10
62
(K10a
41
)
A knot diagram
1
Linearized knot diagam
6 9 7 10 1 3 4 2 8 5
Solving Sequence
5,10
1 6 2
4,8
7 3 9
c
10
c
5
c
1
c
4
c
7
c
3
c
9
c
2
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
18
+ 10u
16
+ ··· + 4b + 4, 2u
18
− 21u
16
+ ··· + 4a − 6, u
19
+ 2u
18
+ ··· + 2u
2
− 2i
I
u
2
= ha
2
+ au + 2b − a + 2, a
3
− 2a
2
+ au + 2a − 2u, u
2
− u − 1i
I
u
3
= hb + 1, 2a + u − 2, u
2
− 2i
I
v
1
= ha, b + 1, v −1i
* 4 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
=
h−u
18
+10u
16
+· · ·+4b+4, 2u
18
−21u
16
+· · ·+4a−6, u
19
+2u
18
+· · ·+2u
2
−2i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
1
=
1
−u
2
a
6
=
u
−u
3
+ u
a
2
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
−u
u
a
8
=
−
1
2
u
18
+
21
4
u
16
+ ··· − 2u +
3
2
1
4
u
18
−
5
2
u
16
+ ··· −
1
2
u − 1
a
7
=
1
4
u
16
−
9
4
u
14
+ ··· −
3
2
u +
1
2
−
1
4
u
18
+
5
2
u
16
+ ··· + 3u
3
− u
a
3
=
1
4
u
16
−
9
4
u
14
+ ··· −
3
2
u +
1
2
−
1
4
u
16
+ 2u
14
+ ··· −
1
2
u
2
+ u
a
9
=
1
4
u
15
− 2u
13
+ ··· −
1
2
u + 1
−
1
4
u
15
+ 2u
13
+ ··· −
1
2
u
2
−
1
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
18
+ 22u
16
− 96u
14
− 2u
13
+ 210u
12
+ 16u
11
− 240u
10
−
46u
9
+ 128u
8
+ 56u
7
+ 12u
6
− 32u
5
− 66u
4
+ 24u
3
+ 20u
2
− 10u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
10
u
19
− 2u
18
+ ··· − 2u
2
+ 2
c
2
, c
8
u
19
+ 2u
18
+ ··· + 5u − 1
c
3
, c
6
, c
7
u
19
− 2u
18
+ ··· − 7u − 1
c
9
u
19
+ 6u
18
+ ··· + 29u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
10
y
19
− 22y
18
+ ··· + 8y −4
c
2
, c
8
y
19
− 6y
18
+ ··· + 29y −1
c
3
, c
6
, c
7
y
19
− 22y
18
+ ··· + 45y −1
c
9
y
19
+ 18y
18
+ ··· + 429y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.833626 + 0.586392I
a = −0.745450 + 0.856359I
b = −0.57278 − 1.50837I
6.16103 − 7.19649I 9.03544 + 6.33971I
u = −0.833626 − 0.586392I
a = −0.745450 − 0.856359I
b = −0.57278 + 1.50837I
6.16103 + 7.19649I 9.03544 − 6.33971I
u = 0.976743 + 0.434841I
a = 1.000180 + 0.545099I
b = 0.281151 − 1.040530I
7.39549 + 1.39372I 11.32275 − 1.16010I
u = 0.976743 − 0.434841I
a = 1.000180 − 0.545099I
b = 0.281151 + 1.040530I
7.39549 − 1.39372I 11.32275 + 1.16010I
u = 0.706968 + 0.375087I
a = −0.23384 − 1.47789I
b = −0.594733 + 0.957959I
0.05288 + 3.91264I 5.51817 − 7.54928I
u = 0.706968 − 0.375087I
a = −0.23384 + 1.47789I
b = −0.594733 − 0.957959I
0.05288 − 3.91264I 5.51817 + 7.54928I
u = −0.109594 + 0.768897I
a = 0.397475 + 0.645275I
b = −0.268744 + 1.200510I
3.98301 + 2.66673I 7.07144 − 2.45976I
u = −0.109594 − 0.768897I
a = 0.397475 − 0.645275I
b = −0.268744 − 1.200510I
3.98301 − 2.66673I 7.07144 + 2.45976I
u = 1.37410
a = 0.844357
b = −0.0493609
6.50526 14.0760
u = −1.43916
a = 1.03336
b = −1.13820
3.34099 2.02410
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.169186 + 0.450873I
a = 0.191882 − 0.311707I
b = −0.742122 − 0.473186I
−1.52268 − 0.97340I −1.44998 + 1.44252I
u = 0.169186 − 0.450873I
a = 0.191882 + 0.311707I
b = −0.742122 + 0.473186I
−1.52268 + 0.97340I −1.44998 − 1.44252I
u = −0.449480
a = 1.73887
b = −0.213083
0.876243 12.3180
u = −1.62272 + 0.09591I
a = 0.13568 + 1.94903I
b = −0.46328 − 1.41274I
8.08934 − 5.62533I 8.31274 + 4.90801I
u = −1.62272 − 0.09591I
a = 0.13568 − 1.94903I
b = −0.46328 + 1.41274I
8.08934 + 5.62533I 8.31274 − 4.90801I
u = 1.66085 + 0.17438I
a = −0.14379 − 1.92619I
b = −0.79811 + 1.82654I
14.6774 + 10.1415I 10.53245 − 5.16770I
u = 1.66085 − 0.17438I
a = −0.14379 + 1.92619I
b = −0.79811 − 1.82654I
14.6774 − 10.1415I 10.53245 + 5.16770I
u = −1.69053 + 0.10897I
a = 0.089568 − 1.231990I
b = 0.85893 + 1.17135I
16.6648 − 3.4892I 12.44780 + 0.95664I
u = −1.69053 − 0.10897I
a = 0.089568 + 1.231990I
b = 0.85893 − 1.17135I
16.6648 + 3.4892I 12.44780 − 0.95664I
6
II. I
u
2
= ha
2
+ au + 2b − a + 2, a
3
− 2a
2
+ au + 2a − 2u, u
2
− u − 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
1
=
1
−u − 1
a
6
=
u
−u − 1
a
2
=
−u
u
a
4
=
−u
u
a
8
=
a
−
1
2
a
2
−
1
2
au +
1
2
a − 1
a
7
=
1
2
a
2
u +
1
2
a
2
−
1
2
au + u + 1
−
1
2
a
2
u − a
2
+
3
2
a − u − 2
a
3
=
1
2
a
2
u +
1
2
a
2
−
1
2
au + u + 1
−a
2
u + au − a − 2u
a
9
=
1
2
a
2
u +
1
2
a
2
−
1
2
au + u + 1
−
1
2
a
2
u − a
2
+
3
2
a − u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
10
(u
2
+ u − 1)
3
c
2
, c
3
, c
6
c
7
, c
8
u
6
− 2u
4
− u
3
+ u
2
+ u − 1
c
9
u
6
+ 4u
5
+ 6u
4
+ 7u
3
+ 7u
2
+ 3u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
10
(y
2
− 3y + 1)
3
c
2
, c
3
, c
6
c
7
, c
8
y
6
− 4y
5
+ 6y
4
− 7y
3
+ 7y
2
− 3y + 1
c
9
y
6
− 4y
5
− 6y
4
+ 13y
3
+ 19y
2
+ 5y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −0.480334
b = −1.50396
0.986960 10.0000
u = −0.618034
a = 1.24017 + 1.01752I
b = −0.248021 − 0.438702I
0.986960 10.0000
u = −0.618034
a = 1.24017 − 1.01752I
b = −0.248021 + 0.438702I
0.986960 10.0000
u = 1.61803
a = 1.21468
b = −2.11309
8.88264 10.0000
u = 1.61803
a = 0.39266 + 1.58428I
b = 0.056543 − 1.111650I
8.88264 10.0000
u = 1.61803
a = 0.39266 − 1.58428I
b = 0.056543 + 1.111650I
8.88264 10.0000
10
III. I
u
3
= hb + 1, 2a + u − 2, u
2
− 2i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
1
=
1
−2
a
6
=
u
−u
a
2
=
−1
0
a
4
=
−u
u
a
8
=
−
1
2
u + 1
−1
a
7
=
1
2
u + 1
−u − 1
a
3
=
−
1
2
u + 1
−1
a
9
=
−
1
2
u + 2
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
10
u
2
− 2
c
2
, c
3
(u − 1)
2
c
6
, c
7
, c
8
c
9
(u + 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
10
(y −2)
2
c
2
, c
3
, c
6
c
7
, c
8
, c
9
(y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = 0.292893
b = −1.00000
4.93480 8.00000
u = −1.41421
a = 1.70711
b = −1.00000
4.93480 8.00000
14
IV. I
v
1
= ha, b + 1, v − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
1
0
a
1
=
1
0
a
6
=
1
0
a
2
=
1
0
a
4
=
1
0
a
8
=
0
−1
a
7
=
1
−1
a
3
=
0
1
a
9
=
1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
10
u
c
2
, c
3
, c
9
u + 1
c
6
, c
7
, c
8
u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
10
y
c
2
, c
3
, c
6
c
7
, c
8
, c
9
y −1
17
(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
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
10
u(u
2
− 2)(u
2
+ u − 1)
3
(u
19
− 2u
18
+ ··· − 2u
2
+ 2)
c
2
((u − 1)
2
)(u + 1)(u
6
− 2u
4
+ ··· + u − 1)(u
19
+ 2u
18
+ ··· + 5u − 1)
c
3
((u − 1)
2
)(u + 1)(u
6
− 2u
4
+ ··· + u − 1)(u
19
− 2u
18
+ ··· − 7u − 1)
c
6
, c
7
(u − 1)(u + 1)
2
(u
6
− 2u
4
+ ··· + u − 1)(u
19
− 2u
18
+ ··· − 7u − 1)
c
8
(u − 1)(u + 1)
2
(u
6
− 2u
4
+ ··· + u − 1)(u
19
+ 2u
18
+ ··· + 5u − 1)
c
9
(u + 1)
3
(u
6
+ 4u
5
+ 6u
4
+ 7u
3
+ 7u
2
+ 3u + 1)
· (u
19
+ 6u
18
+ ··· + 29u + 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
10
y(y −2)
2
(y
2
− 3y + 1)
3
(y
19
− 22y
18
+ ··· + 8y −4)
c
2
, c
8
(y −1)
3
(y
6
− 4y
5
+ 6y
4
− 7y
3
+ 7y
2
− 3y + 1)
· (y
19
− 6y
18
+ ··· + 29y −1)
c
3
, c
6
, c
7
(y −1)
3
(y
6
− 4y
5
+ 6y
4
− 7y
3
+ 7y
2
− 3y + 1)
· (y
19
− 22y
18
+ ··· + 45y −1)
c
9
(y −1)
3
(y
6
− 4y
5
− 6y
4
+ 13y
3
+ 19y
2
+ 5y + 1)
· (y
19
+ 18y
18
+ ··· + 429y −1)
20
