11n
77
(K11n
77
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 11 9 10 4 7 5 10
Solving Sequence
2,4 5,7,10
8 1 3 9 11 6
c
4
c
7
c
1
c
3
c
9
c
11
c
5
c
2
, c
6
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
3
+ u
2
+ 2d + u + 1, −u
3
+ u
2
+ 2c − u + 1, −u
3
+ u
2
+ 2b + u + 1, u
4
− 2u
3
+ 2a + 3, u
5
− u
4
+ 3u + 1i
I
u
2
= h−u
3
+ 2u
2
+ d − 2u + 1, −u
3
+ 2u
2
+ c − 3u + 1, −u
3
+ 2u
2
+ b − 2u + 1, 2u
4
− 4u
3
+ 4u
2
+ a + u,
u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1i
I
u
3
= hu
3
− u
2
+ d + 1, u
4
− 2u
3
+ 2u
2
+ c + u − 1, u
3
− u
2
+ b + 1, a + u − 1, u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1i
I
u
4
= h−u
4
+ 2u
3
+ u
2
+ 4d − 5u + 2, −u
4
+ u
2
+ 4c − 3u, −u
4
+ 2u
3
+ u
2
+ 4b − 5u + 2, −3u
4
− u
2
+ 4a − 9u,
u
5
− u
3
+ 3u
2
− 4i
I
u
5
= hd, c + 1, b, a + 1, u + 1i
I
u
6
= hd + 1, c + 1, b − 1, a, u + 1i
I
u
7
= hd + b, c + b + 1, b
2
− ba + b − 1, u + 1i
I
v
1
= ha, d + 1, c + a + 1, b − 1, v − 1i
* 7 irreducible components of dim
C
= 0, with total 23 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
3
+ u
2
+ 2d + u + 1, −u
3
+ u
2
+ 2c − u + 1, −u
3
+ u
2
+ 2b + u +
1, u
4
− 2u
3
+ 2a + 3, u
5
− u
4
+ 3u + 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
7
=
−
1
2
u
4
+ u
3
−
3
2
1
2
u
3
−
1
2
u
2
−
1
2
u −
1
2
a
10
=
1
2
u
3
−
1
2
u
2
+
1
2
u −
1
2
1
2
u
3
−
1
2
u
2
−
1
2
u −
1
2
a
8
=
−u
4
+
3
2
u
3
−
1
2
u
2
+
1
2
u −
3
2
−
1
2
u
4
+ u
3
−
1
2
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
9
=
−
1
2
u
4
+
1
2
u
3
+ ··· +
1
2
u − 1
−
1
2
u
4
+ u
3
−
1
2
a
11
=
1
2
u
3
−
1
2
u
2
−
1
2
u −
1
2
−
1
2
u
3
−
1
2
u
2
−
1
2
u −
1
2
a
6
=
1
2
u
4
−
1
2
u
3
−
1
2
u
2
−
1
2
u + 1
−
1
2
u
4
−
1
2
u
3
+
1
2
u
2
−
1
2
u
a
6
=
1
2
u
4
−
1
2
u
3
−
1
2
u
2
−
1
2
u + 1
−
1
2
u
4
−
1
2
u
3
+
1
2
u
2
−
1
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
4
+ 2u
3
+ 2u
2
− 2u − 15
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
7
, c
9
c
10
u
5
− u
4
+ 3u + 1
c
2
, c
11
u
5
+ u
4
+ 6u
3
− 2u
2
+ 9u + 1
c
3
, c
8
u
5
− 4u
4
+ 8u
3
− 8u
2
+ 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
7
, c
9
c
10
y
5
− y
4
+ 6y
3
+ 2y
2
+ 9y − 1
c
2
, c
11
y
5
+ 11y
4
+ 58y
3
+ 102y
2
+ 85y − 1
c
3
, c
8
y
5
− 32y
2
+ 64y − 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.629322 + 0.921686I
a = 0.424671 − 0.213935I
b = 0.718690 + 0.275250I
c = 0.089368 + 1.196940I
d = 0.718690 + 0.275250I
1.14410 + 3.50618I −10.79893 − 4.59139I
u = −0.629322 − 0.921686I
a = 0.424671 + 0.213935I
b = 0.718690 − 0.275250I
c = 0.089368 − 1.196940I
d = 0.718690 − 0.275250I
1.14410 − 3.50618I −10.79893 + 4.59139I
u = 1.29342 + 0.87939I
a = −0.15409 + 1.68698I
b = −2.01497 + 0.28960I
c = −0.721553 + 1.168990I
d = −2.01497 + 0.28960I
6.61272 − 11.96040I −13.0958 + 6.1649I
u = 1.29342 − 0.87939I
a = −0.15409 − 1.68698I
b = −2.01497 − 0.28960I
c = −0.721553 − 1.168990I
d = −2.01497 − 0.28960I
6.61272 + 11.96040I −13.0958 − 6.1649I
u = −0.328197
a = −1.54115
b = −0.407434
c = −0.735630
d = −0.407434
−0.709220 −14.2100
5
II. I
u
2
= h−u
3
+ 2u
2
+ d − 2u + 1, −u
3
+ 2u
2
+ c − 3u + 1, −u
3
+ 2u
2
+ b −
2u + 1, 2u
4
− 4u
3
+ 4u
2
+ a + u, u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
7
=
−2u
4
+ 4u
3
− 4u
2
− u
u
3
− 2u
2
+ 2u − 1
a
10
=
u
3
− 2u
2
+ 3u − 1
u
3
− 2u
2
+ 2u − 1
a
8
=
−u
u
4
− 2u
3
+ u
2
+ u − 1
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
9
=
−u
4
+ 2u
3
− u
2
− 2u + 1
u
4
− 2u
3
+ u
2
+ u − 1
a
11
=
u
3
− 2u
2
+ 2u − 1
−2u
2
+ 2u − 1
a
6
=
u
4
− 2u
3
+ 2u
2
− u + 1
−2u
3
+ 3u
2
− u
a
6
=
u
4
− 2u
3
+ 2u
2
− u + 1
−2u
3
+ 3u
2
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
3
− 4u
2
+ 6u − 12
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
10
u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1
c
2
, c
11
u
5
+ 6u
3
+ u
2
− u + 1
c
3
, c
8
u
5
+ u
4
+ 5u
3
+ u
2
+ 2u − 2
c
6
, c
7
, c
9
u
5
− u
3
+ 3u
2
− 4
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
10
y
5
+ 6y
3
− y
2
− y − 1
c
2
, c
11
y
5
+ 12y
4
+ 34y
3
− 13y
2
− y − 1
c
3
, c
8
y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y − 4
c
6
, c
7
, c
9
y
5
− 2y
4
+ y
3
− 9y
2
+ 24y − 16
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.833800
a = −5.23246
b = −4.63772
c = −5.47152
d = −4.63772
−4.49352 −20.9430
u = 0.317129 + 0.618084I
a = −0.36862 − 1.94340I
b = −0.134390 + 0.402477I
c = 0.182739 + 1.020560I
d = −0.134390 + 0.402477I
−1.43849 − 1.10891I −9.63452 + 2.04112I
u = 0.317129 − 0.618084I
a = −0.36862 + 1.94340I
b = −0.134390 − 0.402477I
c = 0.182739 − 1.020560I
d = −0.134390 − 0.402477I
−1.43849 + 1.10891I −9.63452 − 2.04112I
u = 1.09977 + 1.12945I
a = −0.015153 + 0.220489I
b = −1.54675 − 0.05223I
c = −0.446980 + 1.077220I
d = −1.54675 − 0.05223I
8.62005 − 4.12490I −10.89396 + 2.15443I
u = 1.09977 − 1.12945I
a = −0.015153 − 0.220489I
b = −1.54675 + 0.05223I
c = −0.446980 − 1.077220I
d = −1.54675 + 0.05223I
8.62005 + 4.12490I −10.89396 − 2.15443I
9
III. I
u
3
= hu
3
− u
2
+ d + 1, u
4
− 2u
3
+ 2u
2
+ c + u − 1, u
3
− u
2
+ b + 1, a +
u − 1, u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
7
=
−u + 1
−u
3
+ u
2
− 1
a
10
=
−u
4
+ 2u
3
− 2u
2
− u + 1
−u
3
+ u
2
− 1
a
8
=
−u
u
4
− 2u
3
+ u
2
+ u − 1
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
9
=
−u
4
+ 2u
3
− u
2
− 2u + 1
u
4
− 2u
3
+ u
2
+ u − 1
a
11
=
−u
4
+ 3u
3
− 3u
2
+ 2
u
4
− 2u
3
+ u
2
+ u − 2
a
6
=
−u
3
+ u
2
+ u − 2
2u
3
− u
2
+ 2
a
6
=
−u
3
+ u
2
+ u − 2
2u
3
− u
2
+ 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
3
− 4u
2
+ 6u − 12
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
7
, c
9
u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1
c
2
u
5
+ 6u
3
+ u
2
− u + 1
c
3
, c
8
u
5
+ u
4
+ 5u
3
+ u
2
+ 2u − 2
c
5
, c
10
u
5
− u
3
+ 3u
2
− 4
c
11
u
5
+ 2u
4
+ u
3
+ 9u
2
+ 24u + 16
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
7
, c
9
y
5
+ 6y
3
− y
2
− y − 1
c
2
y
5
+ 12y
4
+ 34y
3
− 13y
2
− y − 1
c
3
, c
8
y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y − 4
c
5
, c
10
y
5
− 2y
4
+ y
3
− 9y
2
+ 24y − 16
c
11
y
5
− 2y
4
+ 13y
3
− 97y
2
+ 288y − 256
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.833800
a = 1.83380
b = 0.274898
c = −1.19933
d = 0.274898
−4.49352 −20.9430
u = 0.317129 + 0.618084I
a = 0.682871 − 0.618084I
b = −0.949895 + 0.441667I
c = 0.65713 − 1.28074I
d = −0.949895 + 0.441667I
−1.43849 − 1.10891I −9.63452 + 2.04112I
u = 0.317129 − 0.618084I
a = 0.682871 + 0.618084I
b = −0.949895 − 0.441667I
c = 0.65713 + 1.28074I
d = −0.949895 − 0.441667I
−1.43849 + 1.10891I −9.63452 − 2.04112I
u = 1.09977 + 1.12945I
a = −0.099771 − 1.129450I
b = 1.81245 − 0.17314I
c = 0.442538 − 0.454479I
d = 1.81245 − 0.17314I
8.62005 − 4.12490I −10.89396 + 2.15443I
u = 1.09977 − 1.12945I
a = −0.099771 + 1.129450I
b = 1.81245 + 0.17314I
c = 0.442538 + 0.454479I
d = 1.81245 + 0.17314I
8.62005 + 4.12490I −10.89396 − 2.15443I
13
IV. I
u
4
= h−u
4
+ 2u
3
+ · · · + 4d + 2, −u
4
+ u
2
+ 4c − 3u, −u
4
+ 2u
3
+ · · · +
4b + 2, −3u
4
− u
2
+ 4a − 9u, u
5
− u
3
+ 3u
2
− 4i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
7
=
3
4
u
4
+
1
4
u
2
+
9
4
u
1
4
u
4
−
1
2
u
3
+ ··· +
5
4
u −
1
2
a
10
=
1
4
u
4
−
1
4
u
2
+
3
4
u
1
4
u
4
−
1
2
u
3
+ ··· +
5
4
u −
1
2
a
8
=
7
8
u
4
−
1
4
u
3
+ ··· +
23
8
u −
1
4
1
4
u
4
−
1
2
u
3
+ ··· +
5
4
u −
3
2
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
9
=
5
8
u
4
+
1
4
u
3
+ ··· +
13
8
u +
5
4
1
4
u
4
−
1
2
u
3
+ ··· +
5
4
u −
3
2
a
11
=
1
2
u
3
+
1
2
u +
1
2
1
4
u
4
+
1
2
u
3
+ ··· +
1
4
u +
3
2
a
6
=
−
5
8
u
4
+
1
4
u
3
+ ··· −
9
8
u +
5
4
u
3
− 2u + 1
a
6
=
−
5
8
u
4
+
1
4
u
3
+ ··· −
9
8
u +
5
4
u
3
− 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
3
− 2u − 16
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
5
− u
3
+ 3u
2
− 4
c
2
u
5
+ 2u
4
+ u
3
+ 9u
2
+ 24u + 16
c
3
, c
8
u
5
+ u
4
+ 5u
3
+ u
2
+ 2u − 2
c
5
, c
6
, c
7
c
9
, c
10
u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1
c
11
u
5
+ 6u
3
+ u
2
− u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
5
− 2y
4
+ y
3
− 9y
2
+ 24y − 16
c
2
y
5
− 2y
4
+ 13y
3
− 97y
2
+ 288y − 256
c
3
, c
8
y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y − 4
c
5
, c
6
, c
7
c
9
, c
10
y
5
+ 6y
3
− y
2
− y − 1
c
11
y
5
+ 12y
4
+ 34y
3
− 13y
2
− y − 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.10870
a = 3.93509
b = 0.274898
c = 0.901960
d = 0.274898
−4.49352 −20.9430
u = −1.267020 + 0.176417I
a = −0.748496 − 0.770461I
b = −0.949895 − 0.441667I
c = −0.774241 − 0.107803I
d = −0.949895 − 0.441667I
−1.43849 + 1.10891I −9.63452 − 2.04112I
u = −1.267020 − 0.176417I
a = −0.748496 + 0.770461I
b = −0.949895 + 0.441667I
c = −0.774241 + 0.107803I
d = −0.949895 + 0.441667I
−1.43849 − 1.10891I −9.63452 + 2.04112I
u = 0.71268 + 1.30259I
a = −0.219048 + 0.084129I
b = 1.81245 + 0.17314I
c = 0.323261 − 0.590839I
d = 1.81245 + 0.17314I
8.62005 + 4.12490I −10.89396 − 2.15443I
u = 0.71268 − 1.30259I
a = −0.219048 − 0.084129I
b = 1.81245 − 0.17314I
c = 0.323261 + 0.590839I
d = 1.81245 − 0.17314I
8.62005 − 4.12490I −10.89396 + 2.15443I
17
V. I
u
5
= hd, c + 1, b, a + 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
4
=
1
0
a
5
=
1
1
a
7
=
−1
0
a
10
=
−1
0
a
8
=
−1
0
a
1
=
−1
−1
a
3
=
1
0
a
9
=
−1
0
a
11
=
−2
−1
a
6
=
−1
0
a
6
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u − 1
c
2
, c
4
, c
10
c
11
u + 1
c
3
, c
6
, c
7
c
8
, c
9
u
19
(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
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
c = −1.00000
d = 0
−3.28987 −12.0000
21
VI. I
u
6
= hd + 1, c + 1, b − 1, a, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
4
=
1
0
a
5
=
1
1
a
7
=
0
1
a
10
=
−1
−1
a
8
=
−1
0
a
1
=
−1
−1
a
3
=
1
0
a
9
=
−1
0
a
11
=
−1
−1
a
6
=
1
1
a
6
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
u − 1
c
2
, c
4
, c
9
u + 1
c
3
, c
5
, c
8
c
10
, c
11
u
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
7
, c
9
y − 1
c
3
, c
5
, c
8
c
10
, c
11
y
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 1.00000
c = −1.00000
d = −1.00000
−3.28987 −12.0000
25
VII. I
u
7
= hd + b, c + b + 1, b
2
− ba + b − 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
4
=
1
0
a
5
=
1
1
a
7
=
a
b
a
10
=
−b − 1
−b
a
8
=
−b + a − 1
0
a
1
=
−1
−1
a
3
=
1
0
a
9
=
−b + a − 1
0
a
11
=
−b − 2
−b − 1
a
6
=
−b − 1
−b
a
6
=
−b − 1
−b
(ii) Obstruction class = −1
(iii) Cusp Shapes = a
2
− 2a − 17
(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.
26
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
c = ···
d = ···
−4.93480 −18.4052 − 0.7878I
27
VIII. I
v
1
= ha, d + 1, c + a + 1, b − 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
1
0
a
5
=
1
0
a
7
=
0
1
a
10
=
−1
−1
a
8
=
−1
0
a
1
=
1
0
a
3
=
1
0
a
9
=
−1
0
a
11
=
0
−1
a
6
=
1
1
a
6
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
28
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
8
u
c
5
, c
9
, c
11
u + 1
c
6
, c
7
, c
10
u − 1
29
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
8
y
c
5
, c
6
, c
7
c
9
, c
10
, c
11
y − 1
30
(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 = −1.00000
−3.28987 −12.0000
31
IX. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
u(u − 1)
2
(u
5
− u
3
+ 3u
2
− 4)(u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1)
2
· (u
5
− u
4
+ 3u + 1)
c
2
, c
11
u(u + 1)
2
(u
5
+ 6u
3
+ u
2
− u + 1)
2
(u
5
+ u
4
+ 6u
3
− 2u
2
+ 9u + 1)
· (u
5
+ 2u
4
+ u
3
+ 9u
2
+ 24u + 16)
c
3
, c
8
u
3
(u
5
− 4u
4
+ 8u
3
− 8u
2
+ 4)(u
5
+ u
4
+ 5u
3
+ u
2
+ 2u − 2)
3
c
4
, c
9
u(u + 1)
2
(u
5
− u
3
+ 3u
2
− 4)(u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1)
2
· (u
5
− u
4
+ 3u + 1)
c
5
, c
10
u(u − 1)(u + 1)(u
5
− u
3
+ 3u
2
− 4)(u
5
− 2u
4
+ 2u
3
+ u
2
− u + 1)
2
· (u
5
− u
4
+ 3u + 1)
32
X. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
7
, c
9
c
10
y(y − 1)
2
(y
5
+ 6y
3
− y
2
− y − 1)
2
(y
5
− 2y
4
+ y
3
− 9y
2
+ 24y − 16)
· (y
5
− y
4
+ 6y
3
+ 2y
2
+ 9y − 1)
c
2
, c
11
y(y − 1)
2
(y
5
− 2y
4
+ 13y
3
− 97y
2
+ 288y − 256)
· (y
5
+ 11y
4
+ 58y
3
+ 102y
2
+ 85y − 1)
· (y
5
+ 12y
4
+ 34y
3
− 13y
2
− y − 1)
2
c
3
, c
8
y
3
(y
5
− 32y
2
+ 64y − 16)(y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y − 4)
3
33
