9
24
(K9a
7
)
A knot diagram
1
Linearized knot diagam
7 4 1 8 9 2 6 5 3
Solving Sequence
1,7 2,4
3 6 8 9 5
c
1
c
3
c
6
c
7
c
9
c
5
c
2
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
16
+ 2u
15
+ ··· + 4b − 2, −2u
16
− 3u
15
+ ··· + 4a − 2, u
17
+ 2u
16
+ ··· − 2u − 2i
I
u
2
= ha
2
u − a
2
+ b + 2a − 2, a
3
− 2a
2
u + 3au − u, u
2
− u + 1i
I
v
1
= ha, b − 1, v −1i
* 3 irreducible components of dim
C
= 0, with total 24 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
=
h2u
16
+2u
15
+· · ·+4b−2, −2u
16
−3u
15
+· · ·+4a−2, u
17
+2u
16
+· · ·−2u−2i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
u
2
a
4
=
1
2
u
16
+
3
4
u
15
+ ··· −
1
2
u +
1
2
−
1
2
u
16
−
1
2
u
15
+ ··· + u +
1
2
a
3
=
1
4
u
15
+
1
2
u
13
+ ··· +
1
2
u + 1
−
1
2
u
16
−
1
2
u
15
+ ··· + u +
1
2
a
6
=
u
u
3
+ u
a
8
=
u
3
u
5
+ u
3
+ u
a
9
=
1
2
u
16
+ u
15
+ ··· − u −
1
2
1
4
u
15
+
1
4
u
14
+ ··· −
1
2
u − 1
a
5
=
−
1
4
u
12
−
1
2
u
10
+ ··· +
1
2
u +
1
2
1
2
u
7
+
1
2
u
5
+
3
2
u
3
+
1
2
u
2
+ u
a
5
=
−
1
4
u
12
−
1
2
u
10
+ ··· +
1
2
u +
1
2
1
2
u
7
+
1
2
u
5
+
3
2
u
3
+
1
2
u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
16
+ 4u
15
+ 6u
14
+ 8u
13
+ 8u
12
+ 14u
11
+ 10u
10
+ 12u
9
+ 4u
8
+
10u
7
+ 20u
6
+ 26u
5
+ 16u
4
− 4u
3
− 10u
2
− 8u − 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
17
− 2u
16
+ ··· − 2u + 2
c
2
u
17
+ 8u
16
+ ··· + 3u + 1
c
3
, c
9
u
17
− 2u
16
+ ··· − u + 1
c
4
, c
5
, c
8
u
17
+ 2u
16
+ ··· + 3u + 1
c
7
u
17
− 6u
16
+ ··· + 8u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
17
+ 6y
16
+ ··· + 8y −4
c
2
y
17
+ 4y
16
+ ··· − 13y −1
c
3
, c
9
y
17
− 8y
16
+ ··· + 3y −1
c
4
, c
5
, c
8
y
17
− 16y
16
+ ··· + 19y −1
c
7
y
17
+ 6y
16
+ ··· + 376y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.742615 + 0.650908I
a = 0.456798 − 0.077068I
b = 1.128570 + 0.359117I
−3.65923 − 1.22724I −6.14847 + 0.85505I
u = −0.742615 − 0.650908I
a = 0.456798 + 0.077068I
b = 1.128570 − 0.359117I
−3.65923 + 1.22724I −6.14847 − 0.85505I
u = −0.834865 + 0.265014I
a = 0.636187 + 0.240948I
b = 0.374678 − 0.520641I
2.61956 − 0.43387I 2.56834 − 0.87540I
u = −0.834865 − 0.265014I
a = 0.636187 − 0.240948I
b = 0.374678 + 0.520641I
2.61956 + 0.43387I 2.56834 + 0.87540I
u = 0.976738 + 0.562668I
a = 0.456039 + 0.109653I
b = 1.072950 − 0.498433I
0.61043 + 4.64771I −0.43915 − 4.11695I
u = 0.976738 − 0.562668I
a = 0.456039 − 0.109653I
b = 1.072950 + 0.498433I
0.61043 − 4.64771I −0.43915 + 4.11695I
u = −0.003992 + 0.842342I
a = 1.18580 + 1.31498I
b = −0.621791 − 0.419413I
1.30982 − 1.46955I 3.63583 + 4.66528I
u = −0.003992 − 0.842342I
a = 1.18580 − 1.31498I
b = −0.621791 + 0.419413I
1.30982 + 1.46955I 3.63583 − 4.66528I
u = −0.656745 + 1.004700I
a = −0.46618 − 1.83030I
b = −1.130680 + 0.513073I
−2.57978 + 6.57063I −3.26005 − 6.43452I
u = −0.656745 − 1.004700I
a = −0.46618 + 1.83030I
b = −1.130680 − 0.513073I
−2.57978 − 6.57063I −3.26005 + 6.43452I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.110097 + 1.246510I
a = 0.360483 − 1.280850I
b = −0.796399 + 0.723427I
8.03468 + 2.71165I 5.84242 − 3.13710I
u = −0.110097 − 1.246510I
a = 0.360483 + 1.280850I
b = −0.796399 − 0.723427I
8.03468 − 2.71165I 5.84242 + 3.13710I
u = −0.578864 + 1.116300I
a = 0.568056 + 0.689908I
b = −0.288739 − 0.863831I
5.04981 + 5.51158I 4.25126 − 3.84490I
u = −0.578864 − 1.116300I
a = 0.568056 − 0.689908I
b = −0.288739 + 0.863831I
5.04981 − 5.51158I 4.25126 + 3.84490I
u = 0.718492 + 1.129370I
a = −0.46497 + 1.57649I
b = −1.172120 − 0.583556I
2.40324 − 10.83370I 0.89378 + 7.41261I
u = 0.718492 − 1.129370I
a = −0.46497 − 1.57649I
b = −1.172120 + 0.583556I
2.40324 + 10.83370I 0.89378 − 7.41261I
u = 0.463897
a = 0.535599
b = 0.867068
−1.25812 −8.68790
6
II. I
u
2
= ha
2
u − a
2
+ b + 2a − 2, a
3
− 2a
2
u + 3au − u, u
2
− u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
u − 1
a
4
=
a
−a
2
u + a
2
− 2a + 2
a
3
=
−a
2
u + a
2
− a + 2
−a
2
u + a
2
− 2a + 2
a
6
=
u
u − 1
a
8
=
−1
0
a
9
=
a
2
u − a
2
+ au + 2a − 2
a
2
u − a
2
+ au + a − 2
a
5
=
−a
2
u + a
2
− a + 2
−a
2
u + a
2
− 2a + 2
a
5
=
−a
2
u + a
2
− a + 2
−a
2
u + a
2
− 2a + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u − 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
2
+ u + 1)
3
c
2
u
6
+ 4u
5
+ 6u
4
+ 3u
3
− u
2
− u + 1
c
3
, c
4
, c
5
c
8
, c
9
u
6
− 2u
4
+ u
3
+ u
2
− u + 1
c
7
(u
2
− u + 1)
3
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
7
(y
2
+ y + 1)
3
c
2
y
6
− 4y
5
+ 10y
4
− 11y
3
+ 19y
2
− 3y + 1
c
3
, c
4
, c
5
c
8
, c
9
y
6
− 4y
5
+ 6y
4
− 3y
3
− y
2
+ y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.741145 − 0.632163I
b = −0.218964 + 0.666188I
− 2.02988I 0. + 3.46410I
u = 0.500000 + 0.866025I
a = 0.439111 + 0.046276I
b = 1.252310 − 0.237364I
− 2.02988I 0. + 3.46410I
u = 0.500000 + 0.866025I
a = −0.18026 + 2.31794I
b = −1.033350 − 0.428825I
− 2.02988I 0. + 3.46410I
u = 0.500000 − 0.866025I
a = 0.741145 + 0.632163I
b = −0.218964 − 0.666188I
2.02988I 0. − 3.46410I
u = 0.500000 − 0.866025I
a = 0.439111 − 0.046276I
b = 1.252310 + 0.237364I
2.02988I 0. − 3.46410I
u = 0.500000 − 0.866025I
a = −0.18026 − 2.31794I
b = −1.033350 + 0.428825I
2.02988I 0. − 3.46410I
10
III. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
1
0
a
2
=
1
0
a
4
=
0
1
a
3
=
1
1
a
6
=
1
0
a
8
=
1
0
a
9
=
0
−1
a
5
=
1
1
a
5
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
u
c
2
, c
8
, c
9
u − 1
c
3
, c
4
, c
5
u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
7
y
c
2
, c
3
, c
4
c
5
, c
8
, c
9
y − 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u(u
2
+ u + 1)
3
(u
17
− 2u
16
+ ··· − 2u + 2)
c
2
(u − 1)(u
6
+ 4u
5
+ ··· − u + 1)(u
17
+ 8u
16
+ ··· + 3u + 1)
c
3
(u + 1)(u
6
− 2u
4
+ ··· − u + 1)(u
17
− 2u
16
+ ··· − u + 1)
c
4
, c
5
(u + 1)(u
6
− 2u
4
+ ··· − u + 1)(u
17
+ 2u
16
+ ··· + 3u + 1)
c
7
u(u
2
− u + 1)
3
(u
17
− 6u
16
+ ··· + 8u + 4)
c
8
(u − 1)(u
6
− 2u
4
+ ··· − u + 1)(u
17
+ 2u
16
+ ··· + 3u + 1)
c
9
(u − 1)(u
6
− 2u
4
+ ··· − u + 1)(u
17
− 2u
16
+ ··· − u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
y(y
2
+ y + 1)
3
(y
17
+ 6y
16
+ ··· + 8y − 4)
c
2
(y − 1)(y
6
− 4y
5
+ 10y
4
− 11y
3
+ 19y
2
− 3y + 1)
· (y
17
+ 4y
16
+ ··· − 13y − 1)
c
3
, c
9
(y − 1)(y
6
− 4y
5
+ ··· + y + 1)(y
17
− 8y
16
+ ··· + 3y − 1)
c
4
, c
5
, c
8
(y − 1)(y
6
− 4y
5
+ ··· + y + 1)(y
17
− 16y
16
+ ··· + 19y − 1)
c
7
y(y
2
+ y + 1)
3
(y
17
+ 6y
16
+ ··· + 376y − 16)
16
