10
84
(K10a
50
)
A knot diagram
1
Linearized knot diagam
5 1 7 6 2 9 10 4 3 8
Solving Sequence
1,5
2 3 6
4,8
10 7 9
c
1
c
2
c
5
c
4
c
10
c
7
c
9
c
3
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−26695942110849u
43
+ 24221062450767u
42
+ ··· + 146051535266254b + 143537280527879,
− 958891166678785u
43
+ 1195830341741191u
42
+ ··· + 146051535266254a + 928016870112473,
u
44
− 2u
43
+ ··· − 5u + 1i
I
u
2
= hb + 1, a + 2, u − 1i
* 2 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
= h−2.67×10
13
u
43
+2.42×10
13
u
42
+· · ·+1.46×10
14
b+1.44×10
14
, −9.59×
10
14
u
43
+1.20×10
15
u
42
+· · ·+1.46×10
14
a+9.28×10
14
, u
44
−2u
43
+· · ·−5u+1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
−u
2
+ 1
u
2
a
6
=
−u
−u
3
+ u
a
4
=
u
3
u
5
− u
3
+ u
a
8
=
6.56543u
43
− 8.18773u
42
+ ··· + 35.0046u − 6.35404
0.182784u
43
− 0.165839u
42
+ ··· − 0.0574672u − 0.982785
a
10
=
6.55549u
43
− 8.08325u
42
+ ··· + 36.0390u − 5.60107
0.268862u
43
− 0.336643u
42
+ ··· + 0.229869u − 1.06886
a
7
=
−0.202656u
43
− 0.625196u
42
+ ··· − 1.87375u − 0.511274
−0.827844u
43
+ 1.65839u
42
+ ··· − 3.42533u + 0.827852
a
9
=
4.95923u
43
− 5.84946u
42
+ ··· + 26.4222u − 4.00846
−1.49853u
43
+ 1.58954u
42
+ ··· − 9.24326u + 1.49850
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
2627515223052688
73025767633127
u
43
+
3096365063700750
73025767633127
u
42
+ ··· −
11221793809688154
73025767633127
u +
2703461955268724
73025767633127
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
44
+ 2u
43
+ ··· + 5u + 1
c
2
, c
4
u
44
+ 14u
43
+ ··· − u + 1
c
3
u
44
+ 4u
43
+ ··· − u − 1
c
6
u
44
+ 7u
43
+ ··· − 2u + 2
c
7
, c
10
u
44
− 2u
43
+ ··· − 5u − 1
c
8
u
44
− 2u
43
+ ··· − 17u − 11
c
9
u
44
− 4u
43
+ ··· − 21u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
44
− 14y
43
+ ··· + y + 1
c
2
, c
4
y
44
+ 34y
43
+ ··· + 137y + 1
c
3
y
44
+ 6y
43
+ ··· + y + 1
c
6
y
44
− 9y
43
+ ··· − 40y + 4
c
7
, c
10
y
44
− 26y
43
+ ··· − 71y + 1
c
8
y
44
− 42y
43
+ ··· − 2995y + 121
c
9
y
44
− 38y
43
+ ··· − 123y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.927602 + 0.226351I
a = −0.224310 − 0.062924I
b = −0.115105 + 0.911207I
−1.13001 + 3.88298I −6.50680 − 7.75927I
u = −0.927602 − 0.226351I
a = −0.224310 + 0.062924I
b = −0.115105 − 0.911207I
−1.13001 − 3.88298I −6.50680 + 7.75927I
u = 0.778786 + 0.710214I
a = 1.43805 − 0.91909I
b = −1.049480 + 0.821846I
0.040125 + 0.820231I −5.81896 − 3.03229I
u = 0.778786 − 0.710214I
a = 1.43805 + 0.91909I
b = −1.049480 − 0.821846I
0.040125 − 0.820231I −5.81896 + 3.03229I
u = −0.927070 + 0.063011I
a = −1.49031 − 0.50021I
b = −1.37113 + 0.48363I
−4.92583 + 1.73663I −14.9087 − 4.1335I
u = −0.927070 − 0.063011I
a = −1.49031 + 0.50021I
b = −1.37113 − 0.48363I
−4.92583 − 1.73663I −14.9087 + 4.1335I
u = −0.658922 + 0.846151I
a = −0.183634 − 0.593747I
b = 0.865773 + 0.460561I
3.70844 − 0.99499I 0.63089 + 2.41468I
u = −0.658922 − 0.846151I
a = −0.183634 + 0.593747I
b = 0.865773 − 0.460561I
3.70844 + 0.99499I 0.63089 − 2.41468I
u = 0.878177 + 0.660456I
a = 0.670319 + 1.230810I
b = −1.59639 + 0.09563I
−1.82200 − 2.55706I −9.50147 + 2.98004I
u = 0.878177 − 0.660456I
a = 0.670319 − 1.230810I
b = −1.59639 − 0.09563I
−1.82200 + 2.55706I −9.50147 − 2.98004I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.840203 + 0.731796I
a = 0.53742 + 1.70678I
b = −0.912754 − 0.201576I
1.40942 + 2.09852I −2.80453 − 11.50069I
u = −0.840203 − 0.731796I
a = 0.53742 − 1.70678I
b = −0.912754 + 0.201576I
1.40942 − 2.09852I −2.80453 + 11.50069I
u = 0.772905 + 0.818777I
a = 0.33712 − 1.68078I
b = 0.254394 + 1.136870I
5.49636 + 2.42871I 0. − 2.25678I
u = 0.772905 − 0.818777I
a = 0.33712 + 1.68078I
b = 0.254394 − 1.136870I
5.49636 − 2.42871I 0. + 2.25678I
u = 0.709073 + 0.883385I
a = −0.840026 + 0.862508I
b = 1.27066 − 0.62408I
2.27409 + 8.60569I −2.63926 − 4.58190I
u = 0.709073 − 0.883385I
a = −0.840026 − 0.862508I
b = 1.27066 + 0.62408I
2.27409 − 8.60569I −2.63926 + 4.58190I
u = −1.117850 + 0.238085I
a = 1.25957 + 0.83369I
b = 1.277770 − 0.452951I
−5.29949 + 8.62766I −9.10597 − 7.54655I
u = −1.117850 − 0.238085I
a = 1.25957 − 0.83369I
b = 1.277770 + 0.452951I
−5.29949 − 8.62766I −9.10597 + 7.54655I
u = −0.902384 + 0.723912I
a = −0.42622 − 3.07845I
b = −0.993100 + 0.182802I
1.21777 + 3.45181I 0. + 7.88863I
u = −0.902384 − 0.723912I
a = −0.42622 + 3.07845I
b = −0.993100 − 0.182802I
1.21777 − 3.45181I 0. − 7.88863I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.940648 + 0.702120I
a = −0.12431 + 2.40011I
b = −1.20650 − 0.83342I
−0.45271 − 6.24747I −7.31920 + 8.44159I
u = 0.940648 − 0.702120I
a = −0.12431 − 2.40011I
b = −1.20650 + 0.83342I
−0.45271 + 6.24747I −7.31920 − 8.44159I
u = 1.18227
a = 1.12361
b = 0.834518
−2.71479 5.71830
u = 0.802912 + 0.142507I
a = 0.916574 + 0.518690I
b = −0.174142 − 0.024221I
−1.40557 − 0.34934I −7.47293 + 0.48118I
u = 0.802912 − 0.142507I
a = 0.916574 − 0.518690I
b = −0.174142 + 0.024221I
−1.40557 + 0.34934I −7.47293 − 0.48118I
u = 0.812067
a = −6.68009
b = −1.03778
−2.95636 47.1560
u = 0.098222 + 0.805268I
a = −0.628805 − 0.691139I
b = 1.121430 + 0.435398I
−1.20700 − 5.24815I −2.80453 + 6.18731I
u = 0.098222 − 0.805268I
a = −0.628805 + 0.691139I
b = 1.121430 − 0.435398I
−1.20700 + 5.24815I −2.80453 − 6.18731I
u = 1.133010 + 0.369128I
a = 0.498144 − 0.725878I
b = 1.113320 − 0.253728I
−4.55359 + 1.09231I −9.24999 − 5.05772I
u = 1.133010 − 0.369128I
a = 0.498144 + 0.725878I
b = 1.113320 + 0.253728I
−4.55359 − 1.09231I −9.24999 + 5.05772I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.835443 + 0.852874I
a = −0.056114 + 0.817361I
b = 0.565371 − 0.419340I
4.53169 + 2.82413I 0. − 4.92903I
u = −0.835443 − 0.852874I
a = −0.056114 − 0.817361I
b = 0.565371 + 0.419340I
4.53169 − 2.82413I 0. + 4.92903I
u = −0.940979 + 0.796566I
a = −0.298082 − 0.266590I
b = 0.375731 + 0.414152I
4.19341 + 3.30756I 0
u = −0.940979 − 0.796566I
a = −0.298082 + 0.266590I
b = 0.375731 − 0.414152I
4.19341 − 3.30756I 0
u = 0.973933 + 0.757495I
a = −1.09792 + 1.17097I
b = 0.181664 − 1.203370I
4.87682 − 8.33877I 0. + 7.62816I
u = 0.973933 − 0.757495I
a = −1.09792 − 1.17097I
b = 0.181664 + 1.203370I
4.87682 + 8.33877I 0. − 7.62816I
u = −1.046830 + 0.731985I
a = 0.54736 + 1.47326I
b = 0.992978 − 0.410523I
2.52892 + 6.89763I 0
u = −1.046830 − 0.731985I
a = 0.54736 − 1.47326I
b = 0.992978 + 0.410523I
2.52892 − 6.89763I 0
u = 1.033090 + 0.762400I
a = 0.30235 − 2.16182I
b = 1.32116 + 0.62604I
1.2704 − 14.7099I 0
u = 1.033090 − 0.762400I
a = 0.30235 + 2.16182I
b = 1.32116 − 0.62604I
1.2704 + 14.7099I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.109387 + 0.546973I
a = 0.696572 + 1.005610I
b = 0.218376 − 0.606022I
1.40694 − 1.21023I 2.44144 + 1.67923I
u = −0.109387 − 0.546973I
a = 0.696572 − 1.005610I
b = 0.218376 + 0.606022I
1.40694 + 1.21023I 2.44144 − 1.67923I
u = 0.188748 + 0.259164I
a = 2.94448 + 0.60372I
b = −1.038390 − 0.225035I
−1.92044 − 0.80342I −4.41092 − 0.12174I
u = 0.188748 − 0.259164I
a = 2.94448 − 0.60372I
b = −1.038390 + 0.225035I
−1.92044 + 0.80342I −4.41092 + 0.12174I
9
II. I
u
2
= hb + 1, a + 2, u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
6
=
−1
0
a
4
=
1
1
a
8
=
−2
−1
a
10
=
−1
−1
a
7
=
−1
0
a
9
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
u − 1
c
2
, c
5
, c
8
c
9
, c
10
u + 1
c
6
u
11
(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
, c
10
y −1
c
6
y
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −2.00000
b = −1.00000
−3.28987 −12.0000
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
44
+ 2u
43
+ ··· + 5u + 1)
c
2
(u + 1)(u
44
+ 14u
43
+ ··· − u + 1)
c
3
(u − 1)(u
44
+ 4u
43
+ ··· − u − 1)
c
4
(u − 1)(u
44
+ 14u
43
+ ··· − u + 1)
c
5
(u + 1)(u
44
+ 2u
43
+ ··· + 5u + 1)
c
6
u(u
44
+ 7u
43
+ ··· − 2u + 2)
c
7
(u − 1)(u
44
− 2u
43
+ ··· − 5u − 1)
c
8
(u + 1)(u
44
− 2u
43
+ ··· − 17u − 11)
c
9
(u + 1)(u
44
− 4u
43
+ ··· − 21u + 1)
c
10
(u + 1)(u
44
− 2u
43
+ ··· − 5u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y −1)(y
44
− 14y
43
+ ··· + y + 1)
c
2
, c
4
(y −1)(y
44
+ 34y
43
+ ··· + 137y + 1)
c
3
(y −1)(y
44
+ 6y
43
+ ··· + y + 1)
c
6
y(y
44
− 9y
43
+ ··· − 40y + 4)
c
7
, c
10
(y −1)(y
44
− 26y
43
+ ··· − 71y + 1)
c
8
(y −1)(y
44
− 42y
43
+ ··· − 2995y + 121)
c
9
(y −1)(y
44
− 38y
43
+ ··· − 123y + 1)
15
