10
159
(K10n
34
)
A knot diagram
1
Linearized knot diagam
6 7 10 1 8 9 10 1 2 4
Solving Sequence
1,6 2,8
9 5 4 10 3 7
c
1
c
8
c
5
c
4
c
10
c
3
c
7
c
2
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−31u
8
+ 18u
7
+ 76u
6
− 39u
5
− 208u
4
+ 102u
3
+ 226u
2
+ 25b − 20u − 72,
− 108u
8
+ 49u
7
+ 243u
6
− 77u
5
− 694u
4
+ 261u
3
+ 693u
2
+ 25a + 65u − 196,
u
9
− u
8
− 2u
7
+ 2u
6
+ 6u
5
− 6u
4
− 5u
3
+ 3u
2
+ 2u − 1i
I
u
2
= h1002u
13
− 332u
12
+ ··· + 1889b + 2916, −2310u
13
+ 279u
12
+ ··· + 1889a − 9392,
u
14
+ u
11
+ 3u
10
+ 2u
9
− 7u
8
+ 7u
7
− u
6
+ 11u
5
− 14u
4
+ 9u
3
− 5u
2
+ 5u − 1i
I
u
3
= hu
2
+ b − 1, u
2
+ a + u, u
3
− u + 1i
I
u
4
= hb − u + 1, a + u − 1, u
2
− u − 1i
I
u
5
= hb + 1, a − 1, u + 1i
* 5 irreducible components of dim
C
= 0, with total 29 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−31u
8
+ 18u
7
+ · · · + 25b − 72, −108u
8
+ 49u
7
+ · · · + 25a −
196, u
9
− u
8
+ · · · + 2u − 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
8
=
4.32000u
8
− 1.96000u
7
+ ··· − 2.60000u + 7.84000
31
25
u
8
−
18
25
u
7
+ ··· +
4
5
u +
72
25
a
9
=
5.56000u
8
− 2.68000u
7
+ ··· − 1.80000u + 10.7200
31
25
u
8
−
18
25
u
7
+ ··· +
4
5
u +
72
25
a
5
=
−1.76000u
8
+ 1.28000u
7
+ ··· − 2.20000u − 5.12000
48
25
u
8
−
19
25
u
7
+ ··· −
3
5
u +
76
25
a
4
=
4
25
u
8
+
13
25
u
7
+ ··· −
14
5
u −
52
25
48
25
u
8
−
19
25
u
7
+ ··· −
3
5
u +
76
25
a
10
=
5.56000u
8
− 2.68000u
7
+ ··· − 2.80000u + 10.7200
31
25
u
8
−
18
25
u
7
+ ··· +
4
5
u +
72
25
a
3
=
−2.76000u
8
+ 2.28000u
7
+ ··· − 4.20000u − 9.12000
12
5
u
8
−
6
5
u
7
+ ··· − 2u +
19
5
a
7
=
16
5
u
8
−
3
5
u
7
+ ··· − 6u +
17
5
3.04000u
8
− 1.12000u
7
+ ··· − 1.20000u + 5.48000
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
96
25
u
8
+
13
25
u
7
+
216
25
u
6
−
24
25
u
5
−
653
25
u
4
+
82
25
u
3
+
691
25
u
2
+
26
5
u −
377
25
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
9
u
9
− u
8
− 2u
7
+ 2u
6
+ 6u
5
− 6u
4
− 5u
3
+ 3u
2
+ 2u − 1
c
2
, c
8
u
9
− 4u
7
+ 7u
5
− 2u
4
− 4u
3
− u
2
+ 3u + 1
c
3
, c
4
, c
10
u
9
+ 5u
8
+ 12u
7
+ 12u
6
− 6u
5
− 38u
4
− 57u
3
− 49u
2
− 24u − 5
c
5
, c
7
u
9
+ 6u
7
+ 4u
6
+ 15u
5
+ 18u
4
+ 18u
3
+ 19u
2
+ 7u + 1
c
6
u
9
+ 7u
8
+ 22u
7
+ 44u
6
+ 72u
5
+ 102u
4
+ 103u
3
+ 59u
2
+ 18u + 5
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
9
y
9
− 5y
8
+ 20y
7
− 50y
6
+ 90y
5
− 118y
4
+ 89y
3
− 41y
2
+ 10y −1
c
2
, c
8
y
9
− 8y
8
+ 30y
7
− 64y
6
+ 87y
5
− 84y
4
+ 54y
3
− 21y
2
+ 11y −1
c
3
, c
4
, c
10
y
9
− y
8
+ 12y
7
− 22y
6
+ 22y
5
− 110y
4
− 67y
3
− 45y
2
+ 86y −25
c
5
, c
7
y
9
+ 12y
8
+ ··· + 11y −1
c
6
y
9
− 5y
8
+ 12y
7
+ 10y
6
− 50y
5
− 42y
4
+ 725y
3
− 793y
2
− 266y −25
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.675360 + 0.321360I
a = 0.375927 − 0.035170I
b = −0.490473 + 0.554222I
−1.230240 + 0.388380I −8.56083 − 2.01333I
u = −0.675360 − 0.321360I
a = 0.375927 + 0.035170I
b = −0.490473 − 0.554222I
−1.230240 − 0.388380I −8.56083 + 2.01333I
u = 1.27629
a = −0.656695
b = −0.328475
−6.80161 −15.9820
u = −1.16884 + 0.87463I
a = −0.616776 + 0.922983I
b = 1.299660 + 0.083541I
5.93576 + 3.11393I −2.06870 − 2.32890I
u = −1.16884 − 0.87463I
a = −0.616776 − 0.922983I
b = 1.299660 − 0.083541I
5.93576 − 3.11393I −2.06870 + 2.32890I
u = 0.523277 + 0.089360I
a = −0.56707 − 2.28589I
b = 0.883398 − 0.665684I
0.97258 − 2.76102I −6.12756 + 2.10529I
u = 0.523277 − 0.089360I
a = −0.56707 + 2.28589I
b = 0.883398 + 0.665684I
0.97258 + 2.76102I −6.12756 − 2.10529I
u = 1.18278 + 0.96607I
a = 1.136270 + 0.521152I
b = −1.52834 + 0.58529I
5.94738 − 11.74060I −3.25195 + 6.67016I
u = 1.18278 − 0.96607I
a = 1.136270 − 0.521152I
b = −1.52834 − 0.58529I
5.94738 + 11.74060I −3.25195 − 6.67016I
5
II. I
u
2
= h1002u
13
− 332u
12
+ · · · + 1889b + 2916, −2310u
13
+ 279u
12
+ · · · +
1889a − 9392, u
14
+ u
11
+ · · · + 5u − 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
8
=
1.22287u
13
− 0.147697u
12
+ ··· − 5.55214u + 4.97194
−0.530439u
13
+ 0.175754u
12
+ ··· + 1.11223u − 1.54367
a
9
=
0.692430u
13
+ 0.0280572u
12
+ ··· − 4.43992u + 3.42827
−0.530439u
13
+ 0.175754u
12
+ ··· + 1.11223u − 1.54367
a
5
=
0.967708u
13
− 0.0831128u
12
+ ··· − 6.71572u + 5.12758
−0.582319u
13
− 0.0725251u
12
+ ··· + 2.92959u − 1.74854
a
4
=
0.385389u
13
− 0.155638u
12
+ ··· − 3.78613u + 3.37904
−0.582319u
13
− 0.0725251u
12
+ ··· + 2.92959u − 1.74854
a
10
=
u
13
+ u
10
+ 3u
9
+ 2u
8
− 7u
7
+ 7u
6
− u
5
+ 11u
4
− 14u
3
+ 9u
2
− 5u + 5
−0.307570u
13
+ 0.0280572u
12
+ ··· + 1.56008u − 1.57173
a
3
=
−1.36316u
13
− 0.737957u
12
+ ··· + 1.72155u − 2.43409
−0.238221u
13
+ 0.288512u
12
+ ··· + 2.83483u − 0.124404
a
7
=
0.592377u
13
+ 0.0492324u
12
+ ··· − 5.14929u + 2.67602
0.206988u
13
+ 0.204870u
12
+ ··· + 0.636845u − 0.703017
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1965
1889
u
13
+
5491
1889
u
12
+ ··· −
3549
1889
u +
3230
1889
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
9
u
14
+ u
11
+ ··· + 5u − 1
c
2
, c
8
u
14
− 6u
12
+ ··· − 9u + 1
c
3
, c
4
, c
10
(u
7
− 2u
6
+ 2u
5
+ u
4
− 2u
3
+ 3u
2
− 2u + 1)
2
c
5
, c
7
u
14
− 3u
13
+ ··· − 6u − 1
c
6
(u
7
− 3u
6
+ 3u
5
+ 2u
4
− 6u
3
+ 3u
2
+ 3u − 2)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
9
y
14
+ 6y
12
+ ··· − 15y + 1
c
2
, c
8
y
14
− 12y
13
+ ··· − 63y + 1
c
3
, c
4
, c
10
(y
7
+ 4y
5
− y
4
− 6y
3
− 3y
2
− 2y − 1)
2
c
5
, c
7
y
14
+ 13y
13
+ ··· + 42y + 1
c
6
(y
7
− 3y
6
+ 9y
5
− 16y
4
+ 30y
3
− 37y
2
+ 21y − 4)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877499 + 0.643882I
a = −0.815701 − 0.730313I
b = 1.36033 − 0.47577I
2.12977 − 2.53884I 0.86344 + 1.81085I
u = 0.877499 − 0.643882I
a = −0.815701 + 0.730313I
b = 1.36033 + 0.47577I
2.12977 + 2.53884I 0.86344 − 1.81085I
u = 0.763487 + 0.442848I
a = −0.149425 − 0.086700I
b = 0.33250 − 1.47887I
0.33600 − 4.72329I −7.01907 + 9.17288I
u = 0.763487 − 0.442848I
a = −0.149425 + 0.086700I
b = 0.33250 + 1.47887I
0.33600 + 4.72329I −7.01907 − 9.17288I
u = −0.796980 + 0.997104I
a = 1.285810 − 0.554607I
b = −1.011450 − 0.500189I
0.33600 + 4.72329I −7.01907 − 9.17288I
u = −0.796980 − 0.997104I
a = 1.285810 + 0.554607I
b = −1.011450 + 0.500189I
0.33600 − 4.72329I −7.01907 + 9.17288I
u = −0.775231 + 1.031020I
a = −1.276240 + 0.214140I
b = 1.62238 + 0.39283I
7.17429 + 3.91715I −1.20398 − 3.00324I
u = −0.775231 − 1.031020I
a = −1.276240 − 0.214140I
b = 1.62238 − 0.39283I
7.17429 − 3.91715I −1.20398 + 3.00324I
u = −0.196138 + 0.662538I
a = 0.30408 − 2.08007I
b = 0.162591 + 0.048461I
2.12977 + 2.53884I 0.86344 − 1.81085I
u = −0.196138 − 0.662538I
a = 0.30408 + 2.08007I
b = 0.162591 − 0.048461I
2.12977 − 2.53884I 0.86344 + 1.81085I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.81203 + 1.22658I
a = 0.961913 + 0.622177I
b = −1.361880 − 0.158001I
7.17429 + 3.91715I −1.20398 − 3.00324I
u = 0.81203 − 1.22658I
a = 0.961913 − 0.622177I
b = −1.361880 + 0.158001I
7.17429 − 3.91715I −1.20398 + 3.00324I
u = −1.60968
a = 0.365698
b = −0.746600
−2.83077 1.71920
u = 0.240340
a = 4.01341
b = −1.46232
−2.83077 1.71920
10
III. I
u
3
= hu
2
+ b − 1, u
2
+ a + u, u
3
− u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
8
=
−u
2
− u
−u
2
+ 1
a
9
=
−2u
2
− u + 1
−u
2
+ 1
a
5
=
u
2
− 2
−u
2
a
4
=
−2
−u
2
a
10
=
−2u
2
+ 1
−u
2
+ u
a
3
=
u
2
− 2u − 2
−2u + 1
a
7
=
−2u
2
− 2u − 1
−2u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
2
+ 5u − 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
9
u
3
− u + 1
c
2
, c
8
u
3
+ u
2
− 1
c
3
, c
4
u
3
− 2u
2
+ u − 1
c
5
, c
7
u
3
+ u
2
+ 2u + 1
c
6
u
3
+ 4u
2
+ 7u + 5
c
10
u
3
+ 2u
2
+ u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
9
y
3
− 2y
2
+ y − 1
c
2
, c
8
y
3
− y
2
+ 2y − 1
c
3
, c
4
, c
10
y
3
− 2y
2
− 3y − 1
c
5
, c
7
y
3
+ 3y
2
+ 2y − 1
c
6
y
3
− 2y
2
+ 9y − 25
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.662359 + 0.562280I
a = −0.78492 − 1.30714I
b = 0.877439 − 0.744862I
1.45094 − 3.77083I −1.95284 + 7.28057I
u = 0.662359 − 0.562280I
a = −0.78492 + 1.30714I
b = 0.877439 + 0.744862I
1.45094 + 3.77083I −1.95284 − 7.28057I
u = −1.32472
a = −0.430160
b = −0.754878
−6.19175 −2.09430
14
IV. I
u
4
= hb − u + 1, a + u − 1, u
2
− u − 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
8
=
−u + 1
u − 1
a
9
=
0
u − 1
a
5
=
u − 1
1
a
4
=
u
1
a
10
=
−u + 1
−1
a
3
=
1
0
a
7
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −17
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
8
c
9
u
2
− u − 1
c
3
, c
4
(u + 1)
2
c
5
, c
7
, c
10
(u − 1)
2
c
6
u
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
8
c
9
y
2
− 3y + 1
c
3
, c
4
, c
5
c
7
, c
10
(y − 1)
2
c
6
y
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 1.61803
b = −1.61803
−3.28987 −17.0000
u = 1.61803
a = −0.618034
b = 0.618034
−3.28987 −17.0000
18
V. I
u
5
= hb + 1, a − 1, u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
−1
a
2
=
1
1
a
8
=
1
−1
a
9
=
0
−1
a
5
=
−1
0
a
4
=
−1
0
a
10
=
1
0
a
3
=
−1
0
a
7
=
0
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
8
, c
9
u + 1
c
3
, c
4
, c
6
c
10
u
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
8
, c
9
y − 1
c
3
, c
4
, c
6
c
10
y
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = −1.00000
−1.64493 −6.00000
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
9
(u + 1)(u
2
− u − 1)(u
3
− u + 1)
· (u
9
− u
8
− 2u
7
+ 2u
6
+ 6u
5
− 6u
4
− 5u
3
+ 3u
2
+ 2u − 1)
· (u
14
+ u
11
+ ··· + 5u − 1)
c
2
, c
8
(u + 1)(u
2
− u − 1)(u
3
+ u
2
− 1)(u
9
− 4u
7
+ ··· + 3u + 1)
· (u
14
− 6u
12
+ ··· − 9u + 1)
c
3
, c
4
u(u + 1)
2
(u
3
− 2u
2
+ u − 1)
· (u
7
− 2u
6
+ 2u
5
+ u
4
− 2u
3
+ 3u
2
− 2u + 1)
2
· (u
9
+ 5u
8
+ 12u
7
+ 12u
6
− 6u
5
− 38u
4
− 57u
3
− 49u
2
− 24u − 5)
c
5
, c
7
(u − 1)
2
(u + 1)(u
3
+ u
2
+ 2u + 1)
· (u
9
+ 6u
7
+ 4u
6
+ 15u
5
+ 18u
4
+ 18u
3
+ 19u
2
+ 7u + 1)
· (u
14
− 3u
13
+ ··· − 6u − 1)
c
6
u
3
(u
3
+ 4u
2
+ 7u + 5)(u
7
− 3u
6
+ 3u
5
+ 2u
4
− 6u
3
+ 3u
2
+ 3u − 2)
2
· (u
9
+ 7u
8
+ 22u
7
+ 44u
6
+ 72u
5
+ 102u
4
+ 103u
3
+ 59u
2
+ 18u + 5)
c
10
u(u − 1)
2
(u
3
+ 2u
2
+ u + 1)
· (u
7
− 2u
6
+ 2u
5
+ u
4
− 2u
3
+ 3u
2
− 2u + 1)
2
· (u
9
+ 5u
8
+ 12u
7
+ 12u
6
− 6u
5
− 38u
4
− 57u
3
− 49u
2
− 24u − 5)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
9
(y − 1)(y
2
− 3y + 1)(y
3
− 2y
2
+ y − 1)
· (y
9
− 5y
8
+ 20y
7
− 50y
6
+ 90y
5
− 118y
4
+ 89y
3
− 41y
2
+ 10y − 1)
· (y
14
+ 6y
12
+ ··· − 15y + 1)
c
2
, c
8
(y − 1)(y
2
− 3y + 1)(y
3
− y
2
+ 2y − 1)
· (y
9
− 8y
8
+ 30y
7
− 64y
6
+ 87y
5
− 84y
4
+ 54y
3
− 21y
2
+ 11y − 1)
· (y
14
− 12y
13
+ ··· − 63y + 1)
c
3
, c
4
, c
10
y(y − 1)
2
(y
3
− 2y
2
− 3y − 1)(y
7
+ 4y
5
− y
4
− 6y
3
− 3y
2
− 2y − 1)
2
· (y
9
− y
8
+ 12y
7
− 22y
6
+ 22y
5
− 110y
4
− 67y
3
− 45y
2
+ 86y − 25)
c
5
, c
7
((y − 1)
3
)(y
3
+ 3y
2
+ 2y − 1)(y
9
+ 12y
8
+ ··· + 11y − 1)
· (y
14
+ 13y
13
+ ··· + 42y + 1)
c
6
y
3
(y
3
− 2y
2
+ 9y − 25)
· (y
7
− 3y
6
+ 9y
5
− 16y
4
+ 30y
3
− 37y
2
+ 21y − 4)
2
· (y
9
− 5y
8
+ 12y
7
+ 10y
6
− 50y
5
− 42y
4
+ 725y
3
− 793y
2
− 266y − 25)
24
