11n
44
(K11n
44
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 10 11 4 6 1 8 9
Solving Sequence
1,4
2
5,10
6 9 8 11 7 3
c
1
c
4
c
5
c
9
c
8
c
11
c
6
c
3
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−4.55071 × 10
32
u
40
− 3.04971 × 10
33
u
39
+ ··· + 3.54634 × 10
33
b + 1.97956 × 10
32
,
− 1.27956 × 10
33
u
40
− 7.27594 × 10
33
u
39
+ ··· + 3.54634 × 10
33
a − 1.00211 × 10
34
, u
41
+ 7u
40
+ ··· + 2u + 1i
I
u
2
= hb − a + 1, a
5
− 4a
4
+ 4a
3
+ a
2
− 2a − 1, u − 1i
I
u
3
= hb − 1, −2u
2
+ a − 4u − 3, 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
= h−4.55×10
32
u
40
−3.05×10
33
u
39
+· · ·+3.55×10
33
b+1.98×10
32
, −1.28×
10
33
u
40
−7.28×10
33
u
39
+· · ·+3.55×10
33
a−1.00×10
34
, 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
10
=
0.360812u
40
+ 2.05167u
39
+ ··· − 8.17453u + 2.82576
0.128321u
40
+ 0.859959u
39
+ ··· − 3.73417u − 0.0558198
a
6
=
2.57419u
40
+ 13.6219u
39
+ ··· − 13.2764u + 5.50487
−0.824828u
40
− 4.46073u
39
+ ··· − 3.42158u − 0.703781
a
9
=
0.232491u
40
+ 1.19171u
39
+ ··· − 4.44036u + 2.88158
0.128321u
40
+ 0.859959u
39
+ ··· − 3.73417u − 0.0558198
a
8
=
1.89468u
40
+ 10.8589u
39
+ ··· − 3.67648u + 4.04534
−0.427689u
40
− 2.12225u
39
+ ··· − 3.33185u + 0.204351
a
11
=
−0.101116u
40
− 0.258346u
39
+ ··· + 7.23983u − 3.55682
0.153838u
40
+ 0.879491u
39
+ ··· + 4.81960u + 0.645350
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
3
=
−u
2
+ 1
u
2
(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
+ 16u
39
+ ··· − 1085u − 79
c
6
u
41
+ 4u
40
+ ··· − 237u + 191
c
8
u
41
+ 3u
40
+ ··· − 2u − 1
c
9
, c
11
u
41
+ 5u
40
+ ··· + 119u + 1
c
10
u
41
− 6u
40
+ ··· + 156u − 8
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
+ 32y
40
+ ··· + 183721y −6241
c
6
y
41
+ 8y
40
+ ··· + 999709y −36481
c
8
y
41
− 11y
40
+ ··· + 26y −1
c
9
, c
11
y
41
− 21y
40
+ ··· + 13495y −1
c
10
y
41
− 18y
40
+ ··· + 7824y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.387590 + 0.911908I
a = −0.971331 − 0.325235I
b = 0.764541 − 0.597354I
−2.61027 − 2.03740I −5.72892 + 3.65159I
u = 0.387590 − 0.911908I
a = −0.971331 + 0.325235I
b = 0.764541 + 0.597354I
−2.61027 + 2.03740I −5.72892 − 3.65159I
u = 0.695393 + 0.752192I
a = −0.0733729 + 0.1016160I
b = 0.382217 + 0.951284I
−3.58550 − 3.36599I −6.85826 + 4.39505I
u = 0.695393 − 0.752192I
a = −0.0733729 − 0.1016160I
b = 0.382217 − 0.951284I
−3.58550 + 3.36599I −6.85826 − 4.39505I
u = 0.883212
a = −5.41324
b = −1.04711
0.458131 −57.1150
u = 1.149310 + 0.071261I
a = −2.67006 + 0.97064I
b = −0.783716 + 0.351647I
−0.578838 − 1.255810I 2.38019 + 0.I
u = 1.149310 − 0.071261I
a = −2.67006 − 0.97064I
b = −0.783716 − 0.351647I
−0.578838 + 1.255810I 2.38019 + 0.I
u = 0.508644 + 1.042800I
a = −0.681098 − 0.649137I
b = 1.178240 − 0.659530I
−1.17487 − 9.23550I 0. + 7.03311I
u = 0.508644 − 1.042800I
a = −0.681098 + 0.649137I
b = 1.178240 + 0.659530I
−1.17487 + 9.23550I 0. − 7.03311I
u = −0.817513 + 0.853964I
a = −0.255534 + 0.434819I
b = 0.895543 + 0.128230I
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 = −0.255534 − 0.434819I
b = 0.895543 − 0.128230I
4.46595 − 3.11596I −9.1421 + 11.7493I
u = −0.762796 + 0.059538I
a = 0.685914 + 0.388453I
b = 1.354300 + 0.223700I
4.65237 + 4.48889I 10.07507 − 5.98728I
u = −0.762796 − 0.059538I
a = 0.685914 − 0.388453I
b = 1.354300 − 0.223700I
4.65237 − 4.48889I 10.07507 + 5.98728I
u = 0.858145 + 0.924978I
a = −0.0136488 − 0.1120600I
b = 0.908643 + 0.588751I
−2.16828 + 2.66511I 0
u = 0.858145 − 0.924978I
a = −0.0136488 + 0.1120600I
b = 0.908643 − 0.588751I
−2.16828 − 2.66511I 0
u = 0.737003
a = −0.918962
b = 0.00861004
−1.10369 −8.82470
u = 0.612280 + 0.220916I
a = −0.42857 + 4.33330I
b = −0.908176 − 0.005657I
0.484163 − 0.158339I 13.38590 + 1.00156I
u = 0.612280 − 0.220916I
a = −0.42857 − 4.33330I
b = −0.908176 + 0.005657I
0.484163 + 0.158339I 13.38590 − 1.00156I
u = 0.380775 + 0.454242I
a = 0.670485 − 0.032746I
b = −1.007360 + 0.614689I
1.43126 − 2.56358I 1.01752 + 7.87421I
u = 0.380775 − 0.454242I
a = 0.670485 + 0.032746I
b = −1.007360 − 0.614689I
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 = −0.138455 − 1.392320I
b = 0.596164 − 0.809978I
−5.33009 + 0.54259I 0
u = 1.46523 − 0.11671I
a = −0.138455 + 1.392320I
b = 0.596164 + 0.809978I
−5.33009 − 0.54259I 0
u = −1.48400
a = −0.697477
b = −1.91952
−2.98279 0
u = −1.51445 + 0.11394I
a = −0.47763 − 1.44493I
b = −1.22246 − 1.14875I
−4.95010 + 4.48342I 0
u = −1.51445 − 0.11394I
a = −0.47763 + 1.44493I
b = −1.22246 + 1.14875I
−4.95010 − 4.48342I 0
u = −1.56483 + 0.05519I
a = 0.05994 − 1.52905I
b = −1.049240 − 0.368090I
−6.84034 + 1.09870I 0
u = −1.56483 − 0.05519I
a = 0.05994 + 1.52905I
b = −1.049240 + 0.368090I
−6.84034 − 1.09870I 0
u = −1.53570 + 0.38602I
a = −0.149350 + 1.179900I
b = 1.157160 + 0.624027I
−8.77885 + 6.88775I 0
u = −1.53570 − 0.38602I
a = −0.149350 − 1.179900I
b = 1.157160 − 0.624027I
−8.77885 − 6.88775I 0
u = 1.60618 + 0.13682I
a = 0.489280 − 1.112380I
b = 1.046200 − 0.671367I
−3.96071 − 6.10430I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.60618 − 0.13682I
a = 0.489280 + 1.112380I
b = 1.046200 + 0.671367I
−3.96071 + 6.10430I 0
u = −1.61056 + 0.23381I
a = 0.423955 − 1.249650I
b = 0.32746 − 1.40800I
−11.27280 + 7.05517I 0
u = −1.61056 − 0.23381I
a = 0.423955 + 1.249650I
b = 0.32746 + 1.40800I
−11.27280 − 7.05517I 0
u = −1.58766 + 0.38818I
a = 0.05254 + 1.45672I
b = 1.36138 + 0.75276I
−7.9395 + 14.4828I 0
u = −1.58766 − 0.38818I
a = 0.05254 − 1.45672I
b = 1.36138 − 0.75276I
−7.9395 − 14.4828I 0
u = −1.68426 + 0.19522I
a = 0.197462 − 0.763635I
b = 0.402639 − 0.808887I
−11.04370 + 1.47634I 0
u = −1.68426 − 0.19522I
a = 0.197462 + 0.763635I
b = 0.402639 + 0.808887I
−11.04370 − 1.47634I 0
u = −0.213366 + 0.037411I
a = −1.99525 + 2.11282I
b = −0.146268 + 0.552391I
0.05575 − 1.50352I 0.38191 + 4.17550I
u = −0.213366 − 0.037411I
a = −1.99525 − 2.11282I
b = −0.146268 − 0.552391I
0.05575 + 1.50352I 0.38191 − 4.17550I
u = 0.059485 + 0.186213I
a = 2.78956 − 5.11758I
b = −1.278260 − 0.126441I
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 = 2.78956 + 5.11758I
b = −1.278260 + 0.126441I
2.56340 − 0.10081I 4.27921 − 2.25595I
9
II. I
u
2
= hb − a + 1, a
5
− 4a
4
+ 4a
3
+ a
2
− 2a − 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
10
=
a
a − 1
a
6
=
a
2
− a − 1
a
2
− 2a + 1
a
9
=
1
a − 1
a
8
=
0
a
4
− 5a
3
+ 8a
2
− 3a − 2
a
11
=
a
a
2
− 2a + 1
a
7
=
0
a
4
− 5a
3
+ 8a
2
− 3a − 2
a
3
=
0
1
a
3
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3a
4
− 5a
3
− 5a
2
+ 7a − 5
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
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
8
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
10
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
11
u
5
+ u
4
− 2u
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
, c
11
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
6
, c
10
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
8
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 = 1.30992 + 0.54991I
b = 0.309916 + 0.549911I
−1.31583 + 1.53058I −8.42731 − 4.45807I
u = 1.00000
a = 1.30992 − 0.54991I
b = 0.309916 − 0.549911I
−1.31583 − 1.53058I −8.42731 + 4.45807I
u = 1.00000
a = −0.418784 + 0.219165I
b = −1.41878 + 0.21917I
4.22763 − 4.40083I −8.55516 + 1.78781I
u = 1.00000
a = −0.418784 − 0.219165I
b = −1.41878 − 0.21917I
4.22763 + 4.40083I −8.55516 − 1.78781I
u = 1.00000
a = 2.21774
b = 1.21774
0.756147 3.96490
13
III. I
u
3
= hb − 1, −2u
2
+ a − 4u − 3, 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
10
=
2u
2
+ 4u + 3
1
a
6
=
7u
2
+ 11u + 9
2u
2
+ 3u
a
9
=
2u
2
+ 4u + 2
1
a
8
=
−u
2u
2
+ u − 2
a
11
=
2u
2
+ 4u + 3
1
a
7
=
−u
u
2
+ u − 1
a
3
=
−u
2
+ 1
u
2
a
3
=
−u
2
+ 1
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
3
− 3u
2
+ 2u + 1
c
9
(u + 1)
3
c
10
u
3
c
11
(u − 1)
3
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
y
3
− 5y
2
+ 10y −1
c
9
, c
11
(y −1)
3
c
10
y
3
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.079596 + 0.365165I
b = 1.00000
4.66906 + 2.82812I 4.03193 + 6.06881I
u = −0.877439 − 0.744862I
a = −0.079596 − 0.365165I
b = 1.00000
4.66906 − 2.82812I 4.03193 − 6.06881I
u = 0.754878
a = 7.15919
b = 1.00000
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
+ 16u
39
+ ··· − 1085u − 79)
c
6
(u
3
+ 2u
2
− 3u + 1)(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
· (u
41
+ 4u
40
+ ··· − 237u + 191)
c
7
u
5
(u
3
+ u
2
+ 2u + 1)(u
41
+ 2u
40
+ ··· + 96u + 32)
c
8
(u
3
− 3u
2
+ 2u + 1)(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
· (u
41
+ 3u
40
+ ··· − 2u − 1)
c
9
((u + 1)
3
)(u
5
− u
4
+ ··· + u + 1)(u
41
+ 5u
40
+ ··· + 119u + 1)
c
10
u
3
(u
5
− u
4
+ ··· + u − 1)(u
41
− 6u
40
+ ··· + 156u − 8)
c
11
((u − 1)
3
)(u
5
+ u
4
+ ··· + u − 1)(u
41
+ 5u
40
+ ··· + 119u + 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
− 5y
4
+ 8y
3
− 3y
2
− y −1)
· (y
41
+ 32y
40
+ ··· + 183721y −6241)
c
6
(y
3
− 10y
2
+ 5y −1)(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
· (y
41
+ 8y
40
+ ··· + 999709y −36481)
c
8
(y
3
− 5y
2
+ 10y −1)(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
· (y
41
− 11y
40
+ ··· + 26y −1)
c
9
, c
11
((y −1)
3
)(y
5
− 5y
4
+ ··· − y −1)(y
41
− 21y
40
+ ··· + 13495y −1)
c
10
y
3
(y
5
+ 3y
4
+ ··· − y −1)(y
41
− 18y
40
+ ··· + 7824y −64)
19
