11n
18
(K11n
18
)
A knot diagram
1
Linearized knot diagam
5 1 9 2 3 10 11 3 1 8 7
Solving Sequence
1,7
11 8 10
3,6
5 2 4 9
c
11
c
7
c
10
c
6
c
5
c
1
c
4
c
9
c
2
, c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
21
+ 2u
20
+ ··· + 2b − 1, −2u
21
+ 6u
20
+ ··· + 2a + 5, u
22
− 3u
21
+ ··· − 4u + 1i
I
u
2
= hu
2
a + b + a, u
2
a + a
2
+ au + a − u, u
3
+ u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 28 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−u
21
+2u
20
+· · ·+2b−1, −2u
21
+6u
20
+· · ·+2a+5, u
22
−3u
21
+· · ·−4u+1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
11
=
1
−u
2
a
8
=
−u
u
3
+ u
a
10
=
u
2
+ 1
−u
4
− 2u
2
a
3
=
u
21
− 3u
20
+ ··· + 5u −
5
2
1
2
u
21
− u
20
+ ··· + u +
1
2
a
6
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
5
=
1
2
u
19
− u
18
+ ··· + 5u +
1
2
−
1
2
u
21
+ u
20
+ ··· − u +
1
2
a
2
=
3
2
u
21
− 4u
20
+ ··· + 6u − 2
1
2
u
21
− u
20
+ ··· + u +
1
2
a
4
=
−u
20
+
3
2
u
19
+ ··· + u +
1
2
1
2
u
21
− 2u
20
+ ··· + 3u −
3
2
a
9
=
−u
4
− u
2
+ 1
−u
4
− 2u
2
a
9
=
−u
4
− u
2
+ 1
−u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 3u
21
−
13
2
u
20
+
81
2
u
19
−73u
18
+226u
17
−340u
16
+674u
15
−828u
14
+
2275
2
u
13
−1060u
12
+
1009u
11
−
1069
2
u
10
+
547
2
u
9
+
371
2
u
8
−198u
7
+ 242u
6
−60u
5
+
11
2
u
4
+ 82u
3
−
19
2
u
2
+
23
2
u −
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
22
+ 4u
21
+ ··· + 3u + 1
c
2
u
22
+ 4u
21
+ ··· + 11u + 1
c
3
, c
8
u
22
− u
21
+ ··· − 32u + 64
c
5
u
22
− 4u
21
+ ··· + 1113u + 306
c
6
u
22
+ 3u
21
+ ··· − 105u + 34
c
7
, c
10
, c
11
u
22
− 3u
21
+ ··· − 4u + 1
c
9
u
22
− u
21
+ ··· + 2u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
22
+ 4y
21
+ ··· + 11y + 1
c
2
y
22
+ 32y
21
+ ··· + 11y + 1
c
3
, c
8
y
22
− 35y
21
+ ··· − 17408y + 4096
c
5
y
22
+ 60y
21
+ ··· + 3785751y + 93636
c
6
y
22
+ 19y
21
+ ··· + 18011y + 1156
c
7
, c
10
, c
11
y
22
+ 23y
21
+ ··· + 4y + 1
c
9
y
22
+ 39y
21
+ ··· + 4y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.770283 + 0.589538I
a = 1.125210 − 0.379310I
b = −0.07905 − 1.81869I
10.13960 + 0.88452I −0.211027 + 0.306129I
u = 0.770283 − 0.589538I
a = 1.125210 + 0.379310I
b = −0.07905 + 1.81869I
10.13960 − 0.88452I −0.211027 − 0.306129I
u = 0.804807 + 0.517036I
a = −1.042990 + 0.300031I
b = 0.13147 + 1.87390I
9.90919 − 6.12637I −0.73850 + 4.70880I
u = 0.804807 − 0.517036I
a = −1.042990 − 0.300031I
b = 0.13147 − 1.87390I
9.90919 + 6.12637I −0.73850 − 4.70880I
u = −0.115563 + 1.244550I
a = 0.708694 + 0.469396I
b = 0.634802 − 0.033810I
1.83932 + 1.95875I −3.73580 − 3.68347I
u = −0.115563 − 1.244550I
a = 0.708694 − 0.469396I
b = 0.634802 + 0.033810I
1.83932 − 1.95875I −3.73580 + 3.68347I
u = −0.248700 + 1.353780I
a = 0.267493 − 0.437974I
b = −0.005781 − 0.383501I
3.35457 + 3.66509I 0.212427 − 1.175787I
u = −0.248700 − 1.353780I
a = 0.267493 + 0.437974I
b = −0.005781 + 0.383501I
3.35457 − 3.66509I 0.212427 + 1.175787I
u = −0.597356 + 0.125917I
a = 0.501330 − 0.329651I
b = 0.173251 − 0.149140I
−1.35470 + 0.57102I −7.20802 − 0.39012I
u = −0.597356 − 0.125917I
a = 0.501330 + 0.329651I
b = 0.173251 + 0.149140I
−1.35470 − 0.57102I −7.20802 + 0.39012I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.08081 + 1.44732I
a = 0.03594 + 1.88889I
b = 0.95469 + 1.28056I
5.22365 − 3.91165I 1.62467 + 2.79581I
u = 0.08081 − 1.44732I
a = 0.03594 − 1.88889I
b = 0.95469 − 1.28056I
5.22365 + 3.91165I 1.62467 − 2.79581I
u = −0.02169 + 1.49375I
a = 0.116291 − 1.385550I
b = −0.551917 − 1.006360I
7.30874 + 1.68962I 3.46122 − 1.99684I
u = −0.02169 − 1.49375I
a = 0.116291 + 1.385550I
b = −0.551917 + 1.006360I
7.30874 − 1.68962I 3.46122 + 1.99684I
u = −0.037659 + 0.478054I
a = 1.42737 − 0.64365I
b = 0.056911 − 0.654735I
0.83479 + 1.39529I 1.49278 − 4.06161I
u = −0.037659 − 0.478054I
a = 1.42737 + 0.64365I
b = 0.056911 + 0.654735I
0.83479 − 1.39529I 1.49278 + 4.06161I
u = 0.28918 + 1.53736I
a = −1.00511 + 1.94899I
b = 0.28840 + 2.00471I
16.5973 − 10.1473I 1.94212 + 4.94349I
u = 0.28918 − 1.53736I
a = −1.00511 − 1.94899I
b = 0.28840 − 2.00471I
16.5973 + 10.1473I 1.94212 − 4.94349I
u = 0.25416 + 1.56446I
a = 0.88215 − 1.89631I
b = −0.32453 − 1.88491I
17.2299 − 2.8896I 2.69042 + 0.63603I
u = 0.25416 − 1.56446I
a = 0.88215 + 1.89631I
b = −0.32453 + 1.88491I
17.2299 + 2.8896I 2.69042 − 0.63603I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.321731 + 0.235214I
a = −2.01637 + 0.18504I
b = 0.721764 + 0.861777I
−0.35018 − 2.57282I 0.96973 + 5.85943I
u = 0.321731 − 0.235214I
a = −2.01637 − 0.18504I
b = 0.721764 − 0.861777I
−0.35018 + 2.57282I 0.96973 − 5.85943I
7
II. I
u
2
= hu
2
a + b + a, u
2
a + a
2
+ au + a − u, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
11
=
1
−u
2
a
8
=
−u
−u
2
− u − 1
a
10
=
u
2
+ 1
−u
2
− u − 1
a
3
=
a
−u
2
a − a
a
6
=
−1
0
a
5
=
u
2
+ a + u
−u
2
a − a − 1
a
2
=
−u
2
a
−u
2
a − a
a
4
=
a
−u
2
a − a
a
9
=
−u
−u
2
− u − 1
a
9
=
−u
−u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
a − au − 3u
2
− 3a − 3u − 8
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
3
c
3
, c
8
u
6
c
4
(u
2
− u + 1)
3
c
6
, c
9
(u
3
+ u
2
− 1)
2
c
7
(u
3
− u
2
+ 2u − 1)
2
c
10
, c
11
(u
3
+ u
2
+ 2u + 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
3
c
3
, c
8
y
6
c
6
, c
9
(y
3
− y
2
+ 2y −1)
2
c
7
, c
10
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −0.206350 − 1.132320I
b = 0.500000 − 0.866025I
3.02413 + 0.79824I 1.45566 + 0.28364I
u = −0.215080 + 1.307140I
a = 1.083790 + 0.387453I
b = 0.500000 + 0.866025I
3.02413 + 4.85801I −2.09851 − 6.80481I
u = −0.215080 − 1.307140I
a = −0.206350 + 1.132320I
b = 0.500000 + 0.866025I
3.02413 − 0.79824I 1.45566 − 0.28364I
u = −0.215080 − 1.307140I
a = 1.083790 − 0.387453I
b = 0.500000 − 0.866025I
3.02413 − 4.85801I −2.09851 + 6.80481I
u = −0.569840
a = −0.377439 + 0.653743I
b = 0.500000 − 0.866025I
−1.11345 + 2.02988I −5.85715 − 2.43783I
u = −0.569840
a = −0.377439 − 0.653743I
b = 0.500000 + 0.866025I
−1.11345 − 2.02988I −5.85715 + 2.43783I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
22
+ 4u
21
+ ··· + 3u + 1)
c
2
((u
2
+ u + 1)
3
)(u
22
+ 4u
21
+ ··· + 11u + 1)
c
3
, c
8
u
6
(u
22
− u
21
+ ··· − 32u + 64)
c
4
((u
2
− u + 1)
3
)(u
22
+ 4u
21
+ ··· + 3u + 1)
c
5
((u
2
+ u + 1)
3
)(u
22
− 4u
21
+ ··· + 1113u + 306)
c
6
((u
3
+ u
2
− 1)
2
)(u
22
+ 3u
21
+ ··· − 105u + 34)
c
7
((u
3
− u
2
+ 2u − 1)
2
)(u
22
− 3u
21
+ ··· − 4u + 1)
c
9
((u
3
+ u
2
− 1)
2
)(u
22
− u
21
+ ··· + 2u
2
+ 1)
c
10
, c
11
((u
3
+ u
2
+ 2u + 1)
2
)(u
22
− 3u
21
+ ··· − 4u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
3
)(y
22
+ 4y
21
+ ··· + 11y + 1)
c
2
((y
2
+ y + 1)
3
)(y
22
+ 32y
21
+ ··· + 11y + 1)
c
3
, c
8
y
6
(y
22
− 35y
21
+ ··· − 17408y + 4096)
c
5
((y
2
+ y + 1)
3
)(y
22
+ 60y
21
+ ··· + 3785751y + 93636)
c
6
((y
3
− y
2
+ 2y −1)
2
)(y
22
+ 19y
21
+ ··· + 18011y + 1156)
c
7
, c
10
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
22
+ 23y
21
+ ··· + 4y + 1)
c
9
((y
3
− y
2
+ 2y −1)
2
)(y
22
+ 39y
21
+ ··· + 4y + 1)
13
