10
151
(K10n
8
)
A knot diagram
1
Linearized knot diagam
8 10 2 7 4 10 5 1 7 3
Solving Sequence
3,10
1 2
4,7
6 5 9 8
c
10
c
2
c
3
c
6
c
5
c
9
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−833147u
23
+ 1409387u
22
+ ··· + 10226089b − 1216520,
4990546u
23
− 13216360u
22
+ ··· + 10226089a + 40011410, u
24
− 2u
23
+ ··· − 3u + 1i
I
u
2
= hb, −u
2
+ a + 2u − 1, u
3
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 27 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−8.33 × 10
5
u
23
+ 1.41 × 10
6
u
22
+ · · · + 1.02 × 10
7
b − 1.22 × 10
6
, 4.99 ×
10
6
u
23
−1.32×10
7
u
22
+· · ·+1.02×10
7
a+4.00×10
7
, u
24
−2u
23
+· · · −3u +1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
1
=
1
−u
2
a
2
=
−u
u
a
4
=
u
3
−u
3
+ u
a
7
=
−0.488021u
23
+ 1.29242u
22
+ ··· + 5.71281u − 3.91268
0.0814727u
23
− 0.137823u
22
+ ··· + 1.40274u + 0.118962
a
6
=
−0.569494u
23
+ 1.43024u
22
+ ··· + 4.31007u − 4.03164
0.0814727u
23
− 0.137823u
22
+ ··· + 1.40274u + 0.118962
a
5
=
0.0477684u
23
+ 0.550317u
22
+ ··· + 4.78103u − 4.28575
−0.244418u
23
+ 0.413468u
22
+ ··· + 0.791784u + 0.643113
a
9
=
1.22719u
23
− 2.03273u
22
+ ··· − 1.49320u − 0.914334
−0.824903u
23
+ 1.42138u
22
+ ··· − 1.76223u + 0.824685
a
8
=
1.83274u
23
− 2.23578u
22
+ ··· + 0.231291u − 1.31738
−1.42969u
23
+ 2.03524u
22
+ ··· − 4.18083u + 1.83274
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
65252793
10226089
u
23
+
159009903
10226089
u
22
+ ··· +
182141648
10226089
u −
68504184
10226089
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
24
+ 2u
23
+ ··· − u − 1
c
2
, c
10
u
24
+ 2u
23
+ ··· + 3u + 1
c
3
u
24
− 14u
23
+ ··· − u + 1
c
4
, c
7
u
24
− 4u
23
+ ··· + 10u − 1
c
5
u
24
+ 8u
23
+ ··· + 90u + 1
c
6
, c
9
u
24
+ 3u
23
+ ··· − 4u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
24
+ 6y
23
+ ··· − y + 1
c
2
, c
10
y
24
− 14y
23
+ ··· − y + 1
c
3
y
24
− 6y
23
+ ··· + 11y + 1
c
4
, c
7
y
24
− 8y
23
+ ··· − 90y + 1
c
5
y
24
+ 20y
23
+ ··· − 6310y + 1
c
6
, c
9
y
24
− 21y
23
+ ··· − 1360y + 64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.133944 + 0.985428I
a = −0.133373 − 0.081418I
b = 1.35330 − 0.50270I
1.59272 − 6.31600I 2.35122 + 4.70660I
u = 0.133944 − 0.985428I
a = −0.133373 + 0.081418I
b = 1.35330 + 0.50270I
1.59272 + 6.31600I 2.35122 − 4.70660I
u = −1.032750 + 0.196704I
a = −1.025870 − 0.498775I
b = 0.009347 − 0.679382I
0.987314 − 0.802036I 5.27434 − 1.50428I
u = −1.032750 − 0.196704I
a = −1.025870 + 0.498775I
b = 0.009347 + 0.679382I
0.987314 + 0.802036I 5.27434 + 1.50428I
u = 1.020340 + 0.341153I
a = 0.651605 + 0.756937I
b = −0.08172 − 1.46525I
0.26071 + 4.16679I 3.46466 − 8.01442I
u = 1.020340 − 0.341153I
a = 0.651605 − 0.756937I
b = −0.08172 + 1.46525I
0.26071 − 4.16679I 3.46466 + 8.01442I
u = −0.902544
a = 4.79120
b = −0.343821
−0.317600 45.9600
u = −0.141058 + 0.853854I
a = −0.089032 − 0.200554I
b = −1.319370 + 0.101644I
2.79538 − 0.43178I 4.38138 + 0.30823I
u = −0.141058 − 0.853854I
a = −0.089032 + 0.200554I
b = −1.319370 − 0.101644I
2.79538 + 0.43178I 4.38138 − 0.30823I
u = 0.752210 + 0.267079I
a = −1.94833 + 0.55932I
b = 1.117460 − 0.519931I
−2.35229 + 1.42722I −1.68393 − 3.84628I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.752210 − 0.267079I
a = −1.94833 − 0.55932I
b = 1.117460 + 0.519931I
−2.35229 − 1.42722I −1.68393 + 3.84628I
u = 0.880632 + 0.820126I
a = −0.090055 + 0.319503I
b = 0.608596 + 0.043662I
−4.05969 + 3.04416I 8.04257 − 4.79385I
u = 0.880632 − 0.820126I
a = −0.090055 − 0.319503I
b = 0.608596 − 0.043662I
−4.05969 − 3.04416I 8.04257 + 4.79385I
u = 1.261230 + 0.403008I
a = 1.87210 − 0.55612I
b = −1.74618 − 0.41138I
7.02540 + 4.75296I 7.35135 − 3.93540I
u = 1.261230 − 0.403008I
a = 1.87210 + 0.55612I
b = −1.74618 + 0.41138I
7.02540 − 4.75296I 7.35135 + 3.93540I
u = −1.226420 + 0.541913I
a = 1.26642 + 1.12148I
b = −1.45013 + 0.30367I
6.00062 − 4.69466I 6.29135 + 3.58966I
u = −1.226420 − 0.541913I
a = 1.26642 − 1.12148I
b = −1.45013 − 0.30367I
6.00062 + 4.69466I 6.29135 − 3.58966I
u = −0.651560
a = −0.544856
b = −0.332876
1.00318 10.1720
u = −1.333920 + 0.388157I
a = −1.36316 − 0.85334I
b = 1.45282 + 0.12914I
6.33160 + 1.53755I 6.60463 − 2.15708I
u = −1.333920 − 0.388157I
a = −1.36316 + 0.85334I
b = 1.45282 − 0.12914I
6.33160 − 1.53755I 6.60463 + 2.15708I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.275550 + 0.553583I
a = −1.75052 + 0.73489I
b = 1.54670 + 0.71042I
5.11209 + 11.84300I 4.87428 − 7.23803I
u = 1.275550 − 0.553583I
a = −1.75052 − 0.73489I
b = 1.54670 − 0.71042I
5.11209 − 11.84300I 4.87428 + 7.23803I
u = 0.187302 + 0.360950I
a = −2.01295 + 1.18210I
b = 0.347518 + 0.813420I
−1.83004 − 1.07762I −2.51766 + 1.69232I
u = 0.187302 − 0.360950I
a = −2.01295 − 1.18210I
b = 0.347518 − 0.813420I
−1.83004 + 1.07762I −2.51766 − 1.69232I
7
II. I
u
2
= hb, −u
2
+ a + 2u − 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
1
=
1
−u
2
a
2
=
−u
u
a
4
=
u
2
− 1
−u
2
+ u + 1
a
7
=
u
2
− 2u + 1
0
a
6
=
u
2
− 2u + 1
0
a
5
=
2u
2
− 2u
−u
2
+ u + 1
a
9
=
1
0
a
8
=
−u
2
+ 1
u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
− 4
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
4
(u − 1)
3
c
5
, c
7
(u + 1)
3
c
6
, c
9
u
3
c
8
u
3
+ u
2
+ 2u + 1
c
10
u
3
− u
2
+ 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
8
y
3
+ 3y
2
+ 2y −1
c
2
, c
10
y
3
− y
2
+ 2y −1
c
4
, c
5
, c
7
(y −1)
3
c
6
, c
9
y
3
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.539798 − 0.182582I
b = 0
−4.66906 + 2.82812I −4.21508 − 1.30714I
u = 0.877439 − 0.744862I
a = −0.539798 + 0.182582I
b = 0
−4.66906 − 2.82812I −4.21508 + 1.30714I
u = −0.754878
a = 3.07960
b = 0
−0.531480 −4.56980
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 2u − 1)(u
24
+ 2u
23
+ ··· − u − 1)
c
2
(u
3
+ u
2
− 1)(u
24
+ 2u
23
+ ··· + 3u + 1)
c
3
(u
3
− u
2
+ 2u − 1)(u
24
− 14u
23
+ ··· − u + 1)
c
4
((u − 1)
3
)(u
24
− 4u
23
+ ··· + 10u − 1)
c
5
((u + 1)
3
)(u
24
+ 8u
23
+ ··· + 90u + 1)
c
6
, c
9
u
3
(u
24
+ 3u
23
+ ··· − 4u + 8)
c
7
((u + 1)
3
)(u
24
− 4u
23
+ ··· + 10u − 1)
c
8
(u
3
+ u
2
+ 2u + 1)(u
24
+ 2u
23
+ ··· − u − 1)
c
10
(u
3
− u
2
+ 1)(u
24
+ 2u
23
+ ··· + 3u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y
3
+ 3y
2
+ 2y −1)(y
24
+ 6y
23
+ ··· − y + 1)
c
2
, c
10
(y
3
− y
2
+ 2y −1)(y
24
− 14y
23
+ ··· − y + 1)
c
3
(y
3
+ 3y
2
+ 2y −1)(y
24
− 6y
23
+ ··· + 11y + 1)
c
4
, c
7
((y −1)
3
)(y
24
− 8y
23
+ ··· − 90y + 1)
c
5
((y −1)
3
)(y
24
+ 20y
23
+ ··· − 6310y + 1)
c
6
, c
9
y
3
(y
24
− 21y
23
+ ··· − 1360y + 64)
13
