11n
36
(K11n
36
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 9 10 4 11 1 8 6
Solving Sequence
1,4
2
5,9
6 10 7 3 11 8
c
1
c
4
c
5
c
9
c
6
c
3
c
11
c
8
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h2.74497 × 10
33
u
40
+ 1.38636 × 10
34
u
39
+ ··· + 3.54634 × 10
33
b − 7.84109 × 10
31
,
2.58793 × 10
33
u
40
+ 1.11480 × 10
34
u
39
+ ··· + 3.54634 × 10
33
a − 9.38667 × 10
33
, u
41
+ 7u
40
+ ··· + 2u + 1i
I
u
2
= ha
4
− 6a
3
+ 9a
2
+ b − 8a + 3, a
5
− 6a
4
+ 9a
3
− 8a
2
+ 4a − 1, u − 1i
I
u
3
= hb, 3u
2
+ a + 5u + 4, u
3
+ u
2
− 1i
* 3 irreducible components of dim
C
= 0, with total 49 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
= h2.74 × 10
33
u
40
+ 1.39 × 10
34
u
39
+ · · · + 3.55 × 10
33
b − 7.84 × 10
31
, 2.59 ×
10
33
u
40
+1.11×10
34
u
39
+· · ·+3.55×10
33
a−9.39×10
33
, u
41
+7u
40
+· · ·+2u+1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
9
=
−0.729746u
40
− 3.14351u
39
+ ··· − 16.1294u + 2.64686
−0.774027u
40
− 3.90927u
39
+ ··· − 5.77644u + 0.0221103
a
6
=
−1.47326u
40
− 6.97084u
39
+ ··· + 1.83626u − 0.252120
0.773071u
40
+ 4.41057u
39
+ ··· + 1.49955u + 1.25550
a
10
=
0.0442820u
40
+ 0.765761u
39
+ ··· − 10.3529u + 2.62475
−0.774027u
40
− 3.90927u
39
+ ··· − 5.77644u + 0.0221103
a
7
=
1.89468u
40
+ 10.8589u
39
+ ··· − 3.67648u + 4.04534
−4.09416u
40
− 22.1611u
39
+ ··· − 6.24482u − 2.19947
a
3
=
−u
2
+ 1
u
2
a
11
=
1.60981u
40
+ 8.17572u
39
+ ··· + 9.41348u + 0.620656
0.427689u
40
+ 2.12225u
39
+ ··· + 3.33185u − 0.204351
a
8
=
1.89468u
40
+ 10.8589u
39
+ ··· − 3.67648u + 4.04534
−0.427689u
40
− 2.12225u
39
+ ··· − 3.33185u + 0.204351
a
8
=
1.89468u
40
+ 10.8589u
39
+ ··· − 3.67648u + 4.04534
−0.427689u
40
− 2.12225u
39
+ ··· − 3.33185u + 0.204351
(ii) Obstruction class = −1
(iii) Cusp Shapes = 9.46225u
40
+ 53.2125u
39
+ ··· + 10.2497u + 12.8539
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
41
− 7u
40
+ ··· + 2u − 1
c
2
u
41
+ 43u
40
+ ··· + 12u + 1
c
3
, c
7
u
41
+ 2u
40
+ ··· + 96u + 32
c
5
u
41
+ 4u
40
+ ··· − 237u + 191
c
6
u
41
+ 16u
39
+ ··· − 1085u − 79
c
8
, c
10
u
41
+ 5u
40
+ ··· + 119u + 1
c
9
u
41
− 6u
40
+ ··· + 156u − 8
c
11
u
41
+ 3u
40
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
41
− 43y
40
+ ··· + 12y −1
c
2
y
41
− 83y
40
+ ··· + 1144y −1
c
3
, c
7
y
41
− 30y
40
+ ··· + 3584y −1024
c
5
y
41
+ 8y
40
+ ··· + 999709y −36481
c
6
y
41
+ 32y
40
+ ··· + 183721y −6241
c
8
, c
10
y
41
− 21y
40
+ ··· + 13495y −1
c
9
y
41
− 18y
40
+ ··· + 7824y −64
c
11
y
41
− 11y
40
+ ··· + 26y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.387590 + 0.911908I
a = −0.252509 − 0.478840I
b = 1.071600 + 0.110579I
−2.61027 − 2.03740I −5.72892 + 3.65159I
u = 0.387590 − 0.911908I
a = −0.252509 + 0.478840I
b = 1.071600 − 0.110579I
−2.61027 + 2.03740I −5.72892 − 3.65159I
u = 0.695393 + 0.752192I
a = 0.171525 − 0.630794I
b = 1.287490 + 0.541839I
−3.58550 − 3.36599I −6.85826 + 4.39505I
u = 0.695393 − 0.752192I
a = 0.171525 + 0.630794I
b = 1.287490 − 0.541839I
−3.58550 + 3.36599I −6.85826 − 4.39505I
u = 0.883212
a = −8.08484
b = −0.317773
0.458131 −57.1150
u = 1.149310 + 0.071261I
a = −1.59369 + 0.46729I
b = −0.160687 + 0.786703I
−0.578838 − 1.255810I 2.38019 + 0.I
u = 1.149310 − 0.071261I
a = −1.59369 − 0.46729I
b = −0.160687 − 0.786703I
−0.578838 + 1.255810I 2.38019 + 0.I
u = 0.508644 + 1.042800I
a = −0.333727 + 0.529481I
b = −1.26087 − 0.71395I
−1.17487 − 9.23550I 0. + 7.03311I
u = 0.508644 − 1.042800I
a = −0.333727 − 0.529481I
b = −1.26087 + 0.71395I
−1.17487 + 9.23550I 0. − 7.03311I
u = −0.817513 + 0.853964I
a = −0.249736 − 0.135119I
b = −0.502875 + 0.177164I
4.46595 + 3.11596I −9.1421 − 11.7493I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.817513 − 0.853964I
a = −0.249736 + 0.135119I
b = −0.502875 − 0.177164I
4.46595 − 3.11596I −9.1421 + 11.7493I
u = −0.762796 + 0.059538I
a = −0.517105 + 0.839463I
b = −0.465748 + 1.069230I
4.65237 + 4.48889I 10.07507 − 5.98728I
u = −0.762796 − 0.059538I
a = −0.517105 − 0.839463I
b = −0.465748 − 1.069230I
4.65237 − 4.48889I 10.07507 + 5.98728I
u = 0.858145 + 0.924978I
a = −0.119106 + 0.545272I
b = −1.093570 + 0.304255I
−2.16828 + 2.66511I 0
u = 0.858145 − 0.924978I
a = −0.119106 − 0.545272I
b = −1.093570 − 0.304255I
−2.16828 − 2.66511I 0
u = 0.737003
a = −0.781314
b = −0.0927869
−1.10369 −8.82470
u = 0.612280 + 0.220916I
a = −2.04946 + 4.65031I
b = −0.010448 − 0.399153I
0.484163 − 0.158339I 13.38590 + 1.00156I
u = 0.612280 − 0.220916I
a = −2.04946 − 4.65031I
b = −0.010448 + 0.399153I
0.484163 + 0.158339I 13.38590 − 1.00156I
u = 0.380775 + 0.454242I
a = −0.51545 − 1.79136I
b = −0.444963 + 1.151080I
1.43126 − 2.56358I 1.01752 + 7.87421I
u = 0.380775 − 0.454242I
a = −0.51545 + 1.79136I
b = −0.444963 − 1.151080I
1.43126 + 2.56358I 1.01752 − 7.87421I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.46523 + 0.11671I
a = 1.34696 − 0.69695I
b = 1.262320 − 0.189235I
−5.33009 + 0.54259I 0
u = 1.46523 − 0.11671I
a = 1.34696 + 0.69695I
b = 1.262320 + 0.189235I
−5.33009 − 0.54259I 0
u = −1.48400
a = −2.11095
b = −2.27004
−2.98279 0
u = −1.51445 + 0.11394I
a = −0.301276 − 1.124650I
b = −0.58113 − 2.13843I
−4.95010 + 4.48342I 0
u = −1.51445 − 0.11394I
a = −0.301276 + 1.124650I
b = −0.58113 + 2.13843I
−4.95010 − 4.48342I 0
u = −1.56483 + 0.05519I
a = 0.765814 − 0.615203I
b = 0.519201 + 0.768233I
−6.84034 + 1.09870I 0
u = −1.56483 − 0.05519I
a = 0.765814 + 0.615203I
b = 0.519201 − 0.768233I
−6.84034 − 1.09870I 0
u = −1.53570 + 0.38602I
a = 1.094720 + 0.335992I
b = 1.195690 − 0.685677I
−8.77885 + 6.88775I 0
u = −1.53570 − 0.38602I
a = 1.094720 − 0.335992I
b = 1.195690 + 0.685677I
−8.77885 − 6.88775I 0
u = 1.60618 + 0.13682I
a = −1.43496 − 0.20448I
b = −1.248460 − 0.492822I
−3.96071 − 6.10430I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.60618 − 0.13682I
a = −1.43496 + 0.20448I
b = −1.248460 + 0.492822I
−3.96071 + 6.10430I 0
u = −1.61056 + 0.23381I
a = 1.69093 − 0.02813I
b = 1.81725 − 1.03078I
−11.27280 + 7.05517I 0
u = −1.61056 − 0.23381I
a = 1.69093 + 0.02813I
b = 1.81725 + 1.03078I
−11.27280 − 7.05517I 0
u = −1.58766 + 0.38818I
a = −1.60185 − 0.23694I
b = −1.47850 + 1.01659I
−7.9395 + 14.4828I 0
u = −1.58766 − 0.38818I
a = −1.60185 + 0.23694I
b = −1.47850 − 1.01659I
−7.9395 − 14.4828I 0
u = −1.68426 + 0.19522I
a = −1.133500 − 0.234872I
b = −1.139920 + 0.414386I
−11.04370 + 1.47634I 0
u = −1.68426 − 0.19522I
a = −1.133500 + 0.234872I
b = −1.139920 − 0.414386I
−11.04370 − 1.47634I 0
u = −0.213366 + 0.037411I
a = 0.62944 + 2.88854I
b = 0.480792 + 0.712878I
0.05575 − 1.50352I 0.38191 + 4.17550I
u = −0.213366 − 0.037411I
a = 0.62944 − 2.88854I
b = 0.480792 − 0.712878I
0.05575 + 1.50352I 0.38191 − 4.17550I
u = 0.059485 + 0.186213I
a = 0.39154 − 2.74926I
b = −0.906876 − 0.270893I
2.56340 + 0.10081I 4.27921 + 2.25595I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.059485 − 0.186213I
a = 0.39154 + 2.74926I
b = −0.906876 + 0.270893I
2.56340 − 0.10081I 4.27921 − 2.25595I
9
II. I
u
2
= ha
4
− 6a
3
+ 9a
2
+ b − 8a + 3, a
5
− 6a
4
+ 9a
3
− 8a
2
+ 4a − 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
9
=
a
−a
4
+ 6a
3
− 9a
2
+ 8a − 3
a
6
=
−a
−2a
4
+ 11a
3
− 12a
2
+ 7a − 1
a
10
=
a
4
− 6a
3
+ 9a
2
− 7a + 3
−a
4
+ 6a
3
− 9a
2
+ 8a − 3
a
7
=
0
−3a
4
+ 16a
3
− 15a
2
+ 7a − 1
a
3
=
0
1
a
11
=
a
4
− 6a
3
+ 9a
2
− 7a + 3
−3a
4
+ 16a
3
− 15a
2
+ 7a − 1
a
8
=
0
−3a
4
+ 16a
3
− 15a
2
+ 7a − 1
a
8
=
0
−3a
4
+ 16a
3
− 15a
2
+ 7a − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 9a
4
− 48a
3
+ 48a
2
− 32a
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
c
2
, c
4
(u + 1)
5
c
3
, c
7
u
5
c
5
, c
9
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
6
, c
8
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
10
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
11
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
5
c
3
, c
7
y
5
c
5
, c
9
y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1
c
6
, c
8
, c
10
y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1
c
11
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.313425 + 0.691081I
b = 0.455697 + 1.200150I
4.22763 − 4.40083I −8.55516 + 1.78781I
u = 1.00000
a = 0.313425 − 0.691081I
b = 0.455697 − 1.200150I
4.22763 + 4.40083I −8.55516 − 1.78781I
u = 1.00000
a = 0.542256 + 0.333011I
b = −0.339110 + 0.822375I
−1.31583 + 1.53058I −8.42731 − 4.45807I
u = 1.00000
a = 0.542256 − 0.333011I
b = −0.339110 − 0.822375I
−1.31583 − 1.53058I −8.42731 + 4.45807I
u = 1.00000
a = 4.28864
b = 0.766826
0.756147 3.96490
13
III. I
u
3
= hb, 3u
2
+ a + 5u + 4, u
3
+ u
2
− 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
u
2
+ u − 1
a
9
=
−3u
2
− 5u − 4
0
a
6
=
−9u
2
− 17u − 12
u
2
+ u − 1
a
10
=
−3u
2
− 5u − 4
0
a
7
=
−u
u
2
+ u − 1
a
3
=
−u
2
+ 1
u
2
a
11
=
−3u
2
− 4u − 4
−2u
2
− u + 2
a
8
=
−u
2u
2
+ u − 2
a
8
=
−u
2u
2
+ u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 21u
2
+ 45u + 39
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ u
2
− 1
c
2
, c
7
u
3
+ u
2
+ 2u + 1
c
3
u
3
− u
2
+ 2u − 1
c
4
u
3
− u
2
+ 1
c
5
, c
6
u
3
− 2u
2
− 3u − 1
c
8
(u + 1)
3
c
9
u
3
c
10
(u − 1)
3
c
11
u
3
+ 3u
2
+ 2u − 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
3
− y
2
+ 2y − 1
c
2
, c
3
, c
7
y
3
+ 3y
2
+ 2y − 1
c
5
, c
6
y
3
− 10y
2
+ 5y − 1
c
8
, c
10
(y − 1)
3
c
9
y
3
c
11
y
3
− 5y
2
+ 10y − 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.258045 + 0.197115I
b = 0
4.66906 + 2.82812I 4.03193 + 6.06881I
u = −0.877439 − 0.744862I
a = −0.258045 − 0.197115I
b = 0
4.66906 − 2.82812I 4.03193 − 6.06881I
u = 0.754878
a = −9.48391
b = 0
0.531480 84.9360
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
3
+ u
2
− 1)(u
41
− 7u
40
+ ··· + 2u − 1)
c
2
((u + 1)
5
)(u
3
+ u
2
+ 2u + 1)(u
41
+ 43u
40
+ ··· + 12u + 1)
c
3
u
5
(u
3
− u
2
+ 2u − 1)(u
41
+ 2u
40
+ ··· + 96u + 32)
c
4
((u + 1)
5
)(u
3
− u
2
+ 1)(u
41
− 7u
40
+ ··· + 2u − 1)
c
5
(u
3
− 2u
2
− 3u − 1)(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
· (u
41
+ 4u
40
+ ··· − 237u + 191)
c
6
(u
3
− 2u
2
− 3u − 1)(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
· (u
41
+ 16u
39
+ ··· − 1085u − 79)
c
7
u
5
(u
3
+ u
2
+ 2u + 1)(u
41
+ 2u
40
+ ··· + 96u + 32)
c
8
((u + 1)
3
)(u
5
− u
4
+ ··· + u + 1)(u
41
+ 5u
40
+ ··· + 119u + 1)
c
9
u
3
(u
5
+ u
4
+ ··· + u + 1)(u
41
− 6u
40
+ ··· + 156u − 8)
c
10
((u − 1)
3
)(u
5
+ u
4
+ ··· + u − 1)(u
41
+ 5u
40
+ ··· + 119u + 1)
c
11
(u
3
+ 3u
2
+ 2u − 1)(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
· (u
41
+ 3u
40
+ ··· − 2u − 1)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y − 1)
5
)(y
3
− y
2
+ 2y − 1)(y
41
− 43y
40
+ ··· + 12y − 1)
c
2
((y − 1)
5
)(y
3
+ 3y
2
+ 2y − 1)(y
41
− 83y
40
+ ··· + 1144y − 1)
c
3
, c
7
y
5
(y
3
+ 3y
2
+ 2y − 1)(y
41
− 30y
40
+ ··· + 3584y − 1024)
c
5
(y
3
− 10y
2
+ 5y − 1)(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
· (y
41
+ 8y
40
+ ··· + 999709y − 36481)
c
6
(y
3
− 10y
2
+ 5y − 1)(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
· (y
41
+ 32y
40
+ ··· + 183721y − 6241)
c
8
, c
10
((y − 1)
3
)(y
5
− 5y
4
+ ··· − y − 1)(y
41
− 21y
40
+ ··· + 13495y − 1)
c
9
y
3
(y
5
+ 3y
4
+ ··· − y − 1)(y
41
− 18y
40
+ ··· + 7824y − 64)
c
11
(y
3
− 5y
2
+ 10y − 1)(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
· (y
41
− 11y
40
+ ··· + 26y − 1)
19
