9
28
(K9a
5
)
1
Arc Sequences
9 8 2 7 4 1 5 3 6
Solving Sequence
3,8
9 2
1,5
7 4 6
c
8
c
2
c
1
c
7
c
4
c
5
c
3
, c
6
, c
9
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu 2, b + 1, 2a + 1i
I
u
2
= hu
4
+ 7u
3
+ 15u
2
+ 13u + 13, 2u
3
u
2
+ 39a + 22u, u
3
4u
2
+ 3b 2u 2i
I
u
3
= hu
6
+ 10u
5
+ 38u
4
+ 67u
3
+ 55u
2
+ 18u + 2, u
5
+ 10u
4
+ 38u
3
+ 66u
2
+ b + 49u + 9,
9u
5
+ 88u
4
+ 322u
3
+ 527u
2
+ 2a + 363u + 64i
I
u
4
= hb
16
b
15
4b
14
+ 6b
13
+ 5b
12
13b
11
+ 3b
10
+ 11b
9
12b
8
+ 2b
7
+ 8b
6
8b
5
+ 2b
4
+ 2b
3
2b
2
+ 2b 1,
b
15
b
14
+ 4b
13
+ 3b
12
8b
11
4b
10
+ 7b
9
2b
7
+ 4b
6
b
5
4b
4
2b
3
+ b + a,
b
15
+ 5b
13
2b
12
10b
11
+ 8b
10
+ 7b
9
12b
8
+ 4b
7
+ 6b
6
8b
5
+ 2b
4
+ 2b
3
2b
2
+ b + u 2i
There are 4 irreducible components with 27 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, b + 1, 2a + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
1
2
1
a
9
=
1
2
1
a
2
=
1
2
1
a
1
=
1
2
1
a
5
=
0
2
a
7
=
1
2
1
a
4
=
1
2
1
a
6
=
1
2
1
a
6
=
1
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 2.00000
a = 0.500000
b = 1.00000
3.28987 12.0000
3
II.
I
u
2
= hu
4
+7u
3
+15u
2
+13u+13, 2u
3
u
2
+39a+22u, u
3
4u
2
+3b2u2i
(i) Arc colorings
a
3
=
1
0
a
8
=
2
39
u
3
+
1
39
u
2
22
39
u
1
3
u
3
+
4
3
u
2
+
2
3
u +
2
3
a
9
=
0.282051u
3
1.30769u
2
1.23077u 0.666667
1
3
u
3
+
4
3
u
2
+
2
3
u +
2
3
a
2
=
10
39
u
3
44
39
u
2
46
39
u
2
3
u
3
8
3
u
2
7
3
u
10
3
a
1
=
10
39
u
3
+
31
39
u
2
+
20
39
u +
5
3
4
3
u
3
5u
2
4u
16
3
a
5
=
0
u
a
7
=
2
39
u
3
+
1
39
u
2
22
39
u
2
3
u
3
3u
2
3u
11
3
a
4
=
0.256410u
3
1.12821u
2
1.17949u 1
4
3
u
3
+ 5u
2
+ 4u +
16
3
a
6
=
16
39
u
3
+
20
13
u
2
+
15
13
u +
4
3
u
3
10
3
u
2
5
3
u
10
3
a
6
=
16
39
u
3
+
20
13
u
2
+
15
13
u +
4
3
u
3
10
3
u
2
5
3
u
10
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
+ 16u
2
+ 16u + 14
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 3.26314 0.21087I
a = 0.353103 0.190717I
b = 1.192440 0.547877I
3.85720 + 11.56318I 5.79581 8.26147I
u = 3.26314 + 0.21087I
a = 0.353103 + 0.190717I
b = 1.192440 + 0.547877I
3.85720 11.56318I 5.79581 + 8.26147I
u = 0.236864 1.076893I
a = 0.146897 + 0.675308I
b = 0.692440 + 0.318148I
1.07760 + 1.41376I 4.20419 4.79737I
u = 0.236864 + 1.076893I
a = 0.146897 0.675308I
b = 0.692440 0.318148I
1.07760 1.41376I 4.20419 + 4.79737I
5
III. I
u
3
= hu
6
+ 10u
5
+ 38u
4
+ 67u
3
+ 55u
2
+ 18u + 2, u
5
+ 10u
4
+ 38u
3
+
66u
2
+ b + 49u + 9, 9u
5
+ 88u
4
+ · · · + 2a + 64i
(i) Arc colorings
a
3
=
1
0
a
8
=
9
2
u
5
44u
4
+ ···
363
2
u 32
u
5
10u
4
38u
3
66u
2
49u 9
a
9
=
7
2
u
5
34u
4
+ ···
265
2
u 23
u
5
10u
4
38u
3
66u
2
49u 9
a
2
=
5
2
u
5
+ 25u
4
+ 94u
3
+
319
2
u
2
+
231
2
u + 24
u
4
+ 8u
3
+ 22u
2
+ 22u + 5
a
1
=
3
2
u
5
14u
4
48u
3
143
2
u
2
87
2
u 6
u
5
9u
4
30u
3
44u
2
28u 5
a
5
=
0
u
a
7
=
9
2
u
5
44u
4
+ ···
363
2
u 32
u
5
10u
4
37u
3
60u
2
40u 7
a
4
=
5
2
u
5
+ 25u
4
+ 94u
3
+
319
2
u
2
+
231
2
u + 23
u
5
+ 9u
4
+ 30u
3
+ 44u
2
+ 28u + 5
a
6
=
5
2
u
5
+ 24u
4
+ 86u
3
+
275
2
u
2
+
187
2
u + 17
u
5
+ 9u
4
+ 29u
3
+ 39u
2
+ 21u + 3
a
6
=
5
2
u
5
+ 24u
4
+ 86u
3
+
275
2
u
2
+
187
2
u + 17
u
5
+ 9u
4
+ 29u
3
+ 39u
2
+ 21u + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
5
+ 54u
4
+ 176u
3
+ 242u
2
+ 136u + 24
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 3.21345 0.77320I
a = 0.302137 + 0.219467I
b = 1.140593 + 0.471635I
5.10856 + 5.32947I 7.48262 4.54389I
u = 3.21345 + 0.77320I
a = 0.302137 0.219467I
b = 1.140593 0.471635I
5.10856 5.32947I 7.48262 + 4.54389I
u = 1.51064 0.29222I
a = 0.469620 0.503633I
b = 0.856601 0.623578I
2.72382 + 4.89103I 0.12173 6.59162I
u = 1.51064 + 0.29222I
a = 0.469620 + 0.503633I
b = 0.856601 + 0.623578I
2.72382 4.89103I 0.12173 + 6.59162I
u = 0.275905 0.034779I
a = 1.33252 + 2.40533I
b = 0.283992 + 0.709987I
1.56227 + 1.71504I 1.36090 1.32670I
u = 0.275905 + 0.034779I
a = 1.33252 2.40533I
b = 0.283992 0.709987I
1.56227 1.71504I 1.36090 + 1.32670I
7
IV.
I
u
4
= hb
16
b
15
+ · · · + 2b 1, b
15
b
14
+ · · · + b + a, b
15
+ u + · · · + b 2i
(i) Arc colorings
a
3
=
1
0
a
8
=
b
15
+ b
14
+ ··· + 2b
3
b
b
a
9
=
b
15
+ b
14
+ ··· + 2b
3
2b
b
a
2
=
2b
15
+ 9b
13
+ ··· b
2
+ 2b
b
2
a
1
=
b
15
+ b
14
+ ··· + 2b 1
b
4
a
5
=
0
b
15
5b
13
+ ··· b + 2
a
7
=
b
15
+ b
14
+ ··· + 2b
3
b
2b
15
+ b
14
+ ··· + b 2
a
4
=
b
15
b
14
+ ··· 2b + 3
b
4
a
6
=
2b
15
+ 9b
13
+ ··· + 2b 2
2b
15
8b
13
+ ··· b + 2
a
6
=
2b
15
+ 9b
13
+ ··· + 2b 2
2b
15
8b
13
+ ··· b + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b
12
12b
10
+ 4b
9
+ 16b
8
8b
7
4b
6
+ 8b
5
4b
4
+ 4b
2
2
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 2.54709 0.16160I
a = 0.454208 + 0.137937I
b = 1.242705 0.322774I
5.44928 + 2.57849I 7.72292 3.56796I
u = 2.54709 + 0 .16160I
a = 0.454208 0.137937I
b = 1.242705 + 0.322774I
5.44928 2.57849I 7.72292 + 3.56796I
u = 1.78504
a = 0.383201
b = 1.14767
2.44483 0.105536
u = 0.679466 + 0 .633953I
a = 0.376188 0.557047I
b = 1.082583 0.348383I
2.24921 1.13123I 4.58478 + 0.51079I
u = 0.679466 0.633953I
a = 0.376188 + 0.557047I
b = 1.082583 + 0.348383I
2.24921 + 1.13123I 4.58478 0.51079I
u = 0.679466 0.633953I
a = 1.107538 + 0.520621I
b = 0.097535 0.616980I
2.24921 + 1.13123I 4.58478 0.51079I
u = 0.679466 + 0 .633953I
a = 1.107538 0.520621I
b = 0.097535 + 0.616980I
2.24921 1.13123I 4.58478 + 0.51079I
u = 0.320754 + 0 .851242I
a = 0.982717 + 0.958095I
b = 0.203747 0.848147I
0.91019 6.44354I 2.57155 + 5.29417I
u = 0.320754 0.851242I
a = 0.982717 0.958095I
b = 0.203747 + 0.848147I
0.91019 + 6.44354I 2.57155 5.29417I
u = 1.78504
a = 0.642938
b = 0.684028
2.44483 0.105536
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.120353
a = 5.69573 5.31855I
b = 0.685501 0.640105I
3.21286 1.86404
u = 0.120353
a = 5.69573 + 5.31855I
b = 0.685501 + 0.640105I
3.21286 1.86404
u = 0.320754 0.851242I
a = 0.793509 0.538352I
b = 1.130780 0.529217I
0.91019 + 6.44354I 2.57155 5.29417I
u = 0.320754 + 0 .851242I
a = 0.793509 + 0.538352I
b = 1.130780 + 0.529217I
0.91019 6.44354I 2.57155 + 5.29417I
u = 2.54709 0.16160I
a = 0.477930 + 0.157044I
b = 1.134615 0.424735I
5.44928 + 2.57849I 7.72292 3.56796I
u = 2.54709 + 0 .16160I
a = 0.477930 0.157044I
b = 1.134615 + 0.424735I
5.44928 2.57849I 7.72292 + 3.56796I
10
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
u(u
4
+ 2u
2
3u + 1)(u
6
+ 3u
5
+ 5u
4
+ 7u
3
+ 9u
2
+ 8u + 4)
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
2
c
2
, c
7
(u + 1)(u
4
+ u
3
u
2
u + 1)(u
6
u
4
u
3
+ u
2
+ u + 1)
(u
16
+ u
15
+ ··· 2u 1)
c
3
, c
5
(u + 1)(u
4
+ 3u
3
+ 5u
2
+ 3u + 1)(u
6
+ 2u
5
+ 3u
4
+ u
3
+ u
2
u + 1)
(u
16
+ 9u
15
+ ··· 8u
2
+ 1)
c
4
, c
8
(u 1)(u
4
+ u
3
u
2
u + 1)(u
6
u
4
u
3
+ u
2
+ u + 1)
(u
16
+ u
15
+ ··· 2u 1)
c
6
, c
9
u(u
4
+ 2u
3
+ 2u
2
+ u + 1)(u
6
+ u
5
u
4
3u
3
u
2
+ 2u + 2)
(u
8
u
7
u
6
+ 2u
5
+ u
4
2u
3
+ 2u 1)
2
11
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
y(y
4
+ 4y
3
+ 6y
2
5y + 1)(y
6
+ y
5
+ y
4
+ y
3
+ 9y
2
+ 8y + 16)
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
17y
4
34y
3
22y
2
4y + 1)
2
c
2
, c
4
, c
7
c
8
(y 1)(y
4
3y
3
+ 5y
2
3y + 1)(y
6
2y
5
+ 3y
4
y
3
+ y
2
+ y + 1)
(y
16
9y
15
+ ··· 8y
2
+ 1)
c
3
, c
5
(y 1)(y
4
+ y
3
+ ··· + y + 1)(y
6
+ 2y
5
+ ··· + y + 1)
(y
16
5y
15
+ ··· 16y + 1)
c
6
, c
9
y(y
4
+ 2y
2
+ 3y + 1)(y
6
3y
5
+ 5y
4
7y
3
+ 9y
2
8y + 4)
(y
8
3y
7
+ 7y
6
10y
5
+ 11y
4
10y
3
+ 6y
2
4y + 1)
2
12