10
137
(K10n
2
)
A knot diagram
1
Linearized knot diagam
7 4 10 8 4 2 5 6 2 3
Solving Sequence
4,8
5
6,10
3 2 7 1 9
c
4
c
5
c
3
c
2
c
7
c
1
c
9
c
6
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
15
− 2u
14
+ ··· + 2b − 5u,
u
14
+ 2u
13
+ 7u
12
+ 10u
11
+ 18u
10
+ 23u
9
+ 25u
8
+ 32u
7
+ 22u
6
+ 25u
5
+ 14u
4
+ 6u
3
+ 10u
2
+ 2a − u + 3,
u
16
+ 3u
15
+ ··· + 4u + 1i
I
u
2
= hb + u − 1, a + u + 1, u
2
− u + 1i
I
u
3
= hb − u, a, u
2
− u + 1i
* 3 irreducible components of dim
C
= 0, with total 20 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
=
h−u
15
−2u
14
+· · ·+2b −5u, u
14
+2u
13
+· · ·+2a +3, u
16
+3u
15
+· · ·+4u +1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
5
=
1
u
2
a
6
=
u
2
+ 1
u
2
a
10
=
−
1
2
u
14
− u
13
+ ··· +
1
2
u −
3
2
1
2
u
15
+ u
14
+ ··· +
1
2
u
2
+
5
2
u
a
3
=
−
1
2
u
15
−
3
2
u
14
+ ··· − 7u −
1
2
−
1
2
u
15
− u
14
+ ··· −
5
2
u − 1
a
2
=
−u
15
−
5
2
u
14
+ ··· −
19
2
u −
3
2
−
1
2
u
15
− u
14
+ ··· −
5
2
u − 1
a
7
=
u
u
3
+ u
a
1
=
−u
15
−
7
2
u
14
+ ··· −
21
2
u −
3
2
1
2
u
15
+ 2u
14
+ ··· +
13
2
u
2
+
1
2
u
a
9
=
u
5
+ 2u
3
+ u
u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
5
2
u
15
+ 5u
14
+
35
2
u
13
+ 24u
12
+ 46u
11
+
113
2
u
10
+
141
2
u
9
+ 87u
8
+
71u
7
+
173
2
u
6
+ 56u
5
+ 49u
4
+ 43u
3
+
25
2
u
2
+
37
2
u + 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
16
− u
15
+ ··· + 16u + 16
c
2
u
16
+ 3u
15
+ ··· + 8u + 1
c
3
, c
10
u
16
+ 3u
15
+ ··· + 2u + 1
c
4
, c
7
u
16
− 3u
15
+ ··· − 4u + 1
c
5
u
16
− 11u
15
+ ··· − 8u + 1
c
8
u
16
+ 3u
15
+ ··· + 4u
2
+ 1
c
9
u
16
− 3u
15
+ ··· + 202u + 73
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
16
+ 25y
15
+ ··· + 896y + 256
c
2
y
16
+ 23y
15
+ ··· + 8y + 1
c
3
, c
10
y
16
+ 3y
15
+ ··· + 8y + 1
c
4
, c
7
y
16
+ 11y
15
+ ··· + 8y + 1
c
5
y
16
− 9y
15
+ ··· + 88y + 1
c
8
y
16
− 29y
15
+ ··· + 8y + 1
c
9
y
16
+ 43y
15
+ ··· + 114832y + 5329
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.073480 + 0.057122I
a = −0.540627 − 0.419792I
b = 0.935752 − 0.958508I
8.77898 − 3.44428I −0.71478 + 2.21154I
u = −1.073480 − 0.057122I
a = −0.540627 + 0.419792I
b = 0.935752 + 0.958508I
8.77898 + 3.44428I −0.71478 − 2.21154I
u = −0.186461 + 1.088150I
a = 1.79112 − 0.29650I
b = −0.537019 − 1.088350I
1.80445 + 3.62763I 1.66989 − 3.19198I
u = −0.186461 − 1.088150I
a = 1.79112 + 0.29650I
b = −0.537019 + 1.088350I
1.80445 − 3.62763I 1.66989 + 3.19198I
u = 0.531252 + 0.974365I
a = −1.283580 − 0.440428I
b = 0.361572 − 0.440175I
0.15035 − 2.79885I −1.52268 + 1.51981I
u = 0.531252 − 0.974365I
a = −1.283580 + 0.440428I
b = 0.361572 + 0.440175I
0.15035 + 2.79885I −1.52268 − 1.51981I
u = 0.044881 + 1.189250I
a = 1.25145 − 0.74047I
b = −0.849220 + 0.545637I
3.73547 − 1.61832I 3.41778 + 2.30788I
u = 0.044881 − 1.189250I
a = 1.25145 + 0.74047I
b = −0.849220 − 0.545637I
3.73547 + 1.61832I 3.41778 − 2.30788I
u = 0.460182 + 0.643087I
a = −0.627874 + 0.508017I
b = −0.003649 + 0.625754I
−0.82216 − 1.37285I −5.23267 + 4.39698I
u = 0.460182 − 0.643087I
a = −0.627874 − 0.508017I
b = −0.003649 − 0.625754I
−0.82216 + 1.37285I −5.23267 − 4.39698I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.55660 + 1.34475I
a = −1.82052 + 0.34354I
b = 0.923344 + 1.057020I
12.7882 + 9.2506I 1.44636 − 5.03050I
u = −0.55660 − 1.34475I
a = −1.82052 − 0.34354I
b = 0.923344 − 1.057020I
12.7882 − 9.2506I 1.44636 + 5.03050I
u = −0.48833 + 1.38689I
a = −0.783578 + 0.870931I
b = 1.031440 − 0.889735I
13.35520 + 2.10741I 2.23202 − 0.63352I
u = −0.48833 − 1.38689I
a = −0.783578 − 0.870931I
b = 1.031440 + 0.889735I
13.35520 − 2.10741I 2.23202 + 0.63352I
u = −0.231448 + 0.297600I
a = −1.48639 + 0.77777I
b = −0.362224 + 0.817550I
−0.31203 − 1.54541I −2.29594 + 4.92633I
u = −0.231448 − 0.297600I
a = −1.48639 − 0.77777I
b = −0.362224 − 0.817550I
−0.31203 + 1.54541I −2.29594 − 4.92633I
6
II. I
u
2
= hb + u − 1, a + u + 1, u
2
− u + 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
5
=
1
u − 1
a
6
=
u
u − 1
a
10
=
−u − 1
−u + 1
a
3
=
u − 1
−u
a
2
=
−1
−u
a
7
=
u
u − 1
a
1
=
−1
−u
a
9
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u − 7
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
2
c
2
, c
3
, c
4
c
5
, c
8
, c
9
u
2
− u + 1
c
7
, c
10
u
2
+ u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
2
c
2
, c
3
, c
4
c
5
, c
7
, c
8
c
9
, c
10
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 = −1.50000 − 0.86603I
b = 0.500000 − 0.866025I
− 4.05977I −3.00000 + 6.92820I
u = 0.500000 − 0.866025I
a = −1.50000 + 0.86603I
b = 0.500000 + 0.866025I
4.05977I −3.00000 − 6.92820I
10
III. I
u
3
= hb − u, a, u
2
− u + 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
5
=
1
u − 1
a
6
=
u
u − 1
a
10
=
0
u
a
3
=
1
u − 1
a
2
=
u
u − 1
a
7
=
u
u − 1
a
1
=
u
u − 1
a
9
=
−1
0
(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
u
2
c
2
, c
3
, c
4
c
5
, c
8
, c
9
u
2
− u + 1
c
7
, c
10
u
2
+ u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
2
c
2
, c
3
, c
4
c
5
, c
7
, c
8
c
9
, c
10
y
2
+ y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0
b = 0.500000 + 0.866025I
0 0
u = 0.500000 − 0.866025I
a = 0
b = 0.500000 − 0.866025I
0 0
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u
4
(u
16
− u
15
+ ··· + 16u + 16)
c
2
((u
2
− u + 1)
2
)(u
16
+ 3u
15
+ ··· + 8u + 1)
c
3
((u
2
− u + 1)
2
)(u
16
+ 3u
15
+ ··· + 2u + 1)
c
4
((u
2
− u + 1)
2
)(u
16
− 3u
15
+ ··· − 4u + 1)
c
5
((u
2
− u + 1)
2
)(u
16
− 11u
15
+ ··· − 8u + 1)
c
7
((u
2
+ u + 1)
2
)(u
16
− 3u
15
+ ··· − 4u + 1)
c
8
((u
2
− u + 1)
2
)(u
16
+ 3u
15
+ ··· + 4u
2
+ 1)
c
9
((u
2
− u + 1)
2
)(u
16
− 3u
15
+ ··· + 202u + 73)
c
10
((u
2
+ u + 1)
2
)(u
16
+ 3u
15
+ ··· + 2u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
4
(y
16
+ 25y
15
+ ··· + 896y + 256)
c
2
((y
2
+ y + 1)
2
)(y
16
+ 23y
15
+ ··· + 8y + 1)
c
3
, c
10
((y
2
+ y + 1)
2
)(y
16
+ 3y
15
+ ··· + 8y + 1)
c
4
, c
7
((y
2
+ y + 1)
2
)(y
16
+ 11y
15
+ ··· + 8y + 1)
c
5
((y
2
+ y + 1)
2
)(y
16
− 9y
15
+ ··· + 88y + 1)
c
8
((y
2
+ y + 1)
2
)(y
16
− 29y
15
+ ··· + 8y + 1)
c
9
((y
2
+ y + 1)
2
)(y
16
+ 43y
15
+ ··· + 114832y + 5329)
16
