11n
58
(K11n
58
)
A knot diagram
1
Linearized knot diagam
4 1 9 2 9 10 11 3 1 7 6
Solving Sequence
6,10
7 11 8
1,3
2 9 4 5
c
6
c
10
c
7
c
11
c
2
c
9
c
3
c
4
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
21
− u
20
+ ··· + 5u
2
+ b, −u
14
+ 7u
12
− 18u
10
+ 19u
8
− 6u
6
+ 2u
4
+ 2u
3
− 4u
2
+ a − 4u − 1,
u
22
+ 2u
21
+ ··· − 4u
2
− 1i
I
u
2
= hb + 1, u
3
+ a − 2u, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 27 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
−u
20
+· · · + 5 u
2
+b, −u
14
+7u
12
+· · · + a −1, u
22
+2u
21
+· · · − 4u
2
−1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
−u
2
a
11
=
u
−u
3
+ u
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
−u
3
+ 2u
−u
3
+ u
a
3
=
u
14
− 7u
12
+ 18u
10
− 19u
8
+ 6u
6
− 2u
4
− 2u
3
+ 4u
2
+ 4u + 1
u
21
+ u
20
+ ··· − 9u
3
− 5u
2
a
2
=
u
21
+ u
20
+ ··· + 5u + 1
3u
21
+ 2u
20
+ ··· + u − 1
a
9
=
u
7
− 4u
5
+ 4u
3
u
7
− 3u
5
+ 2u
3
+ u
a
4
=
−2u
21
− 2u
20
+ ··· + 4u + 2
u
21
+ u
20
+ ··· − 5u
2
+ u
a
5
=
u
14
− 7u
12
+ 18u
10
− 19u
8
+ 4u
6
+ 4u
4
− 1
u
14
− 6u
12
+ 13u
10
− 10u
8
− 2u
6
+ 4u
4
+ u
2
a
5
=
u
14
− 7u
12
+ 18u
10
− 19u
8
+ 4u
6
+ 4u
4
− 1
u
14
− 6u
12
+ 13u
10
− 10u
8
− 2u
6
+ 4u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
21
−4u
20
+ 20u
19
+ 37u
18
−87u
17
−136u
16
+ 219u
15
+ 241u
14
−359u
13
−186u
12
+
398u
11
+ 17u
10
− 277u
9
+ 14u
8
+ 98u
7
+ 38u
6
− 34u
5
− 18u
4
+ 36u
3
+ 13u
2
− u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
22
− 6u
21
+ ··· + 6u − 1
c
2
u
22
+ 2u
21
+ ··· + 2u + 1
c
3
, c
8
u
22
− u
21
+ ··· − 64u + 32
c
5
u
22
+ 2u
21
+ ··· + 2996u − 1960
c
6
, c
7
, c
10
u
22
− 2u
21
+ ··· − 4u
2
− 1
c
9
u
22
− 2u
21
+ ··· − 2u + 1
c
11
u
22
+ 6u
21
+ ··· − 64u − 17
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
22
− 2y
21
+ ··· − 2y + 1
c
2
y
22
+ 42y
21
+ ··· − 62y + 1
c
3
, c
8
y
22
− 33y
21
+ ··· − 7680y + 1024
c
5
y
22
+ 66y
21
+ ··· + 61827024y + 3841600
c
6
, c
7
, c
10
y
22
− 22y
21
+ ··· + 8y + 1
c
9
y
22
+ 30y
21
+ ··· + 8y + 1
c
11
y
22
− 14y
21
+ ··· − 2056y + 289
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.547451 + 0.687554I
a = −1.64917 + 1.79616I
b = −0.18168 + 2.30378I
10.04630 − 1.61926I 3.79117 − 0.60262I
u = 0.547451 − 0.687554I
a = −1.64917 − 1.79616I
b = −0.18168 − 2.30378I
10.04630 + 1.61926I 3.79117 + 0.60262I
u = 0.477782 + 0.730631I
a = 1.68752 − 2.10270I
b = 0.07337 − 2.34864I
9.80847 + 6.35147I 3.22096 − 4.88727I
u = 0.477782 − 0.730631I
a = 1.68752 + 2.10270I
b = 0.07337 + 2.34864I
9.80847 − 6.35147I 3.22096 + 4.88727I
u = −1.15891
a = −0.585194
b = 0.132093
1.97038 6.11980
u = −0.253735 + 0.636077I
a = −0.333427 + 0.841858I
b = −0.032077 + 0.372929I
0.10442 − 2.33425I 2.92732 + 5.10863I
u = −0.253735 − 0.636077I
a = −0.333427 − 0.841858I
b = −0.032077 − 0.372929I
0.10442 + 2.33425I 2.92732 − 5.10863I
u = 1.33846
a = 1.06516
b = 1.56525
1.80329 6.37870
u = −1.374360 + 0.085773I
a = −0.046048 − 1.048570I
b = 0.632067 + 0.872611I
3.07940 − 2.15283I 3.96233 + 2.53077I
u = −1.374360 − 0.085773I
a = −0.046048 + 1.048570I
b = 0.632067 − 0.872611I
3.07940 + 2.15283I 3.96233 − 2.53077I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.458175 + 0.412746I
a = −1.044590 − 0.139124I
b = −0.311064 − 0.023201I
1.10436 − 0.93215I 5.59687 + 3.71705I
u = −0.458175 − 0.412746I
a = −1.044590 + 0.139124I
b = −0.311064 + 0.023201I
1.10436 + 0.93215I 5.59687 − 3.71705I
u = 1.384260 + 0.250179I
a = 0.084137 − 0.456690I
b = 0.043050 − 0.643455I
5.30289 + 5.58097I 6.98899 − 5.83204I
u = 1.384260 − 0.250179I
a = 0.084137 + 0.456690I
b = 0.043050 + 0.643455I
5.30289 − 5.58097I 6.98899 + 5.83204I
u = 1.46039 + 0.14631I
a = −0.589431 − 0.029298I
b = −0.810187 + 0.008550I
7.28227 + 3.02618I 8.05288 − 2.57798I
u = 1.46039 − 0.14631I
a = −0.589431 + 0.029298I
b = −0.810187 − 0.008550I
7.28227 − 3.02618I 8.05288 + 2.57798I
u = −1.50300 + 0.26177I
a = 1.77777 + 0.03837I
b = 0.23595 + 2.50634I
16.2359 − 9.9783I 6.35264 + 4.88027I
u = −1.50300 − 0.26177I
a = 1.77777 − 0.03837I
b = 0.23595 − 2.50634I
16.2359 + 9.9783I 6.35264 − 4.88027I
u = −1.52245 + 0.22649I
a = −1.49085 + 0.18083I
b = −0.42745 − 2.45052I
16.8152 − 1.7067I 7.03198 + 0.67482I
u = −1.52245 − 0.22649I
a = −1.49085 − 0.18083I
b = −0.42745 + 2.45052I
16.8152 + 1.7067I 7.03198 − 0.67482I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.152064 + 0.338601I
a = 1.36411 + 1.84123I
b = 0.929357 − 0.322994I
−1.75639 + 0.64723I −5.17438 + 1.08919I
u = 0.152064 − 0.338601I
a = 1.36411 − 1.84123I
b = 0.929357 + 0.322994I
−1.75639 − 0.64723I −5.17438 − 1.08919I
7
II. I
u
2
= hb + 1, u
3
+ a − 2u, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
−u
2
a
11
=
u
−u
3
+ u
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
−u
3
+ 2u
−u
3
+ u
a
3
=
−u
3
+ 2u
−1
a
2
=
−2u
3
+ 4u
−u
3
+ u − 1
a
9
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
−u
3
+ 2u
−1
a
5
=
u
3
− 2u
u
3
− u
a
5
=
u
3
− 2u
u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
3
− u
2
+ 8u + 3
8
(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
8
u
5
c
5
, c
9
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
6
, c
7
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
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
8
y
5
c
5
, c
9
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
6
, c
7
, c
10
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
11
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.21774
a = −0.629714
b = −1.00000
0.756147 −2.80750
u = −0.309916 + 0.549911I
a = −0.871221 + 1.107660I
b = −1.00000
−1.31583 − 1.53058I −0.02714 + 4.76366I
u = −0.309916 − 0.549911I
a = −0.871221 − 1.107660I
b = −1.00000
−1.31583 + 1.53058I −0.02714 − 4.76366I
u = 1.41878 + 0.21917I
a = 0.186078 − 0.874646I
b = −1.00000
4.22763 + 4.40083I 4.43089 − 2.80751I
u = 1.41878 − 0.21917I
a = 0.186078 + 0.874646I
b = −1.00000
4.22763 − 4.40083I 4.43089 + 2.80751I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
22
− 6u
21
+ ··· + 6u − 1)
c
2
((u + 1)
5
)(u
22
+ 2u
21
+ ··· + 2u + 1)
c
3
, c
8
u
5
(u
22
− u
21
+ ··· − 64u + 32)
c
4
((u + 1)
5
)(u
22
− 6u
21
+ ··· + 6u − 1)
c
5
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
22
+ 2u
21
+ ··· + 2996u − 1960)
c
6
, c
7
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
22
− 2u
21
+ ··· − 4u
2
− 1)
c
9
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
22
− 2u
21
+ ··· − 2u + 1)
c
10
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
22
− 2u
21
+ ··· − 4u
2
− 1)
c
11
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)(u
22
+ 6u
21
+ ··· − 64u − 17)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
5
)(y
22
− 2y
21
+ ··· − 2y + 1)
c
2
((y −1)
5
)(y
22
+ 42y
21
+ ··· − 62y + 1)
c
3
, c
8
y
5
(y
22
− 33y
21
+ ··· − 7680y + 1024)
c
5
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
· (y
22
+ 66y
21
+ ··· + 61827024y + 3841600)
c
6
, c
7
, c
10
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
22
− 22y
21
+ ··· + 8y + 1)
c
9
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
22
+ 30y
21
+ ··· + 8y + 1)
c
11
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
22
− 14y
21
+ ··· − 2056y + 289)
13
