10
40
(K10a
30
)
A knot diagram
1
Linearized knot diagam
6 9 10 7 1 5 3 2 8 4
Solving Sequence
2,9
3 8 10 4 7 5 6 1
c
2
c
8
c
9
c
3
c
7
c
4
c
6
c
1
c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
32
+ u
31
+ ··· − u
2
+ 1i
I
u
2
= hu
4
+ u
3
+ 1i
I
u
3
= hu − 1i
* 3 irreducible components of dim
C
= 0, with total 37 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
32
+ u
31
+ · · · − u
2
+ 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
8
=
u
u
a
10
=
−u
3
−u
3
+ u
a
4
=
u
8
− u
6
+ u
4
+ 1
u
8
− 2u
6
+ 2u
4
a
7
=
u
3
u
5
− u
3
+ u
a
5
=
−u
16
+ 3u
14
− 5u
12
+ 4u
10
− u
8
+ 1
−u
18
+ 4u
16
− 9u
14
+ 12u
12
− 11u
10
+ 8u
8
− 6u
6
+ 4u
4
− u
2
a
6
=
u
29
− 6u
27
+ ··· + 2u
3
− u
u
31
− 7u
29
+ ··· − 4u
5
+ u
a
1
=
u
13
− 2u
11
+ 3u
9
− 2u
7
+ 2u
5
− 2u
3
+ u
u
13
− 3u
11
+ 5u
9
− 4u
7
+ 2u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
31
+ 32u
29
+ 4u
28
− 128u
27
− 28u
26
+ 324u
25
+ 104u
24
− 564u
23
− 248u
22
+
696u
21
+ 412u
20
− 616u
19
− 484u
18
+ 404u
17
+ 400u
16
− 228u
15
− 232u
14
+ 136u
13
+
112u
12
− 68u
11
− 68u
10
+ 4u
9
+ 32u
8
+ 20u
7
+ 12u
6
− 16u
5
− 20u
4
+ 8u
3
+ 4u
2
+ 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
32
+ u
31
+ ··· + 2u + 1
c
2
, c
8
u
32
+ u
31
+ ··· − u
2
+ 1
c
3
, c
10
u
32
+ 4u
31
+ ··· + 28u + 4
c
4
, c
6
u
32
− 11u
31
+ ··· − 2u + 1
c
7
u
32
+ 3u
31
+ ··· + 2u + 3
c
9
u
32
+ 15u
31
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
32
− 11y
31
+ ··· − 2y + 1
c
2
, c
8
y
32
− 15y
31
+ ··· − 2y + 1
c
3
, c
10
y
32
− 20y
31
+ ··· − 184y + 16
c
4
, c
6
y
32
+ 21y
31
+ ··· + 2y + 1
c
7
y
32
+ 5y
31
+ ··· + 164y + 9
c
9
y
32
+ 5y
31
+ ··· + 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.961241 + 0.329628I
−1.64326 + 1.19641I −1.57525 − 0.85209I
u = −0.961241 − 0.329628I
−1.64326 − 1.19641I −1.57525 + 0.85209I
u = 0.934575 + 0.495071I
0.10900 − 4.15286I 6.01286 + 7.18864I
u = 0.934575 − 0.495071I
0.10900 + 4.15286I 6.01286 − 7.18864I
u = −1.077140 + 0.188783I
−3.63561 − 0.05779I −1.67435 − 0.61686I
u = −1.077140 − 0.188783I
−3.63561 + 0.05779I −1.67435 + 0.61686I
u = −0.550946 + 0.717103I
3.03384 + 5.05352I 8.11469 − 5.31459I
u = −0.550946 − 0.717103I
3.03384 − 5.05352I 8.11469 + 5.31459I
u = 1.099030 + 0.150244I
−2.66422 + 5.49753I 0.37719 − 4.60034I
u = 1.099030 − 0.150244I
−2.66422 − 5.49753I 0.37719 + 4.60034I
u = −0.473676 + 0.749403I
6.73005 − 1.36697I 11.90065 + 0.55023I
u = −0.473676 − 0.749403I
6.73005 + 1.36697I 11.90065 − 0.55023I
u = −0.407410 + 0.774508I
2.26376 − 7.72193I 6.98438 + 5.32873I
u = −0.407410 − 0.774508I
2.26376 + 7.72193I 6.98438 − 5.32873I
u = 0.399421 + 0.743579I
0.98960 + 2.26361I 5.01894 − 0.67006I
u = 0.399421 − 0.743579I
0.98960 − 2.26361I 5.01894 + 0.67006I
u = −1.104760 + 0.408512I
−5.70053 + 0.95663I −2.35494 − 0.97622I
u = −1.104760 − 0.408512I
−5.70053 − 0.95663I −2.35494 + 0.97622I
u = 1.041040 + 0.566496I
0.08923 − 4.79464I 2.70911 + 5.61871I
u = 1.041040 − 0.566496I
0.08923 + 4.79464I 2.70911 − 5.61871I
u = 1.108350 + 0.436864I
−5.50827 − 6.53878I −1.61404 + 6.99151I
u = 1.108350 − 0.436864I
−5.50827 + 6.53878I −1.61404 − 6.99151I
u = −1.070770 + 0.603221I
4.95901 + 6.50568I 8.96918 − 5.51070I
u = −1.070770 − 0.603221I
4.95901 − 6.50568I 8.96918 + 5.51070I
u = 1.099670 + 0.582909I
−1.06972 − 7.30693I 1.82356 + 4.86883I
u = 1.099670 − 0.582909I
−1.06972 + 7.30693I 1.82356 − 4.86883I
u = −1.105460 + 0.595316I
0.19628 + 12.88870I 3.87677 − 9.41526I
u = −1.105460 − 0.595316I
0.19628 − 12.88870I 3.87677 + 9.41526I
u = 0.527868 + 0.394454I
1.169210 + 0.193186I 9.20830 − 0.78328I
u = 0.527868 − 0.394454I
1.169210 − 0.193186I 9.20830 + 0.78328I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.041447 + 0.613996I
−2.60826 + 2.66625I 2.22295 − 3.31297I
u = 0.041447 − 0.613996I
−2.60826 − 2.66625I 2.22295 + 3.31297I
6
II. I
u
2
= hu
4
+ u
3
+ 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
8
=
u
u
a
10
=
−u
3
−u
3
+ u
a
4
=
u
3
+ 1
u
3
+ u
2
− u
a
7
=
u
3
1
a
5
=
u
u
3
− u
a
6
=
0
u
a
1
=
1
u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
8
u
4
+ u
3
+ 1
c
3
, c
10
(u − 1)
4
c
4
, c
6
u
4
− u
3
+ 2u
2
+ 1
c
7
u
4
− u
2
− 2u + 3
c
9
u
4
+ u
3
+ 2u
2
+ 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
8
y
4
− y
3
+ 2y
2
+ 1
c
3
, c
10
(y −1)
4
c
4
, c
6
, c
9
y
4
+ 3y
3
+ 6y
2
+ 4y + 1
c
7
y
4
− 2y
3
+ 7y
2
− 10y + 9
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.518913 + 0.666610I
1.64493 6.00000
u = 0.518913 − 0.666610I
1.64493 6.00000
u = −1.018910 + 0.602565I
1.64493 6.00000
u = −1.018910 − 0.602565I
1.64493 6.00000
10
III. I
u
3
= hu − 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
1
a
3
=
1
1
a
8
=
1
1
a
10
=
−1
0
a
4
=
2
1
a
7
=
1
1
a
5
=
1
0
a
6
=
0
1
a
1
=
1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
10
u − 1
c
7
u
c
9
u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
9
, c
10
y −1
c
7
y
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
1.64493 6.00000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)(u
4
+ u
3
+ 1)(u
32
+ u
31
+ ··· + 2u + 1)
c
2
, c
8
(u − 1)(u
4
+ u
3
+ 1)(u
32
+ u
31
+ ··· − u
2
+ 1)
c
3
, c
10
((u − 1)
5
)(u
32
+ 4u
31
+ ··· + 28u + 4)
c
4
, c
6
(u − 1)(u
4
− u
3
+ 2u
2
+ 1)(u
32
− 11u
31
+ ··· − 2u + 1)
c
7
u(u
4
− u
2
− 2u + 3)(u
32
+ 3u
31
+ ··· + 2u + 3)
c
9
(u + 1)(u
4
+ u
3
+ 2u
2
+ 1)(u
32
+ 15u
31
+ ··· + 2u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y −1)(y
4
− y
3
+ 2y
2
+ 1)(y
32
− 11y
31
+ ··· − 2y + 1)
c
2
, c
8
(y −1)(y
4
− y
3
+ 2y
2
+ 1)(y
32
− 15y
31
+ ··· − 2y + 1)
c
3
, c
10
((y −1)
5
)(y
32
− 20y
31
+ ··· − 184y + 16)
c
4
, c
6
(y −1)(y
4
+ 3y
3
+ ··· + 4y + 1)(y
32
+ 21y
31
+ ··· + 2y + 1)
c
7
y(y
4
− 2y
3
+ ··· − 10y + 9)(y
32
+ 5y
31
+ ··· + 164y + 9)
c
9
(y −1)(y
4
+ 3y
3
+ ··· + 4y + 1)(y
32
+ 5y
31
+ ··· + 2y + 1)
16
