10
31
(K10a
69
)
A knot diagram
1
Linearized knot diagam
3 8 1 9 10 4 2 7 6 5
Solving Sequence
3,8
2 1 4 7 9 5 6 10
c
2
c
1
c
3
c
7
c
8
c
4
c
6
c
10
c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
28
− u
27
+ ··· − u
2
+ 1i
* 1 irreducible components of dim
C
= 0, with total 28 representations.
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
28
− u
27
+ · · · − u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
4
=
u
4
− u
2
+ 1
u
4
a
7
=
u
−u
3
+ u
a
9
=
−u
3
u
5
− u
3
+ u
a
5
=
u
12
− u
10
+ 3u
8
− 2u
6
+ 2u
4
− u
2
+ 1
−u
14
+ 2u
12
− 5u
10
+ 6u
8
− 6u
6
+ 4u
4
− u
2
a
6
=
u
11
− 2u
9
+ 4u
7
− 4u
5
+ 3u
3
u
11
− u
9
+ 2u
7
− u
5
− u
3
+ u
a
10
=
u
27
− 4u
25
+ ··· + 12u
7
− u
3
u
27
− 3u
25
+ ··· − u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
26
− 4u
25
− 12u
24
+ 16u
23
+ 44u
22
− 52u
21
− 88u
20
+ 116u
19
+
168u
18
− 204u
17
− 236u
16
+ 284u
15
+ 288u
14
− 312u
13
− 280u
12
+ 256u
11
+ 224u
10
−
152u
9
− 136u
8
+ 40u
7
+ 64u
6
+ 16u
5
− 16u
4
− 16u
3
+ 4u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
8
u
28
+ 7u
27
+ ··· + 2u + 1
c
2
, c
7
u
28
− u
27
+ ··· − u
2
+ 1
c
4
u
28
− u
27
+ ··· + 5u + 2
c
5
, c
9
, c
10
u
28
+ u
27
+ ··· + 2u + 1
c
6
u
28
+ 7u
27
+ ··· + 8u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
8
y
28
+ 29y
27
+ ··· + 14y + 1
c
2
, c
7
y
28
− 7y
27
+ ··· − 2y + 1
c
4
y
28
− 3y
27
+ ··· + 19y + 4
c
5
, c
9
, c
10
y
28
+ 25y
27
+ ··· − 2y + 1
c
6
y
28
+ y
27
+ ··· + 62y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.899770 + 0.359295I
−0.52966 − 3.76187I −0.54869 + 7.99757I
u = 0.899770 − 0.359295I
−0.52966 + 3.76187I −0.54869 − 7.99757I
u = 0.954301 + 0.165131I
−6.93655 + 1.29573I −8.16340 + 0.19021I
u = 0.954301 − 0.165131I
−6.93655 − 1.29573I −8.16340 − 0.19021I
u = −0.971170 + 0.356128I
−5.84563 + 6.87695I −5.38448 − 7.29150I
u = −0.971170 − 0.356128I
−5.84563 − 6.87695I −5.38448 + 7.29150I
u = −0.816311 + 0.219669I
−1.41378 + 0.68499I −4.66956 − 0.56233I
u = −0.816311 − 0.219669I
−1.41378 − 0.68499I −4.66956 + 0.56233I
u = −0.894569 + 0.739690I
−1.93517 + 2.81005I −2.61718 − 2.93426I
u = −0.894569 − 0.739690I
−1.93517 − 2.81005I −2.61718 + 2.93426I
u = −0.594944 + 0.540484I
−1.95488 + 1.97473I 0.55963 − 3.90307I
u = −0.594944 − 0.540484I
−1.95488 − 1.97473I 0.55963 + 3.90307I
u = 0.824272 + 0.873080I
2.07406 + 4.77850I 0.63399 − 2.38985I
u = 0.824272 − 0.873080I
2.07406 − 4.77850I 0.63399 + 2.38985I
u = −0.848977 + 0.862822I
7.13238 − 0.98573I 5.20004 + 1.21736I
u = −0.848977 − 0.862822I
7.13238 + 0.98573I 5.20004 − 1.21736I
u = 0.883885 + 0.841772I
4.95278 − 2.93440I 2.09657 + 3.53352I
u = 0.883885 − 0.841772I
4.95278 + 2.93440I 2.09657 − 3.53352I
u = 0.921489 + 0.824235I
4.83159 − 3.27187I 1.73251 + 1.59380I
u = 0.921489 − 0.824235I
4.83159 + 3.27187I 1.73251 − 1.59380I
u = −0.956709 + 0.821698I
6.79399 + 7.24627I 4.35343 − 6.30493I
u = −0.956709 − 0.821698I
6.79399 − 7.24627I 4.35343 + 6.30493I
u = 0.975960 + 0.814541I
1.59839 − 11.04430I −0.28365 + 7.20583I
u = 0.975960 − 0.814541I
1.59839 + 11.04430I −0.28365 − 7.20583I
u = −0.190095 + 0.611771I
−3.43315 − 3.38176I 0.34958 + 2.75424I
u = −0.190095 − 0.611771I
−3.43315 + 3.38176I 0.34958 − 2.75424I
u = 0.313097 + 0.488114I
1.245360 + 0.507461I 6.74123 − 1.23953I
u = 0.313097 − 0.488114I
1.245360 − 0.507461I 6.74123 + 1.23953I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
8
u
28
+ 7u
27
+ ··· + 2u + 1
c
2
, c
7
u
28
− u
27
+ ··· − u
2
+ 1
c
4
u
28
− u
27
+ ··· + 5u + 2
c
5
, c
9
, c
10
u
28
+ u
27
+ ··· + 2u + 1
c
6
u
28
+ 7u
27
+ ··· + 8u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
8
y
28
+ 29y
27
+ ··· + 14y + 1
c
2
, c
7
y
28
− 7y
27
+ ··· − 2y + 1
c
4
y
28
− 3y
27
+ ··· + 19y + 4
c
5
, c
9
, c
10
y
28
+ 25y
27
+ ··· − 2y + 1
c
6
y
28
+ y
27
+ ··· + 62y + 1
7
