 1
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
4
2u
2
a
10
=
1
2
u
15
1
2
u
14
+ ···
7
2
u
3
u
2
u
15
u
14
+ ··· u
2
+ 1
a
5
=
u
3
+ 2u
u
3
+ u
a
4
=
3
2
u
15
5
2
u
14
+ ··· u + 3
u
15
+ 2u
14
+ ··· + 2u
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u
15
14u
14
40u
13
+ 58u
12
+ 88u
11
60u
10
114u
9
46u
8
+
56u
7
+ 120u
6
+ 46u
5
32u
4
56u
3
32u
2
10u
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.47113
a = 0.336214
b = 0.527487
6.93855 11.2726
u = 1.135809 0.292915I
a = 0.357327 + 0.855663I
b = 0.278219 0.783252I
0.17586 4.85157I 10.18415 + 6.53900I
u = 1.135809 + 0.292915I
a = 0.357327 0.855663I
b = 0.278219 + 0.783252I
0.17586 + 4.85157I 10.18415 6.53900I
u = 0.697098 0.499644I
a = 0.07088 1.61327I
b = 0.881957 0.319210I
1.63698 + 4.78532I 12.50670 3.64348I
u = 0.697098 + 0.499644I
a = 0.07088 + 1.61327I
b = 0.881957 + 0.319210I
1.63698 4.78532I 12.50670 + 3.64348I
u = 0.324419 0.762338I
a = 1.92936 0.35560I
b = 1.69353 0.64765I
0.34351 9.16484I 10.75715 + 8.12303I
u = 0.324419 + 0.762338I
a = 1.92936 + 0.35560I
b = 1.69353 + 0.64765I
0.34351 + 9.16484I 10.75715 8.12303I
u = 0.072961 0.717720I
a = 1.172311 + 0.087150I
b = 1.043520 0.306684I
3.06515 + 1.13134I 4.88295 2.50814I
u = 0.072961 + 0.717720I
a = 1.172311 0.087150I
b = 1.043520 + 0.306684I
3.06515 1.13134I 4.88295 + 2.50814I
u = 0.429313
a = 0.367894
b = 0.344564
0.684897 14.2492
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.305322 0.265636I
a = 0.424913 0.397765I
b = 1.44398 + 0.49423I
1.21964 + 2.39915I 9.20728 0.67092I
u = 1.305322 + 0.265636I
a = 0.424913 + 0.397765I
b = 1.44398 0.49423I
1.21964 2.39915I 9.20728 + 0.67092I
u = 1.43956 0.30056I
a = 0.893398 + 0.793757I
b = 2.03588 1.26907I
5.9872 + 13.0293I 14.9902 8.3428I
u = 1.43956 + 0.30056I
a = 0.893398 0.793757I
b = 2.03588 + 1.26907I
5.9872 13.0293I 14.9902 + 8.3428I
u = 1.50631 0.11577I
a = 0.559166 0.622823I
b = 0.137084 0.822008I
8.80698 2.79176I 16.7106 + 5.2072I
u = 1.50631 + 0.11577I
a = 0.559166 + 0.622823I
b = 0.137084 + 0.822008I
8.80698 + 2.79176I 16.7106 5.2072I
10
IV. I
v
1
= hv + 1, b + 1, ai
(i) Arc colorings
a
7
=
1
0
a
9
=
0
1
a
3
=
1
0
a
8
=
1
1
a
2
=
1
0
a
6
=
1
0
a
1
=
1
0
a
10
=
1
1
a
5
=
1
0
a
4
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
11
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
3.28987 12.0000
12
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
6
(u)(u
2
2)(1 + u
2
+ 3u
3
2u
5
6u
6
4u
7
+ 9u
8
+ 4u
9
5u
10
u
11
+ u
12
)
2
(u
16
+ 3u
15
+ ··· 2u 2)
c
3
, c
8
(u 1)
2
(u + 1)(u
16
+ u
15
+ ··· 2u 1)(u
24
+ u
23
+ ··· + 4u + 1)
c
4
, c
10
(u 1)(u + 1)
2
(u
16
+ u
15
+ ··· 2u 1)(u
24
+ u
23
+ ··· + 4u + 1)
c
5
u(u
2
2)
(1 4u + 7u
2
7u
3
+ 14u
4
24u
5
+ 24u
6
14u
7
+ 5u
8
4u
9
+ 5u
10
3u
11
+ u
12
)
2
(u
16
+ 9u
15
+ ··· + 34u + 14)
c
7
, c
9
(u 1)
3
(u
16
+ 7u
15
+ ··· + 10u + 1)(u
24
+ 13u
23
+ ··· + 4u + 1)
13
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
6
y(y 2)
2
(1 + 2y + y
2
21y
3
+ 18y
4
+ 28y
5
12y
6
100y
7
+ 169y
8
126y
9
+ 51y
10
11y
11
+ y
12
)
2
(y
16
15y
15
+ ··· 20y + 4)
c
3
, c
4
, c
8
c
10
(y 1)
3
(y
16
7y
15
+ ··· 10y + 1)(y
24
13y
23
+ ··· 4y + 1)
c
5
y(y 2)
2
(1 2y + 21y
2
+ 3y
3
+ 94y
4
52y
5
+ 36y
6
36y
7
+ 37y
8
2y
9
+ 11y
10
+ y
11
+ y
12
)
2
(y
16
3y
15
+ ··· 1156y + 196)
c
7
, c
9
(y 1)
3
(y
16
+ 9y
15
+ ··· 38y + 1)(y
24
5y
23
+ ··· + 48y + 1)
14