12n
0706
(K12n
0706
)
A knot diagram
1
Linearized knot diagam
4 6 11 9 1 11 4 6 1 8 2 9
Solving Sequence
1,4 2,9
5 6 10 8 12 11 7 3
c
1
c
4
c
5
c
9
c
8
c
12
c
11
c
6
c
3
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
3
+ 2u
2
+ 2b + u, −u
2
+ 2a − 2u − 1, u
4
+ 2u
3
+ 2u
2
− 2u + 1i
I
u
2
= hu
5
− u
4
− 2u
2
+ 4b + 3u − 1, u
5
+ u
4
− 2u
3
+ 12a − 3u + 7, u
6
− 2u
5
+ u
4
+ 3u
2
− 2u + 3i
I
u
3
= hu
5
− 2u
4
− u
3
+ 5u
2
+ 2b + 3u − 4, −2u
5
+ 2u
4
+ 3u
3
− 7u
2
+ 6a − 6u + 3, u
6
− 2u
5
− u
4
+ 6u
3
− 6u + 3i
I
u
4
= h2u
5
− u
4
− 5u
3
+ 6u
2
+ 6b + 9u − 6, −4u
5
+ 5u
4
+ 10u
3
− 21u
2
+ 6a − 15u + 15,
u
6
− 2u
5
− u
4
+ 6u
3
− 6u + 3i
I
u
5
= h−99u
5
− 258u
4
− 363u
3
− 295u
2
+ 82b − 59u − 72,
− 81u
5
− 144u
4
− 174u
3
− 96u
2
+ 82a − 11u − 85, 9u
6
+ 27u
5
+ 48u
4
+ 51u
3
+ 34u
2
+ 16u + 8i
I
u
6
= hb − a, a
2
+ a − 1, u + 1i
I
u
7
= h−a
2
u
2
+ au + b − a − u + 1, u
3
a
2
+ au + u
2
− u + 1i
* 6 irreducible components of dim
C
= 0, with total 30 representations.
* 1 irreducible components of dim
C
= 1
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= hu
3
+ 2u
2
+ 2b + u, −u
2
+ 2a − 2u − 1, u
4
+ 2u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
9
=
1
2
u
2
+ u +
1
2
−
1
2
u
3
− u
2
−
1
2
u
a
5
=
−
1
2
u
2
− u +
1
2
1
2
u
3
+ u
2
+
1
2
u
a
6
=
−
1
2
u
3
−
3
2
u
2
−
3
2
u +
1
2
1
2
u
3
+ u
2
+
1
2
u
a
10
=
1
2
u
3
+
3
2
u
2
+
3
2
u +
1
2
−
1
2
u
3
− u
2
−
1
2
u
a
8
=
1
2
u
3
+ u
2
+
3
2
u
−
1
2
u
3
−
1
2
u
2
−
3
2
u +
1
2
a
12
=
1
2
u
2
+ u +
1
2
−
1
2
u
3
− u
2
+
1
2
u
a
11
=
1
2
u
3
+ u
2
+
3
2
u
1
2
u
2
− u +
1
2
a
7
=
−
1
2
u
3
− u
2
−
3
2
u
−
1
2
u
2
+ 2u −
1
2
a
3
=
−
1
2
u
2
+
1
2
1
2
u
3
+
3
2
u
2
−
1
2
u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
3
+ 9u
2
+ 9u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
4
− 2u
3
+ 2u
2
+ 2u + 1
c
2
, c
3
, c
6
c
7
u
4
− 4u
3
+ 5u
2
− 2u + 1
c
4
, c
5
, c
9
c
12
u
4
+ 4u
3
+ 5u
2
+ 2u + 1
c
10
, c
11
u
4
+ 2u
3
+ 2u
2
− 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
10
c
11
y
4
+ 14y
2
+ 1
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
12
y
4
− 6y
3
+ 11y
2
+ 6y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.366025 + 0.366025I
a = 0.866025 + 0.500000I
b = −0.133975 − 0.500000I
− 1.23808I 0. + 6.00000I
u = 0.366025 − 0.366025I
a = 0.866025 − 0.500000I
b = −0.133975 + 0.500000I
1.23808I 0. − 6.00000I
u = −1.36603 + 1.36603I
a = −0.866025 − 0.500000I
b = −1.86603 + 0.50000I
13.4174I 0. − 6.00000I
u = −1.36603 − 1.36603I
a = −0.866025 + 0.500000I
b = −1.86603 − 0.50000I
− 13.4174I 0. + 6.00000I
5
II. I
u
2
= hu
5
− u
4
− 2u
2
+ 4b + 3u − 1, u
5
+ u
4
− 2u
3
+ 12a − 3u + 7, u
6
−
2u
5
+ u
4
+ 3u
2
− 2u + 3i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
9
=
−
1
12
u
5
−
1
12
u
4
+ ··· +
1
4
u −
7
12
−
1
4
u
5
+
1
4
u
4
+ ··· −
3
4
u +
1
4
a
5
=
1
12
u
5
+
1
12
u
4
+ ··· −
1
4
u −
5
12
1
4
u
5
−
1
4
u
4
+ ··· +
3
4
u −
1
4
a
6
=
−
1
6
u
5
+
1
3
u
4
+ ··· − u −
1
6
1
4
u
5
−
1
4
u
4
+ ··· +
3
4
u −
1
4
a
10
=
1
6
u
5
−
1
3
u
4
+ ··· + u −
5
6
−
1
4
u
5
+
1
4
u
4
+ ··· −
3
4
u +
1
4
a
8
=
−
1
12
u
5
−
1
12
u
4
+ ··· −
1
4
u −
1
12
−
1
2
u
3
+
1
2
u
2
−
1
2
u −
1
2
a
12
=
1
12
u
5
+
1
12
u
4
+ ··· −
1
4
u +
7
12
1
4
u
5
−
1
4
u
4
+ ··· −
1
4
u −
1
4
a
11
=
1
12
u
5
+
1
12
u
4
+ ··· +
1
4
u +
1
12
−
1
4
u
5
+
1
4
u
4
+ ··· −
1
4
u −
1
4
a
7
=
−
1
12
u
5
−
1
12
u
4
+ ··· −
1
4
u −
1
12
1
4
u
5
−
1
4
u
4
+ ··· −
3
4
u +
1
4
a
3
=
1
12
u
5
−
5
12
u
4
+ ··· +
3
4
u +
1
12
−
1
2
u
3
+
3
2
u
2
−
1
2
u +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
5
−
9
2
u
4
+ 3u
2
+ 6u −
3
2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
6
− 2u
5
+ u
4
+ 3u
2
− 2u + 3
c
2
, c
5
, c
6
c
12
u
6
− 3u
5
+ 2u
4
− u
3
+ 2u
2
+ u + 1
c
3
, c
4
, c
7
c
9
u
6
+ 3u
5
+ 2u
4
+ u
3
+ 2u
2
− u + 1
c
8
, c
11
u
6
+ 2u
5
+ u
4
+ 3u
2
+ 2u + 3
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
10
c
11
y
6
− 2y
5
+ 7y
4
+ 4y
3
+ 15y
2
+ 14y + 9
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
12
y
6
− 5y
5
+ 2y
4
+ 15y
3
+ 10y
2
+ 3y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.319448 + 0.816851I
a = −0.649948 + 0.216712I
b = −0.384646 − 0.461682I
− 3.01792I 0. + 8.67149I
u = 0.319448 − 0.816851I
a = −0.649948 − 0.216712I
b = −0.384646 + 0.461682I
3.01792I 0. − 8.67149I
u = −0.814644 + 0.831311I
a = −0.562136 + 0.513813I
b = 0.030802 − 0.885884I
9.18468 + 5.87764I 6.07806 − 4.16480I
u = −0.814644 − 0.831311I
a = −0.562136 − 0.513813I
b = 0.030802 + 0.885884I
9.18468 − 5.87764I 6.07806 + 4.16480I
u = 1.49520 + 0.80186I
a = 1.045420 − 0.362585I
b = 1.85384 + 0.29614I
−9.18468 − 5.87764I −6.07806 + 4.16480I
u = 1.49520 − 0.80186I
a = 1.045420 + 0.362585I
b = 1.85384 − 0.29614I
−9.18468 + 5.87764I −6.07806 − 4.16480I
9
III. I
u
3
= hu
5
− 2u
4
− u
3
+ 5u
2
+ 2b + 3u − 4, −2u
5
+ 2u
4
+ 3u
3
− 7u
2
+ 6a −
6u + 3, u
6
− 2u
5
− u
4
+ 6u
3
− 6u + 3i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
9
=
1
3
u
5
−
1
3
u
4
+ ··· + u −
1
2
−
1
2
u
5
+ u
4
+ ··· −
3
2
u + 2
a
5
=
−
1
6
u
5
+
1
3
u
4
+ ··· −
5
6
u + 1
1
3
u
5
−
1
6
u
4
+ ··· +
3
2
u − 1
a
6
=
−
1
2
u
5
+
1
2
u
4
+ ··· −
7
3
u + 2
1
3
u
5
−
1
6
u
4
+ ··· +
3
2
u − 1
a
10
=
5
6
u
5
−
4
3
u
4
+ ··· +
5
2
u −
5
2
−
1
2
u
5
+ u
4
+ ··· −
3
2
u + 2
a
8
=
5
6
u
5
−
7
6
u
4
+ ··· + 2u − 2
−u
5
+
3
2
u
4
+
3
2
u
3
− 4u
2
−
5
2
u + 3
a
12
=
1
3
u
5
−
1
3
u
4
+ ··· + u −
1
2
1
2
u
5
−
1
2
u
4
−
1
2
u
3
+ u
2
+
1
2
a
11
=
1
3
u
5
−
1
3
u
4
− u
3
+
5
3
u
2
+ 2u − 2
1
2
u
3
−
1
2
u
2
+
1
2
a
7
=
−
5
6
u
5
+
7
6
u
4
+ ··· − 2u + 2
1
2
u
5
−
1
2
u
4
+ ··· + 2u −
3
2
a
3
=
1
3
u
5
−
2
3
u
4
+
4
3
u
2
−
1
3
u
−
1
6
u
5
+
1
3
u
4
+ ··· +
1
2
u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
10
3
u
5
− 4u
4
−
22
3
u
3
+
50
3
u
2
+
38
3
u − 14
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
6
+ 2u
5
− u
4
− 6u
3
+ 6u + 3
c
2
, c
7
u
6
− 3u
5
+ 4u
4
− 9u
3
+ 12u
2
− 4u + 8
c
3
, c
6
3(3u
6
+ 12u
5
+ 15u
4
+ 6u
3
+ 2u
2
+ 2u + 1)
c
4
, c
5
3(3u
6
− 12u
5
+ 15u
4
− 6u
3
+ 2u
2
− 2u + 1)
c
9
, c
12
u
6
+ 3u
5
+ 4u
4
+ 9u
3
+ 12u
2
+ 4u + 8
c
10
, c
11
u
6
− 2u
5
− u
4
+ 6u
3
− 6u + 3
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
10
c
11
y
6
− 6y
5
+ 25y
4
− 54y
3
+ 66y
2
− 36y + 9
c
2
, c
7
, c
9
c
12
y
6
− y
5
− 14y
4
+ 7y
3
+ 136y
2
+ 176y + 64
c
3
, c
4
, c
5
c
6
9(9y
6
− 54y
5
+ 93y
4
− 18y
3
+ 10y
2
+ 1)
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.696323 + 0.248902I
a = 0.555352 + 0.455182I
b = 0.077086 − 0.882809I
− 1.15875I 0. + 5.94444I
u = 0.696323 − 0.248902I
a = 0.555352 − 0.455182I
b = 0.077086 + 0.882809I
1.15875I 0. − 5.94444I
u = −1.213080 + 0.431565I
a = 0.317354 + 0.363091I
b = 0.36468 − 1.56135I
7.57044 + 5.49399I −0.42147 − 2.91709I
u = −1.213080 − 0.431565I
a = 0.317354 − 0.363091I
b = 0.36468 + 1.56135I
7.57044 − 5.49399I −0.42147 + 2.91709I
u = 1.51676 + 1.00438I
a = −0.872706 + 0.406269I
b = −1.94177 − 0.43842I
−7.57044 − 5.49399I 0.42147 + 2.91709I
u = 1.51676 − 1.00438I
a = −0.872706 − 0.406269I
b = −1.94177 + 0.43842I
−7.57044 + 5.49399I 0.42147 − 2.91709I
13
IV. I
u
4
= h2u
5
− u
4
− 5u
3
+ 6u
2
+ 6b + 9u − 6, −4u
5
+ 5u
4
+ 10u
3
− 21u
2
+
6a − 15u + 15, u
6
− 2u
5
− u
4
+ 6u
3
− 6u + 3i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
9
=
2
3
u
5
−
5
6
u
4
+ ··· +
5
2
u −
5
2
−
1
3
u
5
+
1
6
u
4
+ ··· −
3
2
u + 1
a
5
=
−
1
6
u
5
+
5
6
u
4
+ ··· + u + 1
1
3
u
5
−
1
6
u
4
+ ··· +
3
2
u − 1
a
6
=
−
1
2
u
5
+ u
4
+ ··· −
1
2
u + 2
1
3
u
5
−
1
6
u
4
+ ··· +
3
2
u − 1
a
10
=
u
5
− u
4
−
5
2
u
3
+
9
2
u
2
+ 4u −
7
2
−
1
3
u
5
+
1
6
u
4
+ ··· −
3
2
u + 1
a
8
=
1
3
u
5
−
1
3
u
4
− u
3
+
5
3
u
2
+ 2u − 2
−
1
3
u
5
+
1
3
u
4
+ ··· −
4
3
u +
3
2
a
12
=
1
3
u
5
−
1
3
u
4
+ ··· + u −
1
2
1
3
u
4
−
1
6
u
3
−
5
6
u
2
+
1
2
a
11
=
5
6
u
5
−
7
6
u
4
+ ··· + 2u − 2
−
1
6
u
4
−
1
6
u
3
+
2
3
u
2
−
1
2
u
a
7
=
−
1
3
u
5
+
1
3
u
4
+ u
3
−
5
3
u
2
− 2u + 2
1
3
u
5
−
1
3
u
4
+ ··· +
1
3
u −
1
2
a
3
=
−
1
6
u
5
−
1
6
u
4
+ ··· − 2u +
5
2
−
1
6
u
4
+
1
3
u
3
+ ··· −
1
2
u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
10
3
u
5
− 4u
4
−
22
3
u
3
+
50
3
u
2
+
38
3
u − 14
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 2u
5
− u
4
− 6u
3
+ 6u + 3
c
2
, c
6
, c
7
3(3u
6
+ 12u
5
+ 15u
4
+ 6u
3
+ 2u
2
+ 2u + 1)
c
3
u
6
− 3u
5
+ 4u
4
− 9u
3
+ 12u
2
− 4u + 8
c
4
u
6
+ 3u
5
+ 4u
4
+ 9u
3
+ 12u
2
+ 4u + 8
c
5
, c
9
, c
12
3(3u
6
− 12u
5
+ 15u
4
− 6u
3
+ 2u
2
− 2u + 1)
c
8
9(9u
6
− 27u
5
+ 48u
4
− 51u
3
+ 34u
2
− 16u + 8)
c
10
u
6
− 2u
5
− u
4
+ 6u
3
− 6u + 3
c
11
9(9u
6
+ 27u
5
+ 48u
4
+ 51u
3
+ 34u
2
+ 16u + 8)
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
6
− 6y
5
+ 25y
4
− 54y
3
+ 66y
2
− 36y + 9
c
2
, c
5
, c
6
c
7
, c
9
, c
12
9(9y
6
− 54y
5
+ 93y
4
− 18y
3
+ 10y
2
+ 1)
c
3
, c
4
y
6
− y
5
− 14y
4
+ 7y
3
+ 136y
2
+ 176y + 64
c
8
, c
11
81(81y
6
+ 135y
5
+ 162y
4
− 57y
3
+ 292y
2
+ 288y + 64)
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.696323 + 0.248902I
a = 0.303677 + 1.159270I
b = −0.273409 − 0.455182I
− 1.15875I 0. + 5.94444I
u = 0.696323 − 0.248902I
a = 0.303677 − 1.159270I
b = −0.273409 + 0.455182I
1.15875I 0. − 5.94444I
u = −1.213080 + 0.431565I
a = 0.673303 − 1.047560I
b = 0.541674 + 0.303500I
7.57044 + 5.49399I −0.42147 − 2.91709I
u = −1.213080 − 0.431565I
a = 0.673303 + 1.047560I
b = 0.541674 − 0.303500I
7.57044 − 5.49399I −0.42147 + 2.91709I
u = 1.51676 + 1.00438I
a = 1.023020 − 0.388387I
b = 1.73174 + 0.26032I
−7.57044 − 5.49399I 0.42147 + 2.91709I
u = 1.51676 − 1.00438I
a = 1.023020 + 0.388387I
b = 1.73174 − 0.26032I
−7.57044 + 5.49399I 0.42147 − 2.91709I
17
V. I
u
5
= h−99u
5
− 258u
4
+ · · · + 82b − 72, −81u
5
− 144u
4
+ · · · + 82a −
85, 9u
6
+ 27u
5
+ 48u
4
+ 51u
3
+ 34u
2
+ 16u + 8i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
9
=
0.987805u
5
+ 1.75610u
4
+ ··· + 0.134146u + 1.03659
1.20732u
5
+ 3.14634u
4
+ ··· + 0.719512u + 0.878049
a
5
=
2.41463u
5
+ 5.54268u
4
+ ··· + 2.18902u + 1.75610
1.70122u
5
+ 4.77439u
4
+ ··· + 3.53659u + 2.14634
a
6
=
0.713415u
5
+ 0.768293u
4
+ ··· − 1.34756u − 0.390244
1.70122u
5
+ 4.77439u
4
+ ··· + 3.53659u + 2.14634
a
10
=
−0.219512u
5
− 1.39024u
4
+ ··· − 0.585366u + 0.158537
1.20732u
5
+ 3.14634u
4
+ ··· + 0.719512u + 0.878049
a
8
=
−1.04268u
5
− 2.85366u
4
+ ··· − 0.530488u − 0.121951
1.26220u
5
+ 1.99390u
4
+ ··· − 0.634146u − 0.536585
a
12
=
−2.41463u
5
− 5.54268u
4
+ ··· − 2.18902u − 0.756098
−1.70122u
5
− 4.77439u
4
+ ··· − 2.53659u − 2.14634
a
11
=
−1.04268u
5
− 2.85366u
4
+ ··· − 0.530488u − 0.121951
−0.457317u
5
− 0.896341u
4
+ ··· − 1.21951u − 0.878049
a
7
=
1.04268u
5
+ 2.85366u
4
+ ··· + 0.530488u + 0.121951
−2.17683u
5
− 3.78659u
4
+ ··· + 0.195122u + 0.780488
a
3
=
0.466463u
5
+ 1.45427u
4
+ ··· + 0.243902u + 0.475610
−1.26220u
5
− 3.49390u
4
+ ··· − 0.865854u − 0.463415
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
69
41
u
5
−
27
41
u
4
+
116
41
u
3
+
433
41
u
2
+
390
41
u +
166
41
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
9(9u
6
− 27u
5
+ 48u
4
− 51u
3
+ 34u
2
− 16u + 8)
c
2
, c
3
, c
7
3(3u
6
+ 12u
5
+ 15u
4
+ 6u
3
+ 2u
2
+ 2u + 1)
c
4
, c
9
, c
12
3(3u
6
− 12u
5
+ 15u
4
− 6u
3
+ 2u
2
− 2u + 1)
c
5
u
6
+ 3u
5
+ 4u
4
+ 9u
3
+ 12u
2
+ 4u + 8
c
6
u
6
− 3u
5
+ 4u
4
− 9u
3
+ 12u
2
− 4u + 8
c
8
u
6
+ 2u
5
− u
4
− 6u
3
+ 6u + 3
c
10
9(9u
6
+ 27u
5
+ 48u
4
+ 51u
3
+ 34u
2
+ 16u + 8)
c
11
u
6
− 2u
5
− u
4
+ 6u
3
− 6u + 3
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
81(81y
6
+ 135y
5
+ 162y
4
− 57y
3
+ 292y
2
+ 288y + 64)
c
2
, c
3
, c
4
c
7
, c
9
, c
12
9(9y
6
− 54y
5
+ 93y
4
− 18y
3
+ 10y
2
+ 1)
c
5
, c
6
y
6
− y
5
− 14y
4
+ 7y
3
+ 136y
2
+ 176y + 64
c
8
, c
11
y
6
− 6y
5
+ 25y
4
− 54y
3
+ 66y
2
− 36y + 9
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.989374 + 0.463198I
a = 1.53670 + 0.45632I
b = 1.73174 − 0.26032I
−7.57044 + 5.49399I 0.42147 − 2.91709I
u = −0.989374 − 0.463198I
a = 1.53670 − 0.45632I
b = 1.73174 + 0.26032I
−7.57044 − 5.49399I 0.42147 + 2.91709I
u = −0.565978 + 1.232560I
a = −0.036697 + 0.456322I
b = 0.541674 + 0.303500I
7.57044 + 5.49399I −0.42147 − 2.91709I
u = −0.565978 − 1.232560I
a = −0.036697 − 0.456322I
b = 0.541674 − 0.303500I
7.57044 − 5.49399I −0.42147 + 2.91709I
u = 0.055352 + 0.633907I
a = 0.750000 − 0.365819I
b = −0.273409 − 0.455182I
− 1.15875I 0. + 5.94444I
u = 0.055352 − 0.633907I
a = 0.750000 + 0.365819I
b = −0.273409 + 0.455182I
1.15875I 0. − 5.94444I
21
VI. I
u
6
= hb − a, a
2
+ a − 1, u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
−1
a
2
=
1
1
a
9
=
a
a
a
5
=
−a + 1
−a
a
6
=
1
−a
a
10
=
0
a
a
8
=
a + 1
0
a
12
=
a
a − 1
a
11
=
a + 1
a
a
7
=
−a − 1
−a − 1
a
3
=
a + 2
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
(u − 1)
2
c
2
, c
3
, c
6
c
7
u
2
− u − 1
c
4
, c
5
, c
9
c
12
u
2
+ u − 1
c
10
, c
11
(u + 1)
2
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
10
c
11
(y −1)
2
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
12
y
2
− 3y + 1
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.618034
b = 0.618034
3.94784 0
u = −1.00000
a = −1.61803
b = −1.61803
−3.94784 0
25
VII. I
u
7
= h−a
2
u
2
+ au + b − a − u + 1, u
3
a
2
+ au + u
2
− u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
9
=
a
a
2
u
2
− au + a + u − 1
a
5
=
−a
2
u
a
2
u
2
+ a + u
a
6
=
−a
2
u
2
− a
2
u − a − u
a
2
u
2
+ a + u
a
10
=
−a
2
u
2
+ au − u + 1
a
2
u
2
− au + a + u − 1
a
8
=
a
3
u
2
− a
2
u
2
− a
2
u + a
2
+ au − a − u
−a
4
u
2
− a
3
u
2
+ a
3
u + 2a
2
u
2
− a
3
− a
2
− 2au + 3a + 2u − 1
a
12
=
−a
3
u
2
+ a
2
u − a
2
− au + a + 1
−a
3
u
2
+ a
2
u − a
2
− au + u
a
11
=
a
−a
3
u
2
+ a
2
u + u
2
a − a
2
− au + 1
a
7
=
a
3
u
2
− a
2
u
2
− a
2
u + a
2
+ au − a − u
−a
4
u
2
− a
3
u
2
+ a
3
u + 2a
2
u
2
− a
3
+ u
2
a − a
2
− 2au + 3a + 2u
a
3
=
−a
2
u
−a
3
u
2
+ a
2
u − a
2
+ u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
26
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
27
VIII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
9(u − 1)
2
(u
4
− 2u
3
+ 2u
2
+ 2u + 1)(u
6
− 2u
5
+ u
4
+ 3u
2
− 2u + 3)
· (u
6
+ 2u
5
− u
4
− 6u
3
+ 6u + 3)
2
· (9u
6
− 27u
5
+ 48u
4
− 51u
3
+ 34u
2
− 16u + 8)
c
2
, c
6
9(u
2
− u − 1)(u
4
− 4u
3
+ ··· − 2u + 1)(u
6
− 3u
5
+ ··· + u + 1)
· (u
6
− 3u
5
+ 4u
4
− 9u
3
+ 12u
2
− 4u + 8)
· (3u
6
+ 12u
5
+ 15u
4
+ 6u
3
+ 2u
2
+ 2u + 1)
2
c
3
, c
7
9(u
2
− u − 1)(u
4
− 4u
3
+ 5u
2
− 2u + 1)
· (u
6
− 3u
5
+ 4u
4
− 9u
3
+ 12u
2
− 4u + 8)
· (u
6
+ 3u
5
+ 2u
4
+ u
3
+ 2u
2
− u + 1)
· (3u
6
+ 12u
5
+ 15u
4
+ 6u
3
+ 2u
2
+ 2u + 1)
2
c
4
, c
9
9(u
2
+ u − 1)(u
4
+ 4u
3
+ ··· + 2u + 1)(u
6
+ 3u
5
+ ··· − u + 1)
· (u
6
+ 3u
5
+ 4u
4
+ 9u
3
+ 12u
2
+ 4u + 8)
· (3u
6
− 12u
5
+ 15u
4
− 6u
3
+ 2u
2
− 2u + 1)
2
c
5
, c
12
9(u
2
+ u − 1)(u
4
+ 4u
3
+ ··· + 2u + 1)(u
6
− 3u
5
+ ··· + u + 1)
· (u
6
+ 3u
5
+ 4u
4
+ 9u
3
+ 12u
2
+ 4u + 8)
· (3u
6
− 12u
5
+ 15u
4
− 6u
3
+ 2u
2
− 2u + 1)
2
c
8
9(u − 1)
2
(u
4
− 2u
3
+ 2u
2
+ 2u + 1)(u
6
+ 2u
5
− u
4
− 6u
3
+ 6u + 3)
2
· (u
6
+ 2u
5
+ u
4
+ 3u
2
+ 2u + 3)
· (9u
6
− 27u
5
+ 48u
4
− 51u
3
+ 34u
2
− 16u + 8)
c
10
9(u + 1)
2
(u
4
+ 2u
3
+ 2u
2
− 2u + 1)(u
6
− 2u
5
− u
4
+ 6u
3
− 6u + 3)
2
· (u
6
− 2u
5
+ u
4
+ 3u
2
− 2u + 3)
· (9u
6
+ 27u
5
+ 48u
4
+ 51u
3
+ 34u
2
+ 16u + 8)
c
11
9(u + 1)
2
(u
4
+ 2u
3
+ 2u
2
− 2u + 1)(u
6
− 2u
5
− u
4
+ 6u
3
− 6u + 3)
2
· (u
6
+ 2u
5
+ u
4
+ 3u
2
+ 2u + 3)
· (9u
6
+ 27u
5
+ 48u
4
+ 51u
3
+ 34u
2
+ 16u + 8)
28
IX. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
10
c
11
81(y −1)
2
(y
4
+ 14y
2
+ 1)(y
6
− 6y
5
+ ··· − 36y + 9)
2
· (y
6
− 2y
5
+ 7y
4
+ 4y
3
+ 15y
2
+ 14y + 9)
· (81y
6
+ 135y
5
+ 162y
4
− 57y
3
+ 292y
2
+ 288y + 64)
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
12
81(y
2
− 3y + 1)(y
4
− 6y
3
+ 11y
2
+ 6y + 1)
· (y
6
− 5y
5
+ 2y
4
+ 15y
3
+ 10y
2
+ 3y + 1)
· (y
6
− y
5
− 14y
4
+ 7y
3
+ 136y
2
+ 176y + 64)
· (9y
6
− 54y
5
+ 93y
4
− 18y
3
+ 10y
2
+ 1)
2
29
