10
59
(K10a
2
)
A knot diagram
1
Linearized knot diagam
3 9 1 8 4 10 5 6 2 7
Solving Sequence
4,8
5 6
1,9
3 2 7 10
c
4
c
5
c
8
c
3
c
2
c
7
c
10
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
35
− 4u
34
+ ··· + 2b − 4, −6u
35
+ 17u
34
+ ··· + 2a + 7, u
36
− 3u
35
+ ··· − 2u + 1i
I
u
2
= hb + u, a + 1, u
2
+ u + 1i
I
u
3
= hu
2
+ b, −u
2
+ a − 1, u
5
+ u
3
+ u − 1i
I
u
4
= hb − u − 1, a + u, u
2
+ u + 1i
* 4 irreducible components of dim
C
= 0, with total 45 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
35
−4u
34
+· · ·+2b−4, −6u
35
+17u
34
+· · ·+2a+7, u
36
−3u
35
+· · ·−2u+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
1
=
3u
35
−
17
2
u
34
+ ··· +
13
2
u −
7
2
−
1
2
u
35
+ 2u
34
+ ··· −
3
2
u + 2
a
9
=
−u
5
− 2u
3
− u
u
5
+ u
3
+ u
a
3
=
−
1
2
u
34
+ u
33
+ ··· −
1
2
u −
1
2
1
2
u
35
− u
34
+ ··· −
1
2
u
2
+
3
2
u
a
2
=
1
2
u
35
−
3
2
u
34
+ ··· −
3
2
u
2
−
1
2
1
2
u
35
− u
34
+ ··· −
1
2
u
2
+
3
2
u
a
7
=
−u
u
3
+ u
a
10
=
2u
35
−
9
2
u
34
+ ··· +
7
2
u −
5
2
−
3
2
u
35
+ 4u
34
+ ··· −
3
2
u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
13
2
u
35
+ 18u
34
+ ··· −
11
2
u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
36
− 11u
35
+ ··· − 4u + 1
c
2
, c
9
u
36
− 3u
35
+ ··· − 4u + 1
c
4
, c
7
u
36
+ 3u
35
+ ··· + 2u + 1
c
5
u
36
+ 19u
35
+ ··· + 4u + 1
c
6
, c
10
u
36
− 4u
35
+ ··· − 48u + 16
c
8
u
36
− 3u
35
+ ··· − 26u + 17
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
y
36
+ 31y
35
+ ··· + 196y + 1
c
2
, c
9
y
36
+ 11y
35
+ ··· + 4y + 1
c
4
, c
7
y
36
+ 19y
35
+ ··· + 4y + 1
c
5
y
36
− y
35
+ ··· − 12y + 1
c
6
, c
10
y
36
− 20y
35
+ ··· − 128y + 256
c
8
y
36
− 21y
35
+ ··· + 5682y + 289
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.387195 + 0.859809I
a = 0.864957 + 0.109414I
b = −0.0189081 + 0.0958332I
−0.34130 − 1.65777I −2.55644 + 4.36495I
u = −0.387195 − 0.859809I
a = 0.864957 − 0.109414I
b = −0.0189081 − 0.0958332I
−0.34130 + 1.65777I −2.55644 − 4.36495I
u = −0.729583 + 0.777572I
a = −1.03050 + 1.01725I
b = 0.317863 − 1.274650I
−1.45237 − 5.42060I −4.83818 + 6.67480I
u = −0.729583 − 0.777572I
a = −1.03050 − 1.01725I
b = 0.317863 + 1.274650I
−1.45237 + 5.42060I −4.83818 − 6.67480I
u = 0.859716 + 0.267248I
a = −0.901846 + 1.060320I
b = 0.44242 − 1.51885I
−4.44713 − 8.11971I −3.47630 + 5.34748I
u = 0.859716 − 0.267248I
a = −0.901846 − 1.060320I
b = 0.44242 + 1.51885I
−4.44713 + 8.11971I −3.47630 − 5.34748I
u = 0.849597 + 0.216556I
a = 0.853721 − 0.852880I
b = −0.173950 + 1.239740I
−5.28539 − 2.14662I −5.02569 + 0.44253I
u = 0.849597 − 0.216556I
a = 0.853721 + 0.852880I
b = −0.173950 − 1.239740I
−5.28539 + 2.14662I −5.02569 − 0.44253I
u = −0.551728 + 0.987777I
a = −0.415426 − 1.128760I
b = 0.676359 − 0.193589I
1.29309 − 3.12534I 1.43285 + 2.16786I
u = −0.551728 − 0.987777I
a = −0.415426 + 1.128760I
b = 0.676359 + 0.193589I
1.29309 + 3.12534I 1.43285 − 2.16786I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.587634 + 0.555787I
a = −0.996846 + 0.434125I
b = 0.801082 − 0.030150I
2.54802 − 1.41982I 3.82315 + 3.52465I
u = −0.587634 − 0.555787I
a = −0.996846 − 0.434125I
b = 0.801082 + 0.030150I
2.54802 + 1.41982I 3.82315 − 3.52465I
u = −0.424101 + 1.130320I
a = 1.47023 + 1.05531I
b = 0.079663 + 1.259790I
−4.09621 − 1.05243I −6.63369 + 0.71979I
u = −0.424101 − 1.130320I
a = 1.47023 − 1.05531I
b = 0.079663 − 1.259790I
−4.09621 + 1.05243I −6.63369 − 0.71979I
u = 0.515700 + 1.111390I
a = −1.30481 + 0.95936I
b = 1.199870 + 0.507968I
−0.65138 + 7.27213I −2.75984 − 7.42786I
u = 0.515700 − 1.111390I
a = −1.30481 − 0.95936I
b = 1.199870 − 0.507968I
−0.65138 − 7.27213I −2.75984 + 7.42786I
u = 0.445924 + 1.144390I
a = 0.761539 − 0.451840I
b = −0.595300 + 0.157343I
−4.74271 + 4.00295I −9.23293 − 4.01986I
u = 0.445924 − 1.144390I
a = 0.761539 + 0.451840I
b = −0.595300 − 0.157343I
−4.74271 − 4.00295I −9.23293 + 4.01986I
u = −0.471044 + 1.134590I
a = −1.37055 − 1.28746I
b = 0.30745 − 1.38445I
−3.75883 − 6.78157I −5.67848 + 6.13587I
u = −0.471044 − 1.134590I
a = −1.37055 + 1.28746I
b = 0.30745 + 1.38445I
−3.75883 + 6.78157I −5.67848 − 6.13587I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.024315 + 0.768307I
a = 1.47731 − 0.09368I
b = 0.090892 − 0.615872I
−1.10226 − 1.38558I −6.76484 + 3.92520I
u = −0.024315 − 0.768307I
a = 1.47731 + 0.09368I
b = 0.090892 + 0.615872I
−1.10226 + 1.38558I −6.76484 − 3.92520I
u = 0.264173 + 1.234870I
a = 0.373926 − 0.622707I
b = 0.32517 − 1.56431I
−9.31556 − 4.56904I −8.72559 + 2.82656I
u = 0.264173 − 1.234870I
a = 0.373926 + 0.622707I
b = 0.32517 + 1.56431I
−9.31556 + 4.56904I −8.72559 − 2.82656I
u = 0.304638 + 1.233790I
a = −0.105861 + 0.569620I
b = −0.119820 + 1.383530I
−9.88952 + 1.61524I −9.56754 − 2.32735I
u = 0.304638 − 1.233790I
a = −0.105861 − 0.569620I
b = −0.119820 − 1.383530I
−9.88952 − 1.61524I −9.56754 + 2.32735I
u = 0.636789 + 0.302809I
a = −1.62143 + 0.78351I
b = 1.017930 − 0.416218I
1.68165 − 2.75426I 1.42028 + 4.13268I
u = 0.636789 − 0.302809I
a = −1.62143 − 0.78351I
b = 1.017930 + 0.416218I
1.68165 + 2.75426I 1.42028 − 4.13268I
u = 0.549648 + 1.188840I
a = 1.80307 − 0.33069I
b = −0.278564 − 1.254500I
−8.19229 + 7.27945I −7.73458 − 3.93070I
u = 0.549648 − 1.188840I
a = 1.80307 + 0.33069I
b = −0.278564 + 1.254500I
−8.19229 − 7.27945I −7.73458 + 3.93070I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.572738 + 1.180430I
a = −1.98873 + 0.46037I
b = 0.48742 + 1.57860I
−7.1856 + 13.3899I −6.02318 − 8.65555I
u = 0.572738 − 1.180430I
a = −1.98873 − 0.46037I
b = 0.48742 − 1.57860I
−7.1856 − 13.3899I −6.02318 + 8.65555I
u = 0.274519 + 0.624372I
a = −2.10757 + 0.21578I
b = 0.635668 + 1.038860I
−0.04793 + 2.91691I −3.21123 − 0.65680I
u = 0.274519 − 0.624372I
a = −2.10757 − 0.21578I
b = 0.635668 − 1.038860I
−0.04793 − 2.91691I −3.21123 + 0.65680I
u = −0.597841 + 0.093585I
a = −0.261177 + 0.245146I
b = 0.304766 + 1.198650I
−0.94199 + 2.63032I −1.44776 − 2.88489I
u = −0.597841 − 0.093585I
a = −0.261177 − 0.245146I
b = 0.304766 − 1.198650I
−0.94199 − 2.63032I −1.44776 + 2.88489I
8
II. I
u
2
= hb + u, a + 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
1
=
−1
−u
a
9
=
−1
0
a
3
=
u + 1
−u − 1
a
2
=
0
−u − 1
a
7
=
−u
u + 1
a
10
=
−1
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u + 1
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
8
u
2
+ u + 1
c
3
, c
7
, c
9
u
2
− u + 1
c
6
, c
10
u
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
8
, c
9
y
2
+ y + 1
c
6
, c
10
y
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −1.00000
b = 0.500000 − 0.866025I
− 4.05977I −3.00000 + 6.92820I
u = −0.500000 − 0.866025I
a = −1.00000
b = 0.500000 + 0.866025I
4.05977I −3.00000 − 6.92820I
12
III. I
u
3
= hu
2
+ b, −u
2
+ a − 1, u
5
+ u
3
+ u − 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
1
=
u
2
+ 1
−u
2
a
9
=
−u
3
− 1
1
a
3
=
−u
4
− u
2
+ 1
u
4
a
2
=
−u
2
+ u + 1
u
4
− u
a
7
=
−u
u
3
+ u
a
10
=
u
2
+ u + 1
−u
3
− u
2
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
5
− 2u
4
+ 3u
3
− 2u
2
+ u + 1
c
2
, c
4
, c
7
c
9
u
5
+ u
3
+ u + 1
c
5
u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ u − 1
c
6
, c
10
(u + 1)
5
c
8
u
5
+ u
3
− 2u
2
− u + 2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
y
5
+ 2y
4
+ 3y
3
+ 6y
2
+ 5y − 1
c
2
, c
4
, c
7
c
9
y
5
+ 2y
4
+ 3y
3
+ 2y
2
+ y − 1
c
6
, c
10
(y − 1)
5
c
8
y
5
+ 2y
4
− y
3
− 6y
2
+ 9y − 4
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.707729 + 0.841955I
a = 0.79199 − 1.19175I
b = 0.208008 + 1.191750I
−1.64493 −6.00000
u = −0.707729 − 0.841955I
a = 0.79199 + 1.19175I
b = 0.208008 − 1.191750I
−1.64493 −6.00000
u = 0.389287 + 1.070680I
a = 0.005198 + 0.833601I
b = 0.994802 − 0.833601I
−1.64493 −6.00000
u = 0.389287 − 1.070680I
a = 0.005198 − 0.833601I
b = 0.994802 + 0.833601I
−1.64493 −6.00000
u = 0.636883
a = 1.40562
b = −0.405620
−1.64493 −6.00000
16
IV. I
u
4
= hb − u − 1, a + u, 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
1
=
−u
u + 1
a
9
=
−1
0
a
3
=
2
u
a
2
=
u + 2
u
a
7
=
−u
u + 1
a
10
=
−u
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
8
u
2
+ u + 1
c
3
, c
7
, c
9
u
2
− u + 1
c
6
, c
10
u
2
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
8
, c
9
y
2
+ y + 1
c
6
, c
10
y
2
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.500000 − 0.866025I
b = 0.500000 + 0.866025I
0 0
u = −0.500000 − 0.866025I
a = 0.500000 + 0.866025I
b = 0.500000 − 0.866025I
0 0
20
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
2
)(u
5
− 2u
4
+ ··· + u + 1)(u
36
− 11u
35
+ ··· − 4u + 1)
c
2
((u
2
+ u + 1)
2
)(u
5
+ u
3
+ u + 1)(u
36
− 3u
35
+ ··· − 4u + 1)
c
3
((u
2
− u + 1)
2
)(u
5
− 2u
4
+ ··· + u + 1)(u
36
− 11u
35
+ ··· − 4u + 1)
c
4
((u
2
+ u + 1)
2
)(u
5
+ u
3
+ u + 1)(u
36
+ 3u
35
+ ··· + 2u + 1)
c
5
((u
2
+ u + 1)
2
)(u
5
+ 2u
4
+ ··· + u − 1)(u
36
+ 19u
35
+ ··· + 4u + 1)
c
6
, c
10
u
4
(u + 1)
5
(u
36
− 4u
35
+ ··· − 48u + 16)
c
7
((u
2
− u + 1)
2
)(u
5
+ u
3
+ u + 1)(u
36
+ 3u
35
+ ··· + 2u + 1)
c
8
((u
2
+ u + 1)
2
)(u
5
+ u
3
− 2u
2
− u + 2)(u
36
− 3u
35
+ ··· − 26u + 17)
c
9
((u
2
− u + 1)
2
)(u
5
+ u
3
+ u + 1)(u
36
− 3u
35
+ ··· − 4u + 1)
21
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
(y
2
+ y + 1)
2
(y
5
+ 2y
4
+ 3y
3
+ 6y
2
+ 5y − 1)
· (y
36
+ 31y
35
+ ··· + 196y + 1)
c
2
, c
9
((y
2
+ y + 1)
2
)(y
5
+ 2y
4
+ ··· + y − 1)(y
36
+ 11y
35
+ ··· + 4y + 1)
c
4
, c
7
((y
2
+ y + 1)
2
)(y
5
+ 2y
4
+ ··· + y − 1)(y
36
+ 19y
35
+ ··· + 4y + 1)
c
5
((y
2
+ y + 1)
2
)(y
5
+ 2y
4
+ ··· + 5y − 1)(y
36
− y
35
+ ··· − 12y + 1)
c
6
, c
10
y
4
(y − 1)
5
(y
36
− 20y
35
+ ··· − 128y + 256)
c
8
(y
2
+ y + 1)
2
(y
5
+ 2y
4
− y
3
− 6y
2
+ 9y − 4)
· (y
36
− 21y
35
+ ··· + 5682y + 289)
22
