10
89
(K10a
21
)
A knot diagram
1
Linearized knot diagam
4 7 6 1 8 10 9 5 3 2
Solving Sequence
1,5
4
2,9
8 6 3 7 10
c
4
c
1
c
8
c
5
c
3
c
7
c
10
c
2
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, u
8
+ 2u
7
+ 3u
6
+ 2u
5
+ u
4
+ a − 1, u
9
+ 2u
8
+ 4u
7
+ 4u
6
+ 5u
5
+ 4u
4
+ 4u
3
+ 2u
2
+ u − 1i
I
u
2
= h−2.54158 × 10
21
u
39
+ 3.72578 × 10
21
u
38
+ ··· + 1.43109 × 10
22
b − 1.56356 × 10
21
,
− 3.70291 × 10
21
u
39
+ 3.38172 × 10
21
u
38
+ ··· + 1.43109 × 10
22
a − 2.35738 × 10
22
, u
40
− u
39
+ ··· − 4u + 1i
* 2 irreducible components of dim
C
= 0, with total 49 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
= hb + u, u
8
+ 2u
7
+ 3u
6
+ 2u
5
+ u
4
+ a − 1, u
9
+ 2u
8
+ · · · + u − 1i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
4
=
1
u
2
a
2
=
u
u
3
+ u
a
9
=
−u
8
− 2u
7
− 3u
6
− 2u
5
− u
4
+ 1
−u
a
8
=
−u
8
− 2u
7
− 3u
6
− 2u
5
− u
4
+ u + 1
−u
a
6
=
u
7
+ 2u
6
+ 4u
5
+ 4u
4
+ 4u
3
+ 3u
2
+ 2u
−u
2
a
3
=
u
7
+ 3u
6
+ 7u
5
+ 8u
4
+ 7u
3
+ 4u
2
+ u
u
7
+ u
6
+ 2u
5
+ u
4
+ u
3
a
7
=
u
6
+ 2u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ 1
−u
3
− u
a
10
=
u
3
u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
8
− 4u
7
− 8u
6
− 4u
5
− 4u
4
+ 4u
3
+ 8u
2
+ 8u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
8
u
9
+ 2u
8
+ 4u
7
+ 4u
6
+ 5u
5
+ 4u
4
+ 4u
3
+ 2u
2
+ u − 1
c
2
u
9
− 13u
8
+ ··· + 152u − 32
c
3
u
9
− 13u
8
+ ··· + 208u − 32
c
6
, c
9
u
9
− 2u
6
+ 5u
5
+ 4u
3
− 6u
2
+ 3u − 1
c
7
, c
10
u
9
+ 4u
8
+ 10u
7
+ 16u
6
+ 19u
5
+ 20u
4
+ 18u
3
+ 12u
2
+ 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
8
y
9
+ 4y
8
+ 10y
7
+ 16y
6
+ 19y
5
+ 20y
4
+ 18y
3
+ 12y
2
+ 5y −1
c
2
y
9
− 25y
8
+ ··· − 192y −1024
c
3
y
9
− 23y
8
+ ··· + 8960y −1024
c
6
, c
9
y
9
+ 10y
7
+ 4y
6
+ 31y
5
+ 16y
4
+ 42y
3
− 12y
2
− 3y −1
c
7
, c
10
y
9
+ 4y
8
+ 10y
7
− 5y
5
+ 8y
4
+ 66y
3
+ 76y
2
+ 49y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.870256 + 0.574591I
a = 1.51055 − 0.27719I
b = 0.870256 − 0.574591I
4.84938 + 3.79988I 8.45408 − 1.48636I
u = −0.870256 − 0.574591I
a = 1.51055 + 0.27719I
b = 0.870256 + 0.574591I
4.84938 − 3.79988I 8.45408 + 1.48636I
u = 0.547196 + 0.894013I
a = −4.54039 + 0.41851I
b = −0.547196 − 0.894013I
0.19748 + 4.39098I −9.5886 + 15.7654I
u = 0.547196 − 0.894013I
a = −4.54039 − 0.41851I
b = −0.547196 + 0.894013I
0.19748 − 4.39098I −9.5886 − 15.7654I
u = −0.168491 + 1.118820I
a = 1.00104 + 1.15340I
b = 0.168491 − 1.118820I
−6.19752 + 0.38154I −4.67885 − 0.54411I
u = −0.168491 − 1.118820I
a = 1.00104 − 1.15340I
b = 0.168491 + 1.118820I
−6.19752 − 0.38154I −4.67885 + 0.54411I
u = −0.695984 + 1.121930I
a = 2.05153 + 0.69357I
b = 0.695984 − 1.121930I
1.4591 − 15.5661I 3.71332 + 9.69859I
u = −0.695984 − 1.121930I
a = 2.05153 − 0.69357I
b = 0.695984 + 1.121930I
1.4591 + 15.5661I 3.71332 − 9.69859I
u = 0.375070
a = 0.954532
b = −0.375070
1.02805 10.2000
5
II.
I
u
2
= h−2.54×10
21
u
39
+3.73×10
21
u
38
+· · ·+1.43×10
22
b−1.56×10
21
, −3.70×
10
21
u
39
+3.38×10
21
u
38
+· · ·+1.43×10
22
a−2.36×10
22
, u
40
−u
39
+· · ·−4u+1i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
4
=
1
u
2
a
2
=
u
u
3
+ u
a
9
=
0.258747u
39
− 0.236304u
38
+ ··· − 1.86737u + 1.64726
0.177597u
39
− 0.260346u
38
+ ··· − 3.36976u + 0.109256
a
8
=
0.0811501u
39
+ 0.0240417u
38
+ ··· + 1.50239u + 1.53800
0.177597u
39
− 0.260346u
38
+ ··· − 3.36976u + 0.109256
a
6
=
−0.0521006u
39
− 0.686202u
38
+ ··· + 4.91261u + 0.428336
0.110686u
39
− 0.236069u
38
+ ··· − 3.29989u + 1.06601
a
3
=
1.44465u
39
− 1.15627u
38
+ ··· + 1.62073u − 1.83432
0.138363u
39
+ 0.574052u
38
+ ··· + 2.22447u + 0.383801
a
7
=
−0.00240648u
39
− 0.744206u
38
+ ··· + 4.44922u + 0.596332
0.0437786u
39
− 0.281347u
38
+ ··· − 3.16226u + 1.11303
a
10
=
u
3
u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
72408804377930288848424
14310892564212518359243
u
39
−
77968614159801719652396
14310892564212518359243
u
38
+ ··· −
8681622498642290526880
622212720183152972141
u +
76038785248810355101710
14310892564212518359243
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
8
u
40
− u
39
+ ··· − 4u + 1
c
2
(u
20
+ 6u
19
+ ··· − 2u − 1)
2
c
3
(u
20
+ 5u
19
+ ··· − 6u − 1)
2
c
6
, c
9
u
40
+ 5u
39
+ ··· + 4u + 1
c
7
, c
10
u
40
+ 15u
39
+ ··· + 120u
2
+ 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
8
y
40
+ 15y
39
+ ··· + 120y
2
+ 1
c
2
(y
20
− 16y
19
+ ··· − 16y + 1)
2
c
3
(y
20
− 7y
19
+ ··· − 2y + 1)
2
c
6
, c
9
y
40
− 5y
39
+ ··· − 8y + 1
c
7
, c
10
y
40
+ 19y
39
+ ··· + 240y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.548393 + 0.820650I
a = −1.93438 + 2.89472I
b = −0.548393 + 0.820650I
0.434649 −15.8981 + 0.I
u = 0.548393 − 0.820650I
a = −1.93438 − 2.89472I
b = −0.548393 − 0.820650I
0.434649 −15.8981 + 0.I
u = −0.632900 + 0.810710I
a = −0.713489 − 1.128410I
b = −1.003700 − 0.392952I
3.51067 − 0.70102I 13.30095 + 0.29053I
u = −0.632900 − 0.810710I
a = −0.713489 + 1.128410I
b = −1.003700 + 0.392952I
3.51067 + 0.70102I 13.30095 − 0.29053I
u = 0.602510 + 0.849943I
a = 0.252963 − 0.117129I
b = −0.232545 − 0.154995I
0.59509 + 2.36716I 1.43169 − 3.69296I
u = 0.602510 − 0.849943I
a = 0.252963 + 0.117129I
b = −0.232545 + 0.154995I
0.59509 − 2.36716I 1.43169 + 3.69296I
u = 0.378614 + 0.869397I
a = 1.02843 − 2.36154I
b = −0.378614 + 0.869397I
−0.714628 8.43291 + 0.I
u = 0.378614 − 0.869397I
a = 1.02843 + 2.36154I
b = −0.378614 − 0.869397I
−0.714628 8.43291 + 0.I
u = −0.932276 + 0.516877I
a = 1.277650 + 0.549417I
b = 0.693643 + 1.075960I
3.31734 + 9.59937I 6.13875 − 5.98964I
u = −0.932276 − 0.516877I
a = 1.277650 − 0.549417I
b = 0.693643 − 1.075960I
3.31734 − 9.59937I 6.13875 + 5.98964I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.592803 + 0.720077I
a = −0.714830 − 0.969785I
b = −0.783556 − 1.064140I
1.83047 + 2.21575I 9.27050 − 4.60917I
u = −0.592803 − 0.720077I
a = −0.714830 + 0.969785I
b = −0.783556 + 1.064140I
1.83047 − 2.21575I 9.27050 + 4.60917I
u = −0.630140 + 0.869793I
a = −1.50017 − 0.30493I
b = −1.009240 + 0.568343I
3.33020 − 4.24448I 12.4039 + 6.8707I
u = −0.630140 − 0.869793I
a = −1.50017 + 0.30493I
b = −1.009240 − 0.568343I
3.33020 + 4.24448I 12.4039 − 6.8707I
u = 1.003700 + 0.392952I
a = 1.226320 − 0.344870I
b = 0.632900 − 0.810710I
3.51067 − 0.70102I 13.30095 + 0.29053I
u = 1.003700 − 0.392952I
a = 1.226320 + 0.344870I
b = 0.632900 + 0.810710I
3.51067 + 0.70102I 13.30095 − 0.29053I
u = −0.604828 + 0.939285I
a = −2.13314 − 0.62294I
b = −0.729702 + 1.179840I
1.15558 − 6.98661I 6.87126 + 10.77467I
u = −0.604828 − 0.939285I
a = −2.13314 + 0.62294I
b = −0.729702 − 1.179840I
1.15558 + 6.98661I 6.87126 − 10.77467I
u = 0.124209 + 1.127990I
a = 0.383349 + 0.300461I
b = 0.592384 − 0.373525I
−1.87648 + 2.61466I 1.96705 − 3.93297I
u = 0.124209 − 1.127990I
a = 0.383349 − 0.300461I
b = 0.592384 + 0.373525I
−1.87648 − 2.61466I 1.96705 + 3.93297I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.402129 + 1.083400I
a = 0.297918 + 0.759573I
b = 0.116121 − 0.708920I
−1.51323 + 2.73094I 0.60746 − 4.99024I
u = 0.402129 − 1.083400I
a = 0.297918 − 0.759573I
b = 0.116121 + 0.708920I
−1.51323 − 2.73094I 0.60746 + 4.99024I
u = 1.009240 + 0.568343I
a = 1.38205 + 0.32422I
b = 0.630140 + 0.869793I
3.33020 + 4.24448I 12.4039 − 6.8707I
u = 1.009240 − 0.568343I
a = 1.38205 − 0.32422I
b = 0.630140 − 0.869793I
3.33020 − 4.24448I 12.4039 + 6.8707I
u = −0.575991 + 1.044940I
a = −1.005200 − 0.537066I
b = −0.056488 + 1.295430I
−3.62992 − 7.26942I −0.25897 + 8.20898I
u = −0.575991 − 1.044940I
a = −1.005200 + 0.537066I
b = −0.056488 − 1.295430I
−3.62992 + 7.26942I −0.25897 − 8.20898I
u = −0.693643 + 1.075960I
a = 0.527058 + 1.031180I
b = 0.932276 + 0.516877I
3.31734 − 9.59937I 6.13875 + 5.98964I
u = −0.693643 − 1.075960I
a = 0.527058 − 1.031180I
b = 0.932276 − 0.516877I
3.31734 + 9.59937I 6.13875 − 5.98964I
u = −0.116121 + 0.708920I
a = 1.021210 + 0.824545I
b = −0.402129 − 1.083400I
−1.51323 + 2.73094I 0.60746 − 4.99024I
u = −0.116121 − 0.708920I
a = 1.021210 − 0.824545I
b = −0.402129 + 1.083400I
−1.51323 − 2.73094I 0.60746 + 4.99024I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.056488 + 1.295430I
a = 0.609251 − 0.853594I
b = 0.575991 + 1.044940I
−3.62992 + 7.26942I 0. − 8.20898I
u = 0.056488 − 1.295430I
a = 0.609251 + 0.853594I
b = 0.575991 − 1.044940I
−3.62992 − 7.26942I 0. + 8.20898I
u = −0.592384 + 0.373525I
a = 0.709608 − 0.345516I
b = −0.124209 − 1.127990I
−1.87648 + 2.61466I 1.96705 − 3.93297I
u = −0.592384 − 0.373525I
a = 0.709608 + 0.345516I
b = −0.124209 + 1.127990I
−1.87648 − 2.61466I 1.96705 + 3.93297I
u = 0.783556 + 1.064140I
a = 0.540110 − 0.656742I
b = 0.592803 − 0.720077I
1.83047 + 2.21575I 9.27050 − 4.60917I
u = 0.783556 − 1.064140I
a = 0.540110 + 0.656742I
b = 0.592803 + 0.720077I
1.83047 − 2.21575I 9.27050 + 4.60917I
u = 0.729702 + 1.179840I
a = 1.70843 − 0.53284I
b = 0.604828 + 0.939285I
1.15558 + 6.98661I 0. − 10.77467I
u = 0.729702 − 1.179840I
a = 1.70843 + 0.53284I
b = 0.604828 − 0.939285I
1.15558 − 6.98661I 0. + 10.77467I
u = 0.232545 + 0.154995I
a = 1.036860 − 0.069991I
b = −0.602510 − 0.849943I
0.59509 + 2.36716I 1.43169 − 3.69296I
u = 0.232545 − 0.154995I
a = 1.036860 + 0.069991I
b = −0.602510 + 0.849943I
0.59509 − 2.36716I 1.43169 + 3.69296I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
8
(u
9
+ 2u
8
+ 4u
7
+ 4u
6
+ 5u
5
+ 4u
4
+ 4u
3
+ 2u
2
+ u − 1)
· (u
40
− u
39
+ ··· − 4u + 1)
c
2
(u
9
− 13u
8
+ ··· + 152u − 32)(u
20
+ 6u
19
+ ··· − 2u − 1)
2
c
3
(u
9
− 13u
8
+ ··· + 208u − 32)(u
20
+ 5u
19
+ ··· − 6u − 1)
2
c
6
, c
9
(u
9
− 2u
6
+ ··· + 3u − 1)(u
40
+ 5u
39
+ ··· + 4u + 1)
c
7
, c
10
(u
9
+ 4u
8
+ 10u
7
+ 16u
6
+ 19u
5
+ 20u
4
+ 18u
3
+ 12u
2
+ 5u − 1)
· (u
40
+ 15u
39
+ ··· + 120u
2
+ 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
8
(y
9
+ 4y
8
+ 10y
7
+ 16y
6
+ 19y
5
+ 20y
4
+ 18y
3
+ 12y
2
+ 5y −1)
· (y
40
+ 15y
39
+ ··· + 120y
2
+ 1)
c
2
(y
9
− 25y
8
+ ··· − 192y −1024)(y
20
− 16y
19
+ ··· − 16y + 1)
2
c
3
(y
9
− 23y
8
+ ··· + 8960y −1024)(y
20
− 7y
19
+ ··· − 2y + 1)
2
c
6
, c
9
(y
9
+ 10y
7
+ 4y
6
+ 31y
5
+ 16y
4
+ 42y
3
− 12y
2
− 3y −1)
· (y
40
− 5y
39
+ ··· − 8y + 1)
c
7
, c
10
(y
9
+ 4y
8
+ 10y
7
− 5y
5
+ 8y
4
+ 66y
3
+ 76y
2
+ 49y −1)
· (y
40
+ 19y
39
+ ··· + 240y + 1)
14
