10
69
(K10a
38
)
1
Arc Sequences
7 9 1 10 4 2 6 3 8 5
Solving Sequence
1,5
10 4 6
3,8
7 9 2
c
10
c
4
c
5
c
3
c
7
c
9
c
2
c
1
, c
6
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
4
u
2
+ 1, u
3
+ b, u
2
+ a + u 1i
I
u
2
= hb
28
3b
27
+ ··· 70b + 49, 1.14733 × 10
65
u 9.19423 × 10
62
b
27
+ ··· 9.01807 × 10
64
b 8.10091 × 10
64
,
2.82410 × 10
63
b
27
+ 1.05196 × 10
64
b
26
+ ··· + 2.17993 × 10
66
a 9.15323 × 10
65
i
I
u
3
= hu
19
+ 3u
18
+ ··· + 7u + 2, u
18
3u
17
+ ··· + b 3, 3u
18
7u
17
+ ··· + 2a 7i
There are 3 irreducible components with 51 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
4
u
2
+ 1, u
3
+ b, u
2
+ a + u 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
10
=
1
u
2
a
4
=
u
u
3
+ u
a
6
=
u
3
0
a
3
=
u
3
u
3
+ u
a
8
=
u
2
u + 1
u
3
a
7
=
u
3
u
2
u + 1
u
3
a
9
=
u
2
u + 2
u
3
+ u
2
a
2
=
u
2
+ u + 1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.866025 0.500000I
a = 1.36603 0.36603I
b = 1.00000I
3.28987 + 2.02988I 2.00000 3.46410I
u = 0.866025 + 0.500000I
a = 1.36603 + 0.36603I
b = 1.00000I
3.28987 2.02988I 2.00000 + 3.46410I
u = 0.866025 0.500000I
a = 0.36603 + 1.36603I
b = 1.00000I
3.28987 2.02988I 2.00000 + 3.46410I
u = 0.866025 + 0.500000I
a = 0.36603 1.36603I
b = 1.00000I
3.28987 + 2.02988I 2.00000 3.46410I
3
II.
I
u
2
= hb
28
3b
27
+· · ·70b+49, 1.15×10
65
u9.19×10
62
b
27
+· · ·9.02×10
64
b
8.10 × 10
64
, 2.82 × 10
63
b
27
+ 1.05 × 10
64
b
26
+ · · · + 2.18 × 10
66
a 9.15 × 10
65
i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
0.00801359b
27
0.0174233b
26
+ ··· + 0.786005b + 0.706067
a
10
=
1
0.000648960b
27
+ 0.00263653b
26
+ ··· 0.727978b + 0.604115
a
4
=
0.00801359b
27
+ 0.0174233b
26
+ ··· 0.786005b 0.706067
0.0106939b
27
0.0306328b
26
+ ··· + 0.705732b 0.473237
a
6
=
0.00268028b
27
+ 0.0132095b
26
+ ··· + 0.0802729b + 1.17930
0.00768812b
27
+ 0.0227055b
26
+ ··· 1.23555b + 0.997978
a
3
=
0.00268028b
27
0.0132095b
26
+ ··· 0.0802729b 1.17930
0.0106939b
27
0.0306328b
26
+ ··· + 0.705732b 0.473237
a
8
=
0.00129550b
27
0.00482567b
26
+ ··· + 2.24317b + 0.419887
b
a
7
=
0.00230852b
27
0.000220586b
26
+ ··· + 2.42681b + 0.376989
0.00473894b
27
0.0148801b
26
+ ··· + 1.18991b 0.536260
a
9
=
0.00123684b
27
0.00808971b
26
+ ··· + 3.42696b + 1.32634
0.00177094b
27
+ 0.00324259b
26
+ ··· + 0.377865b + 0.530844
a
2
=
0.000701567b
27
+ 0.0104359b
26
+ ··· + 0.0616514b + 2.65184
0.00750187b
27
+ 0.0227812b
26
+ ··· 0.400850b + 1.06901
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0.00234119b
27
+ 0.0161084b
26
+ ··· + 3.13658b + 6.24211
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.403136 0.584808I
a = 0.445488 + 1.297375I
b = 1.48801 1.19980I
4.65252 0.62859I 2.31651 + 1.42251I
u = 0.403136 + 0.584808I
a = 0.445488 1.297375I
b = 1.48801 + 1.19980I
4.65252 + 0.62859I 2.31651 1.42251I
u = 0.403136 + 0.584808I
a = 1.142352 + 0.668190I
b = 0.860151 0.151246I
4.65252 + 0.62859I 2.31651 1.42251I
u = 0.403136 0.584808I
a = 1.142352 0.668190I
b = 0.860151 + 0.151246I
4.65252 0.62859I 2.31651 + 1.42251I
u = 0.268039 + 0.757899I
a = 0.143310 + 0.427216I
b = 0.80984 1.45942I
0.22261 3.62879I 3.66617 + 2.63226I
u = 0.268039 0.757899I
a = 0.143310 0.427216I
b = 0.80984 + 1.45942I
0.22261 + 3.62879I 3.66617 2.63226I
u = 0.728347 0.560551I
a = 0.064777 + 0.599184I
b = 0.371682 0.254174I
1.84948 2.19128I 2.76081 + 3.85718I
u = 0.728347 + 0.560551I
a = 0.064777 0.599184I
b = 0.371682 + 0.254174I
1.84948 + 2.19128I 2.76081 3.85718I
u = 1.157218 + 0.286866I
a = 1.17269 1.74006I
b = 0.06603 1.71504I
4.53640 + 0.47055I 9.32829 + 0.18349I
u = 1.157218 0.286866I
a = 1.17269 + 1.74006I
b = 0.06603 + 1.71504I
4.53640 0.47055I 9.32829 0.18349I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.142594 0.546762I
a = 0.78194 1.24283I
b = 0.06519 1.60824I
2.77434 8.53123I 6.72348 + 6.18031I
u = 1.142594 + 0.546762I
a = 0.78194 + 1.24283I
b = 0.06519 + 1.60824I
2.77434 + 8.53123I 6.72348 6.18031I
u = 0.989783 + 0.381937I
a = 0.75275 1.27344I
b = 0.090790 0.426836I
1.59516 + 1.40484I 5.50927 0.52948I
u = 0.989783 0.381937I
a = 0.75275 + 1.27344I
b = 0.090790 + 0.426836I
1.59516 1.40484I 5.50927 + 0.52948I
u = 0.268039 0.757899I
a = 0.805404 + 0.051418I
b = 0.148756 0.914884I
0.22261 + 3.62879I 3.66617 2.63226I
u = 0.268039 + 0.757899I
a = 0.805404 0.051418I
b = 0.148756 + 0.914884I
0.22261 3.62879I 3.66617 + 2.63226I
u = 0.989783 0.381937I
a = 1.91833 1.38556I
b = 0.20805 2.13390I
1.59516 1.40484I 5.50927 + 0.52948I
u = 0.989783 + 0.381937I
a = 1.91833 + 1.38556I
b = 0.20805 + 2.13390I
1.59516 + 1.40484I 5.50927 0.52948I
u = 1.068409 0.522447I
a = 1.02538 1.04810I
b = 0.439782 0.298160I
2.72606 + 5.07185I 2.32847 6.33126I
u = 1.068409 + 0.522447I
a = 1.02538 + 1.04810I
b = 0.439782 + 0.298160I
2.72606 5.07185I 2.32847 + 6.33126I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.728347 0.560551I
a = 0.912076 + 0.177857I
b = 0.443852 0.575052I
1.84948 2.19128I 2.76081 + 3.85718I
u = 0.728347 + 0.560551I
a = 0.912076 0.177857I
b = 0.443852 + 0.575052I
1.84948 + 2.19128I 2.76081 3.85718I
u = 1.157218 0.286866I
a = 0.208422 0.989667I
b = 0.809510 0.540535I
4.53640 0.47055I 9.32829 0.18349I
u = 1.157218 + 0.286866I
a = 0.208422 + 0.989667I
b = 0.809510 + 0.540535I
4.53640 + 0.47055I 9.32829 + 0.18349I
u = 1.142594 + 0.546762I
a = 0.88693 2.21821I
b = 1.12473 1.96518I
2.77434 + 8.53123I 6.72348 6.18031I
u = 1.142594 0.546762I
a = 0.88693 + 2.21821I
b = 1.12473 + 1.96518I
2.77434 8.53123I 6.72348 + 6.18031I
u = 1.068409 0.522447I
a = 0.47730 2.74473I
b = 1.89542 1.97549I
2.72606 + 5.07185I 2.32847 6.33126I
u = 1.068409 + 0.522447I
a = 0.47730 + 2.74473I
b = 1.89542 + 1.97549I
2.72606 5.07185I 2.32847 + 6.33126I
7
III. I
u
3
=
hu
19
+3u
18
+· · ·+ 7u + 2, u
18
3u
17
+· · ·+ b 3, 3u
18
7u
17
+· · ·+ 2a 7i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
10
=
1
u
2
a
4
=
u
u
3
+ u
a
6
=
u
3
u
5
u
3
+ u
a
3
=
u
3
u
3
+ u
a
8
=
3
2
u
18
+
7
2
u
17
+ ··· +
17
2
u +
7
2
u
18
+ 3u
17
+ ··· + 8u + 3
a
7
=
1
2
u
18
+
3
2
u
17
+ ··· +
7
2
u +
3
2
u
17
+ u
16
+ ··· + 2u + 1
a
9
=
1
2
u
18
3
2
u
17
+ ···
7
2
u
1
2
u
17
u
16
+ ··· 3u 1
a
2
=
3
2
u
18
+
7
2
u
17
+ ··· +
17
2
u +
7
2
u
18
2u
17
+ ··· 4u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12u
18
+ 26u
17
24u
16
116u
15
42u
14
+ 200u
13
+ 222u
12
116u
11
334u
10
108u
9
+ 228u
8
+ 240u
7
4u
6
172u
5
118u
4
+ 12u
3
+ 78u
2
+ 64u + 30
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.143801 0.588812I
a = 1.12767 2.25574I
b = 0.96492 2.22818I
0.26882 + 14.12651I 3.54919 9.60559I
u = 1.143801 + 0.588812I
a = 1.12767 + 2.25574I
b = 0.96492 + 2.22818I
0.26882 14.12651I 3.54919 + 9.60559I
u = 1.141046 0.480142I
a = 0.96822 + 1.32852I
b = 0.00563 + 1.67007I
4.50851 + 3.09886I 9.38086 1.28227I
u = 1.141046 + 0.480142I
a = 0.96822 1.32852I
b = 0.00563 1.67007I
4.50851 3.09886I 9.38086 + 1.28227I
u = 0.954304 0.656562I
a = 1.022244 0.645581I
b = 0.517413 + 0.115037I
2.99077 0.72249I 1.52455 + 2.82827I
u = 0.954304 + 0.656562I
a = 1.022244 + 0.645581I
b = 0.517413 0.115037I
2.99077 + 0.72249I 1.52455 2.82827I
u = 0.695977
a = 0.715081
b = 0.538288
0.927841 11.2944
u = 0.656620 0.736849I
a = 0.530916 0.111769I
b = 0.692991 + 0.514666I
3.87251 + 6.01197I 0.18591 7.59122I
u = 0.656620 + 0.736849I
a = 0.530916 + 0.111769I
b = 0.692991 0.514666I
3.87251 6.01197I 0.18591 + 7.59122I
u = 0.342490 0.822016I
a = 0.423303 + 0.244228I
b = 0.84616 1.72998I
2.12081 8.87474I 0.63360 + 6.11132I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.342490 + 0.822016I
a = 0.423303 0.244228I
b = 0.84616 + 1.72998I
2.12081 + 8.87474I 0.63360 6.11132I
u = 0.085864 0.693927I
a = 0.667057 0.203041I
b = 0.176244 + 0.940079I
1.57783 + 1.22058I 5.73688 3.21713I
u = 0.085864 + 0.693927I
a = 0.667057 + 0.203041I
b = 0.176244 0.940079I
1.57783 1.22058I 5.73688 + 3.21713I
u = 0.833011 0.594872I
a = 0.493073 + 0.284708I
b = 0.042413 0.483034I
1.75185 2.35707I 4.45005 + 4.73717I
u = 0.833011 + 0.594872I
a = 0.493073 0.284708I
b = 0.042413 + 0.483034I
1.75185 + 2.35707I 4.45005 4.73717I
u = 1.160316 0.382174I
a = 0.30429 + 1.44141I
b = 0.996422 + 0.904006I
5.16612 4.98291I 9.41511 + 6.18167I
u = 1.160316 + 0.382174I
a = 0.30429 1.44141I
b = 0.996422 0.904006I
5.16612 + 4.98291I 9.41511 6.18167I
u = 1.178787 0.200823I
a = 1.12805 1.83215I
b = 0.06929 1.61595I
2.86306 + 5.96190I 6.84845 4.63798I
u = 1.178787 + 0.200823I
a = 1.12805 + 1.83215I
b = 0.06929 + 1.61595I
2.86306 5.96190I 6.84845 + 4.63798I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
6
c
8
(u
2
+ 1)
2
(u
19
+ 4u
17
+ ··· + 2u + 1)(u
28
+ u
27
+ ··· 2u + 1)
c
3
(u
4
u
2
+ 1)
(3 7u + 12u
2
12u
3
+ 10u
4
10u
5
+ 14u
6
22u
7
+ 23u
8
19u
9
+ 14u
10
10u
11
+ 7u
12
3u
13
+ u
14
)
2
(u
19
+ 9u
18
+ ··· + 157u + 22)
c
4
, c
10
(u
4
u
2
+ 1)
(1 u + 2u
4
2u
5
2u
6
+ 6u
7
u
8
7u
9
+ 4u
10
+ 4u
11
3u
12
u
13
+ u
14
)
2
(u
19
+ 3u
18
+ ··· + 7u + 2)
c
5
(u
2
u + 1)
2
(1 + u + 4u
2
+ 8u
3
+ 14u
4
+ 18u
5
+ 26u
6
+ 44u
7
+ 73u
8
+ 93u
9
+ 86u
10
+ 56u
11
+ 25u
12
+ 7u
13
+ u
14
)
2
(u
19
+ 9u
18
+ ··· + 5u + 4)
c
7
, c
9
(u + 1)
4
(u
19
+ 8u
18
+ ··· 2u 1)(u
28
+ 15u
27
+ ··· + 10u
2
+ 1)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
6
c
8
(y + 1)
4
(y
19
+ 8y
18
+ ··· 2y 1)(y
28
+ 15y
27
+ ··· + 10y
2
+ 1)
c
3
(y
2
y + 1)
2
(9 + 23y + 36y
2
+ 40y
3
+ 26y
4
+ 22y
5
2y
6
48y
7
23y
8
y
9
+ 34y
10
+ 28y
11
+ 17y
12
+ 5y
13
+ y
14
)
2
(y
19
+ 3y
18
+ ··· + 1461y 484)
c
4
, c
10
(y
2
y + 1)
2
(1 y + 4y
2
8y
3
+ 14y
4
18y
5
+ 26y
6
44y
7
+ 73y
8
93y
9
+ 86y
10
56y
11
+ 25y
12
7y
13
+ y
14
)
2
(y
19
9y
18
+ ··· + 5y 4)
c
5
(y
2
+ y + 1)
2
(1 + 7y + 28y
2
+ 64y
3
+ 174y
4
+ 270y
5
+ 274y
6
+ 212y
7
+ 197y
8
+ 55y
9
+ 66y
10
+ 8y
11
+ 13y
12
+ y
13
+ y
14
)
2
(y
19
+ 3y
18
+ ··· + 129y 16)
c
7
, c
9
(y 1)
4
(y
19
+ 12y
18
+ ··· + 30y 1)(y
28
5y
27
+ ··· + 20y + 1)
12