10
58
(K10a
20
)
A knot diagram
1
Linearized knot diagam
8 5 7 3 10 4 9 1 6 2
Solving Sequence
3,7
4 5 2
6,10
9 8 1
c
3
c
4
c
2
c
6
c
9
c
7
c
1
c
5
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
7
+ u
5
+ 2u
3
+ b + u, −u
6
− u
4
− 2u
2
+ a − 1, u
10
− u
9
+ 2u
8
− u
7
+ 4u
6
− 2u
5
+ 4u
4
− u
3
+ 3u
2
+ u + 1i
I
u
2
= hu
25
+ 2u
24
+ ··· + b + 3, 3u
25
− 7u
24
+ ··· + a − 6, u
26
− 2u
25
+ ··· − u + 1i
I
u
3
= hb + u + 1, a − u, u
2
+ u + 1i
I
u
4
= hb − u, a − 1, u
2
+ u + 1i
* 4 irreducible components of dim
C
= 0, with total 40 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
7
+ u
5
+ 2u
3
+ b + u, −u
6
− u
4
− 2u
2
+ a − 1, u
10
− u
9
+ · · · + u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u
2
a
5
=
u
2
+ 1
−u
2
a
2
=
u
4
+ u
2
+ 1
−u
4
a
6
=
−u
u
3
+ u
a
10
=
u
6
+ u
4
+ 2u
2
+ 1
−u
7
− u
5
− 2u
3
− u
a
9
=
−u
u
8
− u
7
+ 2u
6
− u
5
+ 3u
4
− u
3
+ 3u
2
+ 1
a
8
=
−u
3
−u
6
+ u
5
− u
4
+ u
3
− 2u
2
− 1
a
1
=
u
2
+ 1
−u
7
+ u
6
− u
5
− 2u
3
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
9
− 6u
8
+ 6u
7
− 4u
6
+ 14u
5
− 14u
4
+ 10u
3
− 6u
2
+ 10u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
c
8
u
10
+ u
9
+ 2u
8
+ u
7
+ 4u
6
+ 2u
5
+ 4u
4
+ u
3
+ 3u
2
− u + 1
c
2
, c
4
, c
7
c
10
u
10
+ 3u
9
+ ··· + 5u + 1
c
5
, c
9
u
10
− 5u
9
+ ··· − 8u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
y
10
+ 3y
9
+ ··· + 5y + 1
c
2
, c
4
, c
7
c
10
y
10
+ 11y
9
+ ··· + 13y + 1
c
5
, c
9
y
10
+ 5y
9
+ ··· + 32y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.100577 + 0.954526I
a = −0.658857 − 0.498555I
b = −0.542150 + 0.578753I
−3.48123 − 2.16643I −9.00466 + 4.21901I
u = −0.100577 − 0.954526I
a = −0.658857 + 0.498555I
b = −0.542150 − 0.578753I
−3.48123 + 2.16643I −9.00466 − 4.21901I
u = 0.900362 + 0.768734I
a = −1.68093 + 0.92466I
b = 2.22427 + 0.45966I
10.21950 − 0.19532I 4.14143 − 1.59060I
u = 0.900362 − 0.768734I
a = −1.68093 − 0.92466I
b = 2.22427 − 0.45966I
10.21950 + 0.19532I 4.14143 + 1.59060I
u = −0.774061 + 0.907730I
a = −0.053403 + 0.383357I
b = 0.306648 + 0.345217I
4.50100 − 5.87397I 1.27770 + 5.35715I
u = −0.774061 − 0.907730I
a = −0.053403 − 0.383357I
b = 0.306648 − 0.345217I
4.50100 + 5.87397I 1.27770 − 5.35715I
u = 0.782324 + 1.035710I
a = 1.19625 − 1.47594I
b = −2.46450 − 0.08430I
8.4959 + 12.7213I 1.50029 − 7.98966I
u = 0.782324 − 1.035710I
a = 1.19625 + 1.47594I
b = −2.46450 + 0.08430I
8.4959 − 12.7213I 1.50029 + 7.98966I
u = −0.308049 + 0.477623I
a = 0.696944 − 0.500305I
b = −0.024265 − 0.486995I
0.004061 − 1.246020I 0.08524 + 5.02615I
u = −0.308049 − 0.477623I
a = 0.696944 + 0.500305I
b = −0.024265 + 0.486995I
0.004061 + 1.246020I 0.08524 − 5.02615I
5
II.
I
u
2
= hu
25
+2u
24
+· · ·+b +3, 3u
25
−7u
24
+· · ·+a −6, u
26
−2u
25
+· · ·−u +1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u
2
a
5
=
u
2
+ 1
−u
2
a
2
=
u
4
+ u
2
+ 1
−u
4
a
6
=
−u
u
3
+ u
a
10
=
−3u
25
+ 7u
24
+ ··· − 4u + 6
−u
25
− 2u
24
+ ··· − 3u − 3
a
9
=
−u
25
+ 3u
24
+ ··· − 2u + 4
−u
22
− 3u
20
+ ··· − 3u − 1
a
8
=
−u
25
+ u
24
+ ··· − 3u − 1
u
25
− 2u
24
+ ··· + 3u − 1
a
1
=
−u
25
+ 3u
24
+ ··· − u + 5
−u
25
− 4u
23
+ ··· − 3u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 5u
25
− 12u
24
+ 25u
23
− 43u
22
+ 75u
21
− 109u
20
+ 126u
19
−
175u
18
+ 168u
17
− 200u
16
+ 98u
15
− 164u
14
− 6u
13
− 49u
12
− 144u
11
− 10u
10
− 179u
9
+
26u
8
− 112u
7
− 28u
6
− 47u
5
− 26u
4
+ 9u
3
− 16u
2
+ 15u − 5
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
c
8
u
26
+ 2u
25
+ ··· + u + 1
c
2
, c
4
, c
7
c
10
u
26
+ 8u
25
+ ··· + 13u + 1
c
5
, c
9
(u
13
+ 2u
12
+ ··· + 3u + 2)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
y
26
+ 8y
25
+ ··· + 13y + 1
c
2
, c
4
, c
7
c
10
y
26
+ 20y
25
+ ··· − 11y + 1
c
5
, c
9
(y
13
+ 10y
12
+ ··· − 7y −4)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.752045 + 0.803934I
a = 1.97258 − 1.23710I
b = −2.47801 − 0.65547I
1.88524 − 0.96841I 0.413632 + 1.140295I
u = 0.752045 − 0.803934I
a = 1.97258 + 1.23710I
b = −2.47801 + 0.65547I
1.88524 + 0.96841I 0.413632 − 1.140295I
u = −0.578645 + 0.950081I
a = 0.030039 + 0.285319I
b = 0.288458 + 0.136559I
−0.80957 − 3.02973I −5.16840 + 1.62282I
u = −0.578645 − 0.950081I
a = 0.030039 − 0.285319I
b = 0.288458 − 0.136559I
−0.80957 + 3.02973I −5.16840 − 1.62282I
u = −0.496478 + 0.720203I
a = 0.299461 − 0.224790I
b = −0.013218 − 0.327276I
0.00150 − 1.41503I −1.90513 + 4.60201I
u = −0.496478 − 0.720203I
a = 0.299461 + 0.224790I
b = −0.013218 + 0.327276I
0.00150 + 1.41503I −1.90513 − 4.60201I
u = −0.335785 + 1.109920I
a = 0.267955 + 0.444237I
b = 0.583042 − 0.148240I
1.88524 − 0.96841I 0.413632 + 1.140295I
u = −0.335785 − 1.109920I
a = 0.267955 − 0.444237I
b = 0.583042 + 0.148240I
1.88524 + 0.96841I 0.413632 − 1.140295I
u = 0.905446 + 0.730041I
a = 1.71372 − 0.85399I
b = −2.17513 − 0.47784I
9.45063 − 6.48172I 3.04187 + 3.27257I
u = 0.905446 − 0.730041I
a = 1.71372 + 0.85399I
b = −2.17513 + 0.47784I
9.45063 + 6.48172I 3.04187 − 3.27257I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.269616 + 1.131670I
a = −0.308288 − 0.503667I
b = −0.653102 + 0.213082I
1.46125 − 6.61332I −1.15142 + 6.72912I
u = −0.269616 − 1.131670I
a = −0.308288 + 0.503667I
b = −0.653102 − 0.213082I
1.46125 + 6.61332I −1.15142 − 6.72912I
u = −0.786233 + 0.860060I
a = 0.072545 − 0.387289I
b = −0.276054 − 0.366893I
4.64840 1.75564 + 0.I
u = −0.786233 − 0.860060I
a = 0.072545 + 0.387289I
b = −0.276054 + 0.366893I
4.64840 1.75564 + 0.I
u = −0.819468 + 0.042718I
a = 0.021039 − 0.644673I
b = −0.010298 − 0.529188I
5.42596 − 2.97283I 4.39163 + 2.88376I
u = −0.819468 − 0.042718I
a = 0.021039 + 0.644673I
b = −0.010298 + 0.529188I
5.42596 + 2.97283I 4.39163 − 2.88376I
u = 0.791857 + 0.886903I
a = −1.65384 + 1.34745I
b = 2.50466 + 0.39980I
5.42596 + 2.97283I 4.39163 − 2.88376I
u = 0.791857 − 0.886903I
a = −1.65384 − 1.34745I
b = 2.50466 − 0.39980I
5.42596 − 2.97283I 4.39163 + 2.88376I
u = 0.732196 + 0.941652I
a = 1.54369 − 1.64448I
b = −2.67881 − 0.24953I
1.46125 + 6.61332I −1.15142 − 6.72912I
u = 0.732196 − 0.941652I
a = 1.54369 + 1.64448I
b = −2.67881 + 0.24953I
1.46125 − 6.61332I −1.15142 + 6.72912I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.156803 + 0.747604I
a = −1.50536 − 0.52470I
b = −0.156223 + 1.207680I
−0.80957 + 3.02973I −5.16840 − 1.62282I
u = 0.156803 − 0.747604I
a = −1.50536 + 0.52470I
b = −0.156223 − 1.207680I
−0.80957 − 3.02973I −5.16840 + 1.62282I
u = 0.799863 + 1.014160I
a = −1.26195 + 1.42943I
b = 2.45906 + 0.13647I
9.45063 + 6.48172I 3.04187 − 3.27257I
u = 0.799863 − 1.014160I
a = −1.26195 − 1.42943I
b = 2.45906 − 0.13647I
9.45063 − 6.48172I 3.04187 + 3.27257I
u = 0.148015 + 0.419312I
a = 1.80840 − 0.30217I
b = −0.394374 − 0.713561I
0.00150 − 1.41503I −1.90513 + 4.60201I
u = 0.148015 − 0.419312I
a = 1.80840 + 0.30217I
b = −0.394374 + 0.713561I
0.00150 + 1.41503I −1.90513 − 4.60201I
11
III. I
u
3
= hb + u + 1, a − u, u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u + 1
a
5
=
−u
u + 1
a
2
=
0
−u
a
6
=
−u
u + 1
a
10
=
u
−u − 1
a
9
=
u
−u − 1
a
8
=
−1
0
a
1
=
u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u + 4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
10
u
2
− u + 1
c
3
, c
4
, c
8
u
2
+ u + 1
c
5
, c
9
u
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
y
2
+ y + 1
c
5
, c
9
y
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = −0.500000 − 0.866025I
− 4.05977I 0. + 6.92820I
u = −0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = −0.500000 + 0.866025I
4.05977I 0. − 6.92820I
15
IV. I
u
4
= hb − u, a − 1, u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u + 1
a
5
=
−u
u + 1
a
2
=
0
−u
a
6
=
−u
u + 1
a
10
=
1
u
a
9
=
1
u
a
8
=
−u
2u + 1
a
1
=
1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
10
u
2
− u + 1
c
3
, c
4
, c
8
u
2
+ u + 1
c
5
, c
9
u
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
y
2
+ y + 1
c
5
, c
9
y
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 1.00000
b = −0.500000 + 0.866025I
0 −3.00000
u = −0.500000 − 0.866025I
a = 1.00000
b = −0.500000 − 0.866025I
0 −3.00000
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
((u
2
− u + 1)
2
)(u
10
+ u
9
+ ··· − u + 1)
· (u
26
+ 2u
25
+ ··· + u + 1)
c
2
, c
7
, c
10
((u
2
− u + 1)
2
)(u
10
+ 3u
9
+ ··· + 5u + 1)(u
26
+ 8u
25
+ ··· + 13u + 1)
c
3
, c
8
((u
2
+ u + 1)
2
)(u
10
+ u
9
+ ··· − u + 1)
· (u
26
+ 2u
25
+ ··· + u + 1)
c
4
((u
2
+ u + 1)
2
)(u
10
+ 3u
9
+ ··· + 5u + 1)(u
26
+ 8u
25
+ ··· + 13u + 1)
c
5
, c
9
u
4
(u
10
− 5u
9
+ ··· − 8u + 4)(u
13
+ 2u
12
+ ··· + 3u + 2)
2
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
((y
2
+ y + 1)
2
)(y
10
+ 3y
9
+ ··· + 5y + 1)(y
26
+ 8y
25
+ ··· + 13y + 1)
c
2
, c
4
, c
7
c
10
((y
2
+ y + 1)
2
)(y
10
+ 11y
9
+ ··· + 13y + 1)(y
26
+ 20y
25
+ ··· − 11y + 1)
c
5
, c
9
y
4
(y
10
+ 5y
9
+ ··· + 32y + 16)(y
13
+ 10y
12
+ ··· − 7y −4)
2
21
