11n
116
(K11n
116
)
A knot diagram
1
Linearized knot diagam
7 1 8 10 9 2 10 1 5 8 4
Solving Sequence
3,8 4,10
5 11 1 2 7 6 9
c
3
c
4
c
10
c
11
c
2
c
7
c
6
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h8u
7
+ 2u
6
+ 43u
5
− 42u
4
+ 38u
3
− 164u
2
+ 17b + 20u − 27,
28u
7
+ 58u
6
+ 244u
5
+ 278u
4
+ 490u
3
+ 191u
2
+ 17a + 53u + 33,
u
8
+ 2u
7
+ 9u
6
+ 10u
5
+ 20u
4
+ 8u
3
+ 7u
2
+ u + 1i
I
u
2
= h56u
7
− 6u
6
+ 421u
5
+ 108u
4
+ 1168u
3
+ 208u
2
+ 397b + 934u + 377,
− 418u
7
+ 300u
6
− 2788u
5
+ 158u
4
− 4408u
3
− 475u
2
+ 1191a − 251u − 985,
u
8
+ 7u
6
+ 4u
5
+ 16u
4
+ 10u
3
+ 11u
2
+ 7u + 3i
* 2 irreducible components of dim
C
= 0, with total 16 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
= h8u
7
+2u
6
+· · ·+17b−27, 28u
7
+58u
6
+· · ·+17a+33, u
8
+2u
7
+· · ·+u+1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
10
=
−1.64706u
7
− 3.41176u
6
+ ··· − 3.11765u − 1.94118
−0.470588u
7
− 0.117647u
6
+ ··· − 1.17647u + 1.58824
a
5
=
1.35294u
7
+ 0.588235u
6
+ ··· + 6.88235u − 2.94118
−2.70588u
7
− 5.17647u
6
+ ··· − 6.76471u − 3.11765
a
11
=
−1.64706u
7
− 3.41176u
6
+ ··· − 3.11765u − 1.94118
−1.17647u
7
− 1.29412u
6
+ ··· − 2.94118u + 1.47059
a
1
=
−1.17647u
7
− 3.29412u
6
+ ··· − 1.94118u − 3.52941
−1.11765u
7
− 1.52941u
6
+ ··· − 3.29412u + 0.647059
a
2
=
−7.58824u
7
− 14.6471u
6
+ ··· − 17.4706u − 8.76471
0.176471u
7
+ 2.29412u
6
+ ··· − 5.05882u + 8.52941
a
7
=
−1.64706u
7
− 1.41176u
6
+ ··· − 7.11765u + 3.05882
2.41176u
7
+ 4.35294u
6
+ ··· + 6.52941u + 3.23529
a
6
=
−3.64706u
7
− 12.4118u
6
+ ··· − 1.11765u − 20.9412
−6.70588u
7
− 9.17647u
6
+ ··· − 21.7647u + 2.88235
a
9
=
6.52941u
7
+ 11.8824u
6
+ ··· + 15.8235u + 6.58824
−0.529412u
7
− 2.88235u
6
+ ··· + 3.17647u − 7.58824
a
9
=
6.52941u
7
+ 11.8824u
6
+ ··· + 15.8235u + 6.58824
−0.529412u
7
− 2.88235u
6
+ ··· + 3.17647u − 7.58824
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
19
17
u
7
−
60
17
u
6
−
219
17
u
5
−
389
17
u
4
−
613
17
u
3
−
571
17
u
2
−
260
17
u −
57
17
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
8
+ 3u
7
+ 12u
6
+ 15u
5
+ 32u
4
+ 9u
3
+ 27u
2
− 21u + 11
c
2
u
8
+ 15u
7
+ ··· + 153u + 121
c
3
u
8
+ 2u
7
+ 9u
6
+ 10u
5
+ 20u
4
+ 8u
3
+ 7u
2
+ u + 1
c
4
, c
5
, c
9
u
8
+ 8u
6
− 4u
5
+ 36u
4
− 29u
3
+ 41u
2
+ 17u + 19
c
7
, c
10
u
8
+ 2u
7
+ 8u
6
+ 3u
5
+ 38u
4
− 38u
3
+ 58u
2
− 12u + 7
c
8
u
8
− 2u
7
+ 10u
6
+ 6u
5
+ 168u
4
+ 139u
3
+ 116u
2
− 60u + 47
c
11
u
8
− 4u
7
+ 13u
6
− 28u
5
+ 90u
4
+ 60u
3
+ 85u
2
+ 23u + 11
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
8
+ 15y
7
+ ··· + 153y + 121
c
2
y
8
+ 11y
7
+ ··· + 414853y + 14641
c
3
y
8
+ 14y
7
+ 81y
6
+ 242y
5
+ 364y
4
+ 214y
3
+ 73y
2
+ 13y + 1
c
4
, c
5
, c
9
y
8
+ 16y
7
+ ··· + 1269y + 361
c
7
, c
10
y
8
+ 12y
7
+ ··· + 668y + 49
c
8
y
8
+ 16y
7
+ ··· + 7304y + 2209
c
11
y
8
+ 10y
7
+ ··· + 1341y + 121
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.270939 + 0.522049I
a = −3.45142 + 1.23452I
b = −0.44567 − 2.88321I
4.85825 − 1.12061I 1.61393 + 0.60117I
u = −0.270939 − 0.522049I
a = −3.45142 − 1.23452I
b = −0.44567 + 2.88321I
4.85825 + 1.12061I 1.61393 − 0.60117I
u = 0.104875 + 0.438980I
a = 0.849602 + 0.031731I
b = −0.136204 + 0.426385I
0.137633 + 1.005540I 2.51610 − 6.63610I
u = 0.104875 − 0.438980I
a = 0.849602 − 0.031731I
b = −0.136204 − 0.426385I
0.137633 − 1.005540I 2.51610 + 6.63610I
u = −0.66203 + 1.74906I
a = −0.565751 − 0.130138I
b = −0.274053 − 0.658590I
−5.40704 − 2.79901I 1.66145 + 4.10976I
u = −0.66203 − 1.74906I
a = −0.565751 + 0.130138I
b = −0.274053 + 0.658590I
−5.40704 + 2.79901I 1.66145 − 4.10976I
u = −0.17191 + 2.00694I
a = 1.16757 + 0.88058I
b = 2.85593 + 2.12606I
−19.3281 − 7.7545I 1.70851 + 2.41364I
u = −0.17191 − 2.00694I
a = 1.16757 − 0.88058I
b = 2.85593 − 2.12606I
−19.3281 + 7.7545I 1.70851 − 2.41364I
5
II. I
u
2
= h56u
7
− 6u
6
+ · · · + 397b + 377, −418u
7
+ 300u
6
+ · · · + 1191a −
985, u
8
+ 7u
6
+ 4u
5
+ 16u
4
+ 10u
3
+ 11u
2
+ 7u + 3i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
10
=
0.350966u
7
− 0.251889u
6
+ ··· + 0.210747u + 0.827036
−0.141058u
7
+ 0.0151134u
6
+ ··· − 2.35264u − 0.949622
a
5
=
−0.316541u
7
+ 0.141058u
6
+ ··· − 2.95802u + 1.13686
−0.251889u
7
− 0.115869u
6
+ ··· − 1.62972u − 2.05290
a
11
=
0.350966u
7
− 0.251889u
6
+ ··· + 0.210747u + 0.827036
−0.0251889u
7
− 0.211587u
6
+ ··· − 3.06297u − 1.70529
a
1
=
0.492024u
7
− 0.267003u
6
+ ··· + 2.56339u + 1.77666
−0.0982368u
7
− 0.0251889u
6
+ ··· − 2.74559u − 1.75063
a
2
=
0.306465u
7
− 0.425693u
6
+ ··· + 2.93283u + 1.58102
−0.365239u
7
− 0.0680101u
6
+ ··· − 4.41310u − 1.22670
a
7
=
−0.0772460u
7
+ 0.151134u
6
+ ··· + 2.14022u + 1.83711
−0.410579u
7
+ 0.151134u
6
+ ··· − 0.526448u − 0.496222
a
6
=
−0.518892u
7
− 0.158690u
6
+ ··· − 3.29723u − 2.52897
−0.0251889u
7
− 0.211587u
6
+ ··· − 1.06297u + 0.294710
a
9
=
−0.590260u
7
+ 0.241814u
6
+ ··· − 5.30898u − 2.52729
0.425693u
7
− 0.224181u
6
+ ··· + 2.56423u + 0.919395
a
9
=
−0.590260u
7
+ 0.241814u
6
+ ··· − 5.30898u − 2.52729
0.425693u
7
− 0.224181u
6
+ ··· + 2.56423u + 0.919395
(ii) Obstruction class = 1
(iii) Cusp Shapes =
543
397
u
7
−
44
397
u
6
+
3749
397
u
5
+
1983
397
u
4
+
8433
397
u
3
+
5363
397
u
2
+
5526
397
u +
4485
397
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ u
7
+ 4u
6
+ 3u
5
+ 6u
4
+ 3u
3
+ 5u
2
+ u + 1
c
2
u
8
+ 7u
7
+ 22u
6
+ 43u
5
+ 58u
4
+ 53u
3
+ 31u
2
+ 9u + 1
c
3
u
8
+ 7u
6
+ 4u
5
+ 16u
4
+ 10u
3
+ 11u
2
+ 7u + 3
c
4
, c
5
u
8
+ 4u
6
+ 6u
4
− u
3
+ 5u
2
− u + 1
c
6
u
8
− u
7
+ 4u
6
− 3u
5
+ 6u
4
− 3u
3
+ 5u
2
− u + 1
c
7
u
8
+ 2u
7
+ 2u
6
+ u
5
− 2u
4
− 2u
3
+ 1
c
8
u
8
− 2u
5
− 2u
4
+ u
3
+ 2u
2
+ 2u + 1
c
9
u
8
+ 4u
6
+ 6u
4
+ u
3
+ 5u
2
+ u + 1
c
10
u
8
− 2u
7
+ 2u
6
− u
5
− 2u
4
+ 2u
3
+ 1
c
11
u
8
− 2u
7
+ u
6
− 2u
4
+ 2u
3
+ u
2
− u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
8
+ 7y
7
+ 22y
6
+ 43y
5
+ 58y
4
+ 53y
3
+ 31y
2
+ 9y + 1
c
2
y
8
− 5y
7
− 2y
6
+ 23y
5
+ 46y
4
+ 57y
3
+ 123y
2
− 19y + 1
c
3
y
8
+ 14y
7
+ 81y
6
+ 230y
5
+ 336y
4
+ 238y
3
+ 77y
2
+ 17y + 9
c
4
, c
5
, c
9
y
8
+ 8y
7
+ 28y
6
+ 58y
5
+ 78y
4
+ 67y
3
+ 35y
2
+ 9y + 1
c
7
, c
10
y
8
− 4y
6
− y
5
+ 10y
4
− 4y
2
+ 1
c
8
y
8
− 4y
6
+ 10y
4
− y
3
− 4y
2
+ 1
c
11
y
8
− 2y
7
− 3y
6
+ 6y
5
+ 4y
4
− 6y
3
+ y
2
+ y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.186474 + 0.912486I
a = 0.013874 − 1.371960I
b = −0.307955 − 0.595773I
2.35558 − 2.73711I 0.76054 + 3.80045I
u = 0.186474 − 0.912486I
a = 0.013874 + 1.371960I
b = −0.307955 + 0.595773I
2.35558 + 2.73711I 0.76054 − 3.80045I
u = −0.456155 + 0.354859I
a = 1.219070 + 0.568101I
b = −0.188647 − 1.156400I
6.52667 − 1.60807I 7.81804 + 3.92468I
u = −0.456155 − 0.354859I
a = 1.219070 − 0.568101I
b = −0.188647 + 1.156400I
6.52667 + 1.60807I 7.81804 − 3.92468I
u = −0.26697 + 1.43177I
a = −0.909444 − 0.062444I
b = −1.377940 + 0.156538I
−2.57180 − 1.45446I 1.40377 + 1.96166I
u = −0.26697 − 1.43177I
a = −0.909444 + 0.062444I
b = −1.377940 − 0.156538I
−2.57180 + 1.45446I 1.40377 − 1.96166I
u = 0.53665 + 2.14327I
a = 0.343167 + 0.006141I
b = 0.874540 + 0.277933I
−6.31045 + 2.33823I −5.48234 − 1.59126I
u = 0.53665 − 2.14327I
a = 0.343167 − 0.006141I
b = 0.874540 − 0.277933I
−6.31045 − 2.33823I −5.48234 + 1.59126I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
8
+ u
7
+ 4u
6
+ 3u
5
+ 6u
4
+ 3u
3
+ 5u
2
+ u + 1)
· (u
8
+ 3u
7
+ 12u
6
+ 15u
5
+ 32u
4
+ 9u
3
+ 27u
2
− 21u + 11)
c
2
(u
8
+ 7u
7
+ 22u
6
+ 43u
5
+ 58u
4
+ 53u
3
+ 31u
2
+ 9u + 1)
· (u
8
+ 15u
7
+ ··· + 153u + 121)
c
3
(u
8
+ 7u
6
+ 4u
5
+ 16u
4
+ 10u
3
+ 11u
2
+ 7u + 3)
· (u
8
+ 2u
7
+ 9u
6
+ 10u
5
+ 20u
4
+ 8u
3
+ 7u
2
+ u + 1)
c
4
, c
5
(u
8
+ 4u
6
+ 6u
4
− u
3
+ 5u
2
− u + 1)
· (u
8
+ 8u
6
− 4u
5
+ 36u
4
− 29u
3
+ 41u
2
+ 17u + 19)
c
6
(u
8
− u
7
+ 4u
6
− 3u
5
+ 6u
4
− 3u
3
+ 5u
2
− u + 1)
· (u
8
+ 3u
7
+ 12u
6
+ 15u
5
+ 32u
4
+ 9u
3
+ 27u
2
− 21u + 11)
c
7
(u
8
+ 2u
7
+ 2u
6
+ u
5
− 2u
4
− 2u
3
+ 1)
· (u
8
+ 2u
7
+ 8u
6
+ 3u
5
+ 38u
4
− 38u
3
+ 58u
2
− 12u + 7)
c
8
(u
8
− 2u
5
− 2u
4
+ u
3
+ 2u
2
+ 2u + 1)
· (u
8
− 2u
7
+ 10u
6
+ 6u
5
+ 168u
4
+ 139u
3
+ 116u
2
− 60u + 47)
c
9
(u
8
+ 4u
6
+ 6u
4
+ u
3
+ 5u
2
+ u + 1)
· (u
8
+ 8u
6
− 4u
5
+ 36u
4
− 29u
3
+ 41u
2
+ 17u + 19)
c
10
(u
8
− 2u
7
+ 2u
6
− u
5
− 2u
4
+ 2u
3
+ 1)
· (u
8
+ 2u
7
+ 8u
6
+ 3u
5
+ 38u
4
− 38u
3
+ 58u
2
− 12u + 7)
c
11
(u
8
− 4u
7
+ 13u
6
− 28u
5
+ 90u
4
+ 60u
3
+ 85u
2
+ 23u + 11)
· (u
8
− 2u
7
+ u
6
− 2u
4
+ 2u
3
+ u
2
− u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
8
+ 7y
7
+ 22y
6
+ 43y
5
+ 58y
4
+ 53y
3
+ 31y
2
+ 9y + 1)
· (y
8
+ 15y
7
+ ··· + 153y + 121)
c
2
(y
8
− 5y
7
− 2y
6
+ 23y
5
+ 46y
4
+ 57y
3
+ 123y
2
− 19y + 1)
· (y
8
+ 11y
7
+ ··· + 414853y + 14641)
c
3
(y
8
+ 14y
7
+ 81y
6
+ 230y
5
+ 336y
4
+ 238y
3
+ 77y
2
+ 17y + 9)
· (y
8
+ 14y
7
+ 81y
6
+ 242y
5
+ 364y
4
+ 214y
3
+ 73y
2
+ 13y + 1)
c
4
, c
5
, c
9
(y
8
+ 8y
7
+ 28y
6
+ 58y
5
+ 78y
4
+ 67y
3
+ 35y
2
+ 9y + 1)
· (y
8
+ 16y
7
+ ··· + 1269y + 361)
c
7
, c
10
(y
8
− 4y
6
− y
5
+ 10y
4
− 4y
2
+ 1)(y
8
+ 12y
7
+ ··· + 668y + 49)
c
8
(y
8
− 4y
6
+ 10y
4
− y
3
− 4y
2
+ 1)(y
8
+ 16y
7
+ ··· + 7304y + 2209)
c
11
(y
8
− 2y
7
− 3y
6
+ 6y
5
+ 4y
4
− 6y
3
+ y
2
+ y + 1)
· (y
8
+ 10y
7
+ ··· + 1341y + 121)
11
