10
54
(K10a
48
)
A knot diagram
1
Linearized knot diagam
8 5 6 10 3 9 1 7 4 2
Solving Sequence
1,8 2,5
3 7 9 6 10 4
c
1
c
2
c
7
c
8
c
6
c
10
c
4
c
3
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
16
+ 2u
14
− 6u
12
+ 8u
10
− 2u
9
− 10u
8
+ 2u
7
+ 8u
6
− 6u
5
− 4u
4
+ 4u
3
+ b − 2u, u
23
+ u
22
+ ··· + a + 2,
u
26
+ 2u
25
+ ··· + 2u − 1i
I
u
2
= hu
2
+ b, a + u, u
3
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 29 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
16
+2u
14
+· · ·+b−2u, u
23
+u
22
+· · ·+a+2, u
26
+2u
25
+· · ·+2u−1i
(i) Arc colorings
a
1
=
1
0
a
8
=
0
u
a
2
=
1
u
2
a
5
=
−u
23
− u
22
+ ··· + 2u − 2
u
16
− 2u
14
+ ··· − 4u
3
+ 2u
a
3
=
−u
25
− u
24
+ ··· − 2u + 3
−u
25
− 2u
24
+ ··· − 4u + 1
a
7
=
u
u
a
9
=
−u
3
−u
3
+ u
a
6
=
u
5
+ u
u
5
− u
3
+ u
a
10
=
−u
2
+ 1
−u
4
a
4
=
2u
25
+ 2u
24
+ ··· + 2u − 3
2u
25
+ 4u
24
+ ··· + 7u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
25
+ 6u
24
− 4u
23
− 20u
22
+ 15u
21
+ 67u
20
− 6u
19
− 133u
18
+
9u
17
+ 243u
16
+ 26u
15
− 312u
14
− 14u
13
+ 380u
12
+ 8u
11
− 325u
10
+ 66u
9
+ 275u
8
−
98u
7
− 154u
6
+ 121u
5
+ 79u
4
− 58u
3
− 13u
2
+ 27u + 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
26
+ 2u
25
+ ··· + 2u − 1
c
2
, c
3
, c
5
u
26
+ 4u
25
+ ··· − u − 1
c
4
, c
9
u
26
− u
25
+ ··· − 12u + 8
c
6
, c
8
, c
10
u
26
+ 6u
25
+ ··· + 14u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
26
− 6y
25
+ ··· − 14y + 1
c
2
, c
3
, c
5
y
26
− 28y
25
+ ··· + 9y + 1
c
4
, c
9
y
26
− 21y
25
+ ··· − 272y + 64
c
6
, c
8
, c
10
y
26
+ 30y
25
+ ··· − 38y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.846572 + 0.426560I
a = 0.369102 + 1.145660I
b = −0.275280 + 0.265581I
0.25133 − 3.55563I 0.67279 + 7.82227I
u = 0.846572 − 0.426560I
a = 0.369102 − 1.145660I
b = −0.275280 − 0.265581I
0.25133 + 3.55563I 0.67279 − 7.82227I
u = −1.05838
a = 0.930276
b = −0.383659
3.31147 2.10670
u = 1.024210 + 0.483667I
a = −0.41844 − 1.77157I
b = −0.06027 − 1.68353I
6.23030 − 6.31822I 4.39684 + 5.98052I
u = 1.024210 − 0.483667I
a = −0.41844 + 1.77157I
b = −0.06027 + 1.68353I
6.23030 + 6.31822I 4.39684 − 5.98052I
u = 0.352335 + 0.784080I
a = −1.72547 − 0.09649I
b = −0.633711 − 1.002200I
8.43955 + 1.72575I 8.31886 − 0.55186I
u = 0.352335 − 0.784080I
a = −1.72547 + 0.09649I
b = −0.633711 + 1.002200I
8.43955 − 1.72575I 8.31886 + 0.55186I
u = −0.714859 + 0.468666I
a = 1.90202 − 1.71328I
b = 0.66236 − 1.66931I
2.60764 + 1.82411I 3.14672 − 3.41167I
u = −0.714859 − 0.468666I
a = 1.90202 + 1.71328I
b = 0.66236 + 1.66931I
2.60764 − 1.82411I 3.14672 + 3.41167I
u = 0.884681 + 0.778751I
a = −0.886815 − 0.322575I
b = −0.169423 − 1.226160I
3.71424 − 2.93248I −1.57920 + 3.07432I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.884681 − 0.778751I
a = −0.886815 + 0.322575I
b = −0.169423 + 1.226160I
3.71424 + 2.93248I −1.57920 − 3.07432I
u = −0.782649 + 0.135062I
a = −0.896199 + 0.591232I
b = −0.443229 + 0.258658I
−1.320760 + 0.339413I −6.54496 − 0.64162I
u = −0.782649 − 0.135062I
a = −0.896199 − 0.591232I
b = −0.443229 − 0.258658I
−1.320760 − 0.339413I −6.54496 + 0.64162I
u = −0.890496 + 0.876738I
a = 0.197188 − 0.399123I
b = 0.018430 − 1.188940I
8.31406 + 0.26926I 5.67547 + 0.24692I
u = −0.890496 − 0.876738I
a = 0.197188 + 0.399123I
b = 0.018430 + 1.188940I
8.31406 − 0.26926I 5.67547 − 0.24692I
u = −0.851371 + 0.929645I
a = −1.80201 + 0.27660I
b = −0.91806 + 3.09384I
15.7394 − 4.0044I 7.52896 + 1.00327I
u = −0.851371 − 0.929645I
a = −1.80201 − 0.27660I
b = −0.91806 − 3.09384I
15.7394 + 4.0044I 7.52896 − 1.00327I
u = 0.920092 + 0.872965I
a = 2.37362 + 0.94576I
b = 0.20685 + 3.87193I
10.46160 − 3.23113I 6.21855 + 2.44261I
u = 0.920092 − 0.872965I
a = 2.37362 − 0.94576I
b = 0.20685 − 3.87193I
10.46160 + 3.23113I 6.21855 − 2.44261I
u = −0.942244 + 0.855193I
a = 1.174670 − 0.368934I
b = 0.172482 − 1.056320I
8.15003 + 6.14753I 5.18996 − 5.20017I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.942244 − 0.855193I
a = 1.174670 + 0.368934I
b = 0.172482 + 1.056320I
8.15003 − 6.14753I 5.18996 + 5.20017I
u = −0.996075 + 0.858678I
a = −1.86455 + 1.56620I
b = 0.61540 + 3.28212I
15.2731 + 10.5913I 6.79989 − 5.68919I
u = −0.996075 − 0.858678I
a = −1.86455 − 1.56620I
b = 0.61540 − 3.28212I
15.2731 − 10.5913I 6.79989 + 5.68919I
u = 0.493543 + 0.417386I
a = −0.243260 − 0.166657I
b = 0.698144 + 0.266835I
1.336670 + 0.113896I 6.51816 + 0.27618I
u = 0.493543 − 0.417386I
a = −0.243260 + 0.166657I
b = 0.698144 − 0.266835I
1.336670 − 0.113896I 6.51816 − 0.27618I
u = 0.370909
a = −1.28999
b = 0.636266
1.14285 10.2090
7
II. I
u
2
= hu
2
+ b, a + u, u
3
− u
2
+ 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
2
a
3
=
−u + 1
0
a
7
=
u
u
a
9
=
−u
2
+ 1
−u
2
+ u + 1
a
6
=
−1
−u
2
a
10
=
−u
2
+ 1
−u
2
+ u + 1
a
4
=
−u
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
2
+ 7u + 2
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
− u
2
+ 1
c
2
, c
3
(u + 1)
3
c
4
, c
9
u
3
c
5
(u − 1)
3
c
6
, c
10
u
3
− u
2
+ 2u − 1
c
7
u
3
+ u
2
− 1
c
8
u
3
+ u
2
+ 2u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
3
− y
2
+ 2y −1
c
2
, c
3
, c
5
(y −1)
3
c
4
, c
9
y
3
c
6
, c
8
, c
10
y
3
+ 3y
2
+ 2y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.877439 − 0.744862I
b = −0.215080 − 1.307140I
4.66906 − 2.82812I 7.71191 + 2.59975I
u = 0.877439 − 0.744862I
a = −0.877439 + 0.744862I
b = −0.215080 + 1.307140I
4.66906 + 2.82812I 7.71191 − 2.59975I
u = −0.754878
a = 0.754878
b = −0.569840
0.531480 −4.42380
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 1)(u
26
+ 2u
25
+ ··· + 2u − 1)
c
2
, c
3
((u + 1)
3
)(u
26
+ 4u
25
+ ··· − u − 1)
c
4
, c
9
u
3
(u
26
− u
25
+ ··· − 12u + 8)
c
5
((u − 1)
3
)(u
26
+ 4u
25
+ ··· − u − 1)
c
6
, c
10
(u
3
− u
2
+ 2u − 1)(u
26
+ 6u
25
+ ··· + 14u + 1)
c
7
(u
3
+ u
2
− 1)(u
26
+ 2u
25
+ ··· + 2u − 1)
c
8
(u
3
+ u
2
+ 2u + 1)(u
26
+ 6u
25
+ ··· + 14u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
3
− y
2
+ 2y −1)(y
26
− 6y
25
+ ··· − 14y + 1)
c
2
, c
3
, c
5
((y −1)
3
)(y
26
− 28y
25
+ ··· + 9y + 1)
c
4
, c
9
y
3
(y
26
− 21y
25
+ ··· − 272y + 64)
c
6
, c
8
, c
10
(y
3
+ 3y
2
+ 2y −1)(y
26
+ 30y
25
+ ··· − 38y + 1)
13
