11n
74
(K11n
74
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 11 9 4 6 8 1 6
Solving Sequence
6,8 4,9 1,3
2 5 7 11 10
c
8
c
3
c
2
c
4
c
7
c
11
c
10
c
1
, c
5
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
4
+ 2u
3
− 2u
2
+ 2d + 1, −u
6
+ 3u
5
− 5u
4
+ 3u
3
+ 2c − 6u − 4, u
3
− u
2
+ 2b + u + 1,
− 2u
6
+ 5u
5
− 7u
4
+ 6u
2
+ 4a − 13u − 13, u
7
− 3u
6
+ 5u
5
− 3u
4
− u
3
+ 7u
2
+ 3u − 1i
I
u
2
= hu
3
+ 4d − u − 2, 3u
3
− 4u
2
+ 8c + 9u + 18, b + u − 1, −u
3
+ 8a − 3u − 2, u
4
− 2u
3
+ 3u
2
+ 4u − 4i
I
u
3
= hd, c + 1, b − 1, a, u + 1i
I
u
4
= hd, c − 1, b, a − 1, u + 1i
I
u
5
= hd, cb + 1, a − 1, u + 1i
I
v
1
= ha, d, c − 1, b + 1, v − 1i
* 5 irreducible components of dim
C
= 0, with total 14 representations.
* 1 irreducible components of dim
C
= 1
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
4
+ 2u
3
− 2u
2
+ 2d + 1, −u
6
+ 3u
5
+ · · · + 2c − 4, u
3
− u
2
+ 2b +
u + 1, −2u
6
+ 5u
5
+ · · · + 4a − 13, u
7
− 3u
6
+ · · · + 3u − 1i
(i) Arc colorings
a
6
=
0
u
a
8
=
1
0
a
4
=
1
2
u
6
−
3
2
u
5
+ ··· + 3u + 2
1
2
u
4
− u
3
+ u
2
−
1
2
a
9
=
1
u
2
a
1
=
1
2
u
6
−
5
4
u
5
+ ··· +
13
4
u +
13
4
−
1
2
u
3
+
1
2
u
2
−
1
2
u −
1
2
a
3
=
1
2
u
6
−
3
2
u
5
+ ··· + 3u +
5
2
1
2
u
4
− u
3
+ u
2
−
1
2
a
2
=
1
2
u
6
−
5
4
u
5
+ ··· +
11
4
u +
11
4
−
1
2
u
5
+ u
4
−
3
2
u
3
+
1
2
u
2
−
1
2
a
5
=
3
4
u
6
−
9
4
u
5
+ ··· +
17
4
u +
5
2
−
1
2
u
6
+ u
5
+ ··· −
1
2
u −
1
2
a
7
=
−u
−u
3
+ u
a
11
=
1
2
u
6
−
5
4
u
5
+ ··· +
13
4
u +
13
4
1
4
u
5
−
1
4
u
4
+ ··· −
3
4
u −
1
4
a
10
=
−u
2
+ 1
u
2
a
10
=
−u
2
+ 1
u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
6
−
19
2
u
5
+
35
2
u
4
− 14u
3
+ 3u
2
+
39
2
u +
9
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
8
u
7
− 3u
6
+ 5u
5
− 3u
4
− u
3
+ 7u
2
+ 3u − 1
c
2
, c
9
u
7
− u
6
+ 5u
5
− 29u
4
+ 67u
3
+ 61u
2
+ 23u + 1
c
3
, c
7
u
7
− 6u
5
+ 4u
4
+ 32u
3
− 12u
2
+ 16u − 8
c
5
, c
11
u
7
+ u
6
− 4u
5
+ 15u
3
+ 3u
2
− 8u − 4
c
10
u
7
− 9u
6
+ 46u
5
− 142u
4
+ 297u
3
− 249u
2
+ 88u − 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
y
7
+ y
6
+ 5y
5
+ 29y
4
+ 67y
3
− 61y
2
+ 23y − 1
c
2
, c
9
y
7
+ 9y
6
+ 101y
5
− 3y
4
+ 8259y
3
− 581y
2
+ 407y − 1
c
3
, c
7
y
7
− 12y
6
+ 100y
5
− 368y
4
+ 928y
3
+ 944y
2
+ 64y − 64
c
5
, c
11
y
7
− 9y
6
+ 46y
5
− 142y
4
+ 297y
3
− 249y
2
+ 88y − 16
c
10
y
7
+ 11y
6
+ 154y
5
+ 2854y
4
+ 25301y
3
− 14273y
2
− 224y − 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.643564 + 0.238013I
a = 0.616252 + 0.619029I
b = 0.079132 − 0.413310I
c = 0.317102 − 0.524945I
d = 0.031685 − 0.698136I
−1.11796 + 1.29283I −4.63450 − 5.74515I
u = −0.643564 − 0.238013I
a = 0.616252 − 0.619029I
b = 0.079132 + 0.413310I
c = 0.317102 + 0.524945I
d = 0.031685 + 0.698136I
−1.11796 − 1.29283I −4.63450 + 5.74515I
u = 0.46828 + 1.59550I
a = −0.405220 − 1.031160I
b = −0.16054 + 1.45536I
c = −0.812628 − 0.339128I
d = 2.23667 + 1.02998I
5.28066 − 2.46552I 0.37200 + 1.61165I
u = 0.46828 − 1.59550I
a = −0.405220 + 1.031160I
b = −0.16054 − 1.45536I
c = −0.812628 + 0.339128I
d = 2.23667 − 1.02998I
5.28066 + 2.46552I 0.37200 − 1.61165I
u = 0.222829
a = 3.90340
b = −0.592120
c = 2.65729
d = −0.460179
1.26042 8.87750
u = 1.56387 + 1.00084I
a = −0.662734 + 0.809308I
b = −0.12253 − 2.10558I
c = 0.666881 + 0.919602I
d = −2.03826 + 1.30990I
14.9463 − 10.4045I −1.17625 + 4.09895I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.56387 − 1.00084I
a = −0.662734 − 0.809308I
b = −0.12253 + 2.10558I
c = 0.666881 − 0.919602I
d = −2.03826 − 1.30990I
14.9463 + 10.4045I −1.17625 − 4.09895I
6
II. I
u
2
= hu
3
+ 4d − u − 2, 3u
3
− 4u
2
+ 8c + 9u + 18, b + u − 1, −u
3
+ 8a −
3u − 2, u
4
− 2u
3
+ 3u
2
+ 4u − 4i
(i) Arc colorings
a
6
=
0
u
a
8
=
1
0
a
4
=
−
3
8
u
3
+
1
2
u
2
−
9
8
u −
9
4
−
1
4
u
3
+
1
4
u +
1
2
a
9
=
1
u
2
a
1
=
1
8
u
3
+
3
8
u +
1
4
−u + 1
a
3
=
−
1
8
u
3
+
1
2
u
2
−
11
8
u −
11
4
−
1
4
u
3
+
1
4
u +
1
2
a
2
=
−
1
2
u
3
+
1
2
u
2
−
1
2
u −
5
2
1
4
u
3
+ u
2
−
1
4
u +
1
2
a
5
=
−
1
2
u
2
+
1
2
−
5
4
u
3
+ 3u
2
+
5
4
u −
5
2
a
7
=
−u
−u
3
+ u
a
11
=
1
8
u
3
+
3
8
u +
1
4
1
2
u
3
− u
2
−
3
2
u + 2
a
10
=
−u
2
+ 1
u
2
a
10
=
−u
2
+ 1
u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
8
u
4
− 2u
3
+ 3u
2
+ 4u − 4
c
2
, c
9
u
4
− 2u
3
+ 17u
2
+ 40u + 16
c
3
, c
7
(u
2
+ 4u + 2)
2
c
5
, c
11
(u
2
+ 2u − 1)
2
c
10
(u
2
− 6u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
y
4
+ 2y
3
+ 17y
2
− 40y + 16
c
2
, c
9
y
4
+ 30y
3
+ 481y
2
− 1056y + 256
c
3
, c
7
(y
2
− 12y + 4)
2
c
5
, c
11
(y
2
− 6y + 1)
2
c
10
(y
2
− 34y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.14055
a = −0.363169
b = 2.14055
c = 0.239938
d = 0.585786
−2.46740 0
u = 0.726339
a = 0.570276
b = 0.273661
c = −2.94704
d = 0.585786
−2.46740 0
u = 1.20711 + 1.83612I
a = −0.603553 + 0.918058I
b = −0.20711 − 1.83612I
c = −0.646447 − 0.537786I
d = 3.41421
17.2718 0
u = 1.20711 − 1.83612I
a = −0.603553 − 0.918058I
b = −0.20711 + 1.83612I
c = −0.646447 + 0.537786I
d = 3.41421
17.2718 0
10
III. I
u
3
= hd, c + 1, b − 1, a, u + 1i
(i) Arc colorings
a
6
=
0
−1
a
8
=
1
0
a
4
=
−1
0
a
9
=
1
1
a
1
=
0
1
a
3
=
−1
0
a
2
=
−1
1
a
5
=
0
−1
a
7
=
1
0
a
11
=
0
1
a
10
=
0
1
a
10
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u − 1
c
2
, c
4
, c
8
c
9
u + 1
c
3
, c
5
, c
7
c
10
, c
11
u
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
8
, c
9
y − 1
c
3
, c
5
, c
7
c
10
, c
11
y
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 1.00000
c = −1.00000
d = 0
−3.28987 −12.0000
14
IV. I
u
4
= hd, c − 1, b, a − 1, u + 1i
(i) Arc colorings
a
6
=
0
−1
a
8
=
1
0
a
4
=
1
0
a
9
=
1
1
a
1
=
1
0
a
3
=
1
0
a
2
=
1
0
a
5
=
1
0
a
7
=
1
0
a
11
=
1
1
a
10
=
0
1
a
10
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
u
c
5
, c
6
u − 1
c
8
, c
9
, c
10
c
11
u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
y
c
5
, c
6
, c
8
c
9
, c
10
, c
11
y − 1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = 0
c = 1.00000
d = 0
0 0
18
V. I
u
5
= hd, cb + 1, a − 1, u + 1i
(i) Arc colorings
a
6
=
0
−1
a
8
=
1
0
a
4
=
c
0
a
9
=
1
1
a
1
=
1
b
a
3
=
c
0
a
2
=
c + 1
b
a
5
=
−1
−b
a
7
=
1
0
a
11
=
1
b + 1
a
10
=
0
1
a
10
=
0
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −c
2
− b
2
− 4
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
19
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
c = ···
d = ···
−1.64493 −1.58105 + 0.82889I
20
VI. I
v
1
= ha, d, c − 1, b + 1, v − 1i
(i) Arc colorings
a
6
=
1
0
a
8
=
1
0
a
4
=
1
0
a
9
=
1
0
a
1
=
0
−1
a
3
=
1
0
a
2
=
1
−1
a
5
=
0
1
a
7
=
1
0
a
11
=
1
−1
a
10
=
1
0
a
10
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u − 1
c
2
, c
4
, c
5
c
10
u + 1
c
3
, c
6
, c
7
c
8
, c
9
u
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
10
, c
11
y − 1
c
3
, c
6
, c
7
c
8
, c
9
y
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
c = 1.00000
d = 0
0 0
24
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u(u − 1)
2
(u
4
− 2u
3
+ 3u
2
+ 4u − 4)
· (u
7
− 3u
6
+ 5u
5
− 3u
4
− u
3
+ 7u
2
+ 3u − 1)
c
2
, c
9
u(u + 1)
2
(u
4
− 2u
3
+ 17u
2
+ 40u + 16)
· (u
7
− u
6
+ 5u
5
− 29u
4
+ 67u
3
+ 61u
2
+ 23u + 1)
c
3
, c
7
u
3
(u
2
+ 4u + 2)
2
(u
7
− 6u
5
+ 4u
4
+ 32u
3
− 12u
2
+ 16u − 8)
c
4
, c
8
u(u + 1)
2
(u
4
− 2u
3
+ 3u
2
+ 4u − 4)
· (u
7
− 3u
6
+ 5u
5
− 3u
4
− u
3
+ 7u
2
+ 3u − 1)
c
5
, c
11
u(u − 1)(u + 1)(u
2
+ 2u − 1)
2
(u
7
+ u
6
+ ··· − 8u − 4)
c
10
u(u + 1)
2
(u
2
− 6u + 1)
2
· (u
7
− 9u
6
+ 46u
5
− 142u
4
+ 297u
3
− 249u
2
+ 88u − 16)
25
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
y(y − 1)
2
(y
4
+ 2y
3
+ 17y
2
− 40y + 16)
· (y
7
+ y
6
+ 5y
5
+ 29y
4
+ 67y
3
− 61y
2
+ 23y − 1)
c
2
, c
9
y(y − 1)
2
(y
4
+ 30y
3
+ 481y
2
− 1056y + 256)
· (y
7
+ 9y
6
+ 101y
5
− 3y
4
+ 8259y
3
− 581y
2
+ 407y − 1)
c
3
, c
7
y
3
(y
2
− 12y + 4)
2
· (y
7
− 12y
6
+ 100y
5
− 368y
4
+ 928y
3
+ 944y
2
+ 64y − 64)
c
5
, c
11
y(y − 1)
2
(y
2
− 6y + 1)
2
· (y
7
− 9y
6
+ 46y
5
− 142y
4
+ 297y
3
− 249y
2
+ 88y − 16)
c
10
y(y − 1)
2
(y
2
− 34y + 1)
2
· (y
7
+ 11y
6
+ 154y
5
+ 2854y
4
+ 25301y
3
− 14273y
2
− 224y − 256)
26
