10
158
(K10n
41
)
A knot diagram
1
Linearized knot diagam
4 5 10 8 3 1 5 1 4 6
Solving Sequence
6,10
1
4,7
3 5 2 9 8
c
10
c
6
c
3
c
5
c
2
c
9
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h17u
11
+ 20u
10
− 54u
9
− 54u
8
+ 116u
7
+ 92u
6
− 101u
5
− 48u
4
− u
3
+ 16u
2
+ 37b + 5u − 10,
− 17u
11
− 20u
10
+ 54u
9
+ 54u
8
− 116u
7
− 92u
6
+ 101u
5
+ 48u
4
+ u
3
− 16u
2
+ 37a − 5u − 27,
u
12
+ u
11
− 3u
10
− 3u
9
+ 7u
8
+ 6u
7
− 6u
6
− 6u
5
+ 3u
4
+ 3u
3
+ 1i
I
u
2
= h2201978u
15
− 5194678u
14
+ ··· + 19920857b + 76411393,
− 7295235u
15
+ 80490581u
14
+ ··· + 378496283a − 1195533446, u
16
− u
15
+ ··· + 2u + 19i
I
u
3
= h−u
4
+ u
3
+ u
2
+ b − u, u
4
− u
3
− u
2
+ a + u + 1, u
6
− u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1i
* 3 irreducible components of dim
C
= 0, with total 34 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
= h17u
11
+ 20u
10
+ · · · + 37b − 10, −17u
11
− 20u
10
+ · · · + 37a −
27, u
12
+ u
11
+ · · · + 3u
3
+ 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
1
=
1
−u
2
a
4
=
0.459459u
11
+ 0.540541u
10
+ ··· + 0.135135u + 0.729730
−0.459459u
11
− 0.540541u
10
+ ··· − 0.135135u + 0.270270
a
7
=
u
−u
3
+ u
a
3
=
1
−0.459459u
11
− 0.540541u
10
+ ··· − 0.135135u + 0.270270
a
5
=
−u
0.0810811u
11
− 0.0810811u
10
+ ··· + 0.729730u − 0.459459
a
2
=
−u
2
+ 1
−0.621622u
11
− 0.378378u
10
+ ··· − 0.594595u + 0.189189
a
9
=
−0.945946u
11
− 0.0540541u
10
+ ··· + 0.486486u + 0.0270270
1.40541u
11
+ 0.594595u
10
+ ··· − 0.351351u + 0.702703
a
8
=
−0.324324u
11
+ 0.324324u
10
+ ··· + 1.08108u − 0.162162
0.918919u
11
+ 0.0810811u
10
+ ··· + 0.270270u + 0.459459
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
50
37
u
11
−
135
37
u
10
+
87
37
u
9
+
383
37
u
8
−
191
37
u
7
−
843
37
u
6
+
25
37
u
5
+
657
37
u
4
+
127
37
u
3
−
404
37
u
2
−
117
37
u−
25
37
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
12
− u
11
+ ··· − 3u + 1
c
2
, c
5
, c
6
c
10
u
12
+ u
11
− 3u
10
− 3u
9
+ 7u
8
+ 6u
7
− 6u
6
− 6u
5
+ 3u
4
+ 3u
3
+ 1
c
3
, c
9
u
12
+ 9u
11
+ ··· + 96u + 16
c
4
, c
7
u
12
− 6u
11
+ ··· − 10u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
12
− 15y
11
+ ··· + 7y + 1
c
2
, c
5
, c
6
c
10
y
12
− 7y
11
+ ··· + 6y
2
+ 1
c
3
, c
9
y
12
+ 5y
11
+ ··· + 896y + 256
c
4
, c
7
y
12
+ 6y
11
+ ··· + 20y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.932110 + 0.403591I
a = 0.86095 − 1.68780I
b = 0.13905 + 1.68780I
7.74885 − 1.69313I −0.90926 + 4.65688I
u = −0.932110 − 0.403591I
a = 0.86095 + 1.68780I
b = 0.13905 − 1.68780I
7.74885 + 1.69313I −0.90926 − 4.65688I
u = 0.964469 + 0.359565I
a = −0.466137 − 0.540935I
b = 1.46614 + 0.54093I
−1.16607 + 4.31349I 1.88826 − 4.73148I
u = 0.964469 − 0.359565I
a = −0.466137 + 0.540935I
b = 1.46614 − 0.54093I
−1.16607 − 4.31349I 1.88826 + 4.73148I
u = −0.581296 + 0.573734I
a = −0.118591 + 0.442092I
b = 1.118590 − 0.442092I
−3.10204 + 1.08202I −2.61157 + 1.33940I
u = −0.581296 − 0.573734I
a = −0.118591 − 0.442092I
b = 1.118590 + 0.442092I
−3.10204 − 1.08202I −2.61157 − 1.33940I
u = 1.157820 + 0.740786I
a = 0.319275 + 1.332450I
b = 0.68073 − 1.33245I
−0.09105 + 5.46102I −0.77116 − 3.85424I
u = 1.157820 − 0.740786I
a = 0.319275 − 1.332450I
b = 0.68073 + 1.33245I
−0.09105 − 5.46102I −0.77116 + 3.85424I
u = 0.256008 + 0.492477I
a = 0.735049 + 0.459069I
b = 0.264951 − 0.459069I
−0.090701 + 1.098140I −1.42722 − 6.18957I
u = 0.256008 − 0.492477I
a = 0.735049 − 0.459069I
b = 0.264951 + 0.459069I
−0.090701 − 1.098140I −1.42722 + 6.18957I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.36489 + 0.70235I
a = 0.169451 − 1.349320I
b = 0.83055 + 1.34932I
1.63582 − 12.27120I 1.33095 + 7.21681I
u = −1.36489 − 0.70235I
a = 0.169451 + 1.349320I
b = 0.83055 − 1.34932I
1.63582 + 12.27120I 1.33095 − 7.21681I
6
II.
I
u
2
= h2.20 × 10
6
u
15
− 5.19 × 10
6
u
14
+ · · · + 1.99 × 10
7
b + 7.64 × 10
7
, −7.30 ×
10
6
u
15
+8.05×10
7
u
14
+· · ·+3.78×10
8
a−1.20×10
9
, u
16
−u
15
+· · ·+2u +19i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
1
=
1
−u
2
a
4
=
0.0192743u
15
− 0.212659u
14
+ ··· + 0.532412u + 3.15864
−0.110536u
15
+ 0.260766u
14
+ ··· + 2.55924u − 3.83575
a
7
=
u
−u
3
+ u
a
3
=
−0.0912621u
15
+ 0.0481069u
14
+ ··· + 3.09165u − 0.677108
−0.110536u
15
+ 0.260766u
14
+ ··· + 2.55924u − 3.83575
a
5
=
−0.0769980u
15
+ 0.306127u
14
+ ··· − 1.09454u + 0.395782
0.245201u
15
− 0.534471u
14
+ ··· − 4.85223u + 6.14044
a
2
=
−0.489701u
15
+ 0.863952u
14
+ ··· + 6.26646u − 3.26339
0.294483u
15
− 0.508467u
14
+ ··· − 6.40955u + 4.64021
a
9
=
−0.212645u
15
+ 0.307616u
14
+ ··· + 2.68411u − 2.66285
−0.110536u
15
+ 0.260766u
14
+ ··· + 2.55924u − 2.83575
a
8
=
−0.507128u
15
+ 0.816083u
14
+ ··· + 9.09366u − 7.30305
0.0690837u
15
− 0.0661749u
14
+ ··· − 2.60797u + 1.22994
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
187108568
378496283
u
15
+
381114024
378496283
u
14
+ ··· +
1298769712
378496283
u −
136693666
19920857
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
16
− 3u
15
+ ··· − 10u + 1
c
2
, c
5
, c
6
c
10
u
16
− u
15
+ ··· + 2u + 19
c
3
, c
9
(u
2
− u + 1)
8
c
4
, c
7
(u
4
+ u
3
+ u
2
+ 1)
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
16
− 5y
15
+ ··· + 88y + 1
c
2
, c
5
, c
6
c
10
y
16
− 9y
15
+ ··· − 1980y + 361
c
3
, c
9
(y
2
+ y + 1)
8
c
4
, c
7
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.921978 + 0.154671I
a = −1.18718 + 0.84702I
b = −0.500000 − 0.866025I
5.14581 − 0.61478I 3.82674 − 1.44464I
u = −0.921978 − 0.154671I
a = −1.18718 − 0.84702I
b = −0.500000 + 0.866025I
5.14581 + 0.61478I 3.82674 + 1.44464I
u = −1.000120 + 0.458209I
a = 0.23948 + 2.07179I
b = −0.500000 − 0.866025I
−1.85594 − 5.19385I 0.17326 + 6.02890I
u = −1.000120 − 0.458209I
a = 0.23948 − 2.07179I
b = −0.500000 + 0.866025I
−1.85594 + 5.19385I 0.17326 − 6.02890I
u = 0.740779 + 0.385723I
a = 0.60451 − 2.36642I
b = −0.500000 + 0.866025I
−1.85594 − 1.13408I 0.173262 − 0.899303I
u = 0.740779 − 0.385723I
a = 0.60451 + 2.36642I
b = −0.500000 − 0.866025I
−1.85594 + 1.13408I 0.173262 + 0.899303I
u = 0.656157 + 1.071140I
a = 0.546203 + 0.202750I
b = −0.500000 − 0.866025I
−1.85594 + 1.13408I 0.173262 + 0.899303I
u = 0.656157 − 1.071140I
a = 0.546203 − 0.202750I
b = −0.500000 + 0.866025I
−1.85594 − 1.13408I 0.173262 − 0.899303I
u = −0.291942 + 1.325280I
a = 0.328801 − 0.073667I
b = −0.500000 + 0.866025I
−1.85594 + 5.19385I 0.17326 − 6.02890I
u = −0.291942 − 1.325280I
a = 0.328801 + 0.073667I
b = −0.500000 − 0.866025I
−1.85594 − 5.19385I 0.17326 + 6.02890I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.126160 + 0.776883I
a = −0.398018 − 0.407492I
b = −0.500000 + 0.866025I
5.14581 + 3.44499I 3.82674 − 8.37284I
u = 1.126160 − 0.776883I
a = −0.398018 + 0.407492I
b = −0.500000 − 0.866025I
5.14581 − 3.44499I 3.82674 + 8.37284I
u = −1.367540 + 0.181274I
a = −0.383277 + 1.317030I
b = −0.500000 − 0.866025I
5.14581 − 3.44499I 3.82674 + 8.37284I
u = −1.367540 − 0.181274I
a = −0.383277 − 1.317030I
b = −0.500000 + 0.866025I
5.14581 + 3.44499I 3.82674 − 8.37284I
u = 1.55848 + 0.24344I
a = −0.092631 − 0.872701I
b = −0.500000 + 0.866025I
5.14581 + 0.61478I 3.82674 + 1.44464I
u = 1.55848 − 0.24344I
a = −0.092631 + 0.872701I
b = −0.500000 − 0.866025I
5.14581 − 0.61478I 3.82674 − 1.44464I
11
III.
I
u
3
= h−u
4
+u
3
+u
2
+b−u, u
4
−u
3
−u
2
+a+u+1, u
6
−u
5
−2u
4
+2u
3
+u
2
−u+1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
1
=
1
−u
2
a
4
=
−u
4
+ u
3
+ u
2
− u − 1
u
4
− u
3
− u
2
+ u
a
7
=
u
−u
3
+ u
a
3
=
−1
u
4
− u
3
− u
2
+ u
a
5
=
−u
u
5
− u
4
− u
3
+ u
2
+ u
a
2
=
u
2
− 1
−u
2
+ 1
a
9
=
u
4
− u
3
− 2u
2
+ 2u + 1
u
2
− u
a
8
=
u
4
− u
3
− u
2
+ 2u
−u
4
+ 2u
2
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
5
− u
4
+ 6u
3
+ 3u
2
− 7u + 3
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
6
− u
5
− 2u
3
+ 2u
2
+ 1
c
2
, c
6
u
6
+ u
5
− 2u
4
− 2u
3
+ u
2
+ u + 1
c
3
u
6
+ 2u
4
+ 2u
3
+ u + 1
c
4
u
6
− u
5
+ 3u
4
− 2u
3
+ 3u
2
+ 1
c
5
, c
10
u
6
− u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1
c
7
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 3u
2
+ 1
c
9
u
6
+ 2u
4
− 2u
3
− u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
6
− y
5
− 2y
3
+ 4y
2
+ 4y + 1
c
2
, c
5
, c
6
c
10
y
6
− 5y
5
+ 10y
4
− 8y
3
+ y
2
+ y + 1
c
3
, c
9
y
6
+ 4y
5
+ 4y
4
− 2y
3
− y + 1
c
4
, c
7
y
6
+ 5y
5
+ 11y
4
+ 16y
3
+ 15y
2
+ 6y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.099190 + 0.287563I
a = −0.69782 + 1.52185I
b = −0.30218 − 1.52185I
8.54916 − 1.24964I 7.95941 − 0.00232I
u = −1.099190 − 0.287563I
a = −0.69782 − 1.52185I
b = −0.30218 + 1.52185I
8.54916 + 1.24964I 7.95941 + 0.00232I
u = 0.264925 + 0.576623I
a = −1.74836 − 0.18113I
b = 0.748359 + 0.181129I
−2.37427 + 2.84527I −1.26269 − 3.26816I
u = 0.264925 − 0.576623I
a = −1.74836 + 0.18113I
b = 0.748359 − 0.181129I
−2.37427 − 2.84527I −1.26269 + 3.26816I
u = 1.334260 + 0.378781I
a = −0.553818 − 0.708238I
b = −0.446182 + 0.708238I
5.33965 + 2.32699I 5.80328 − 1.20156I
u = 1.334260 − 0.378781I
a = −0.553818 + 0.708238I
b = −0.446182 − 0.708238I
5.33965 − 2.32699I 5.80328 + 1.20156I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
8
(u
6
− u
5
− 2u
3
+ 2u
2
+ 1)(u
12
− u
11
+ ··· − 3u + 1)
· (u
16
− 3u
15
+ ··· − 10u + 1)
c
2
, c
6
(u
6
+ u
5
− 2u
4
− 2u
3
+ u
2
+ u + 1)
· (u
12
+ u
11
− 3u
10
− 3u
9
+ 7u
8
+ 6u
7
− 6u
6
− 6u
5
+ 3u
4
+ 3u
3
+ 1)
· (u
16
− u
15
+ ··· + 2u + 19)
c
3
((u
2
− u + 1)
8
)(u
6
+ 2u
4
+ 2u
3
+ u + 1)(u
12
+ 9u
11
+ ··· + 96u + 16)
c
4
(u
4
+ u
3
+ u
2
+ 1)
4
(u
6
− u
5
+ 3u
4
− 2u
3
+ 3u
2
+ 1)
· (u
12
− 6u
11
+ ··· − 10u + 4)
c
5
, c
10
(u
6
− u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1)
· (u
12
+ u
11
− 3u
10
− 3u
9
+ 7u
8
+ 6u
7
− 6u
6
− 6u
5
+ 3u
4
+ 3u
3
+ 1)
· (u
16
− u
15
+ ··· + 2u + 19)
c
7
(u
4
+ u
3
+ u
2
+ 1)
4
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 3u
2
+ 1)
· (u
12
− 6u
11
+ ··· − 10u + 4)
c
9
((u
2
− u + 1)
8
)(u
6
+ 2u
4
− 2u
3
− u + 1)(u
12
+ 9u
11
+ ··· + 96u + 16)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y
6
− y
5
− 2y
3
+ 4y
2
+ 4y + 1)(y
12
− 15y
11
+ ··· + 7y + 1)
· (y
16
− 5y
15
+ ··· + 88y + 1)
c
2
, c
5
, c
6
c
10
(y
6
− 5y
5
+ 10y
4
− 8y
3
+ y
2
+ y + 1)(y
12
− 7y
11
+ ··· + 6y
2
+ 1)
· (y
16
− 9y
15
+ ··· − 1980y + 361)
c
3
, c
9
(y
2
+ y + 1)
8
(y
6
+ 4y
5
+ 4y
4
− 2y
3
− y + 1)
· (y
12
+ 5y
11
+ ··· + 896y + 256)
c
4
, c
7
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
4
(y
6
+ 5y
5
+ 11y
4
+ 16y
3
+ 15y
2
+ 6y + 1)
· (y
12
+ 6y
11
+ ··· + 20y + 16)
17
