11n
81
(K11n
81
)
A knot diagram
1
Linearized knot diagam
4 1 9 2 11 1 10 4 7 8 6
Solving Sequence
1,4 2,6 7,9
3 8 11 5 10
c
1
c
6
c
3
c
8
c
11
c
5
c
10
c
2
, c
4
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
3
+ u
2
+ 2d + 3u − 1, u
4
+ 2u
3
− 4u
2
+ 2c − 8u + 1, b − u, −u
4
− u
3
+ 3u
2
+ 2a + 3u,
u
5
+ u
4
− 4u
3
− 4u
2
+ 3u − 1i
I
u
2
= h−u
4
+ 2u
2
+ d − 2u, u
4
+ u
3
− 2u
2
+ c + 2, b − u, u
3
+ 2u
2
+ a − u − 1, u
5
+ 2u
4
− 2u
3
− 3u
2
+ 3u + 1i
I
u
3
= h−u
4
+ 2u
2
+ d − 2u, u
4
+ u
3
− 2u
2
+ c + 2, −u
4
− u
3
+ 2u
2
+ b + u − 1, −u
4
− 2u
3
+ 2u
2
+ a + 3u − 3,
u
5
+ 2u
4
− 2u
3
− 3u
2
+ 3u + 1i
I
u
4
= h−5u
4
+ 6u
3
− 3u
2
+ 4d − 9u + 14, 3u
4
− 2u
3
+ u
2
+ 8c + 3u − 10, u
4
− 2u
3
− u
2
+ 4b + 5u − 2,
u
4
− u
2
+ 4a + 3u, u
5
− u
3
+ 3u
2
− 4i
I
u
5
= hd + 1, c, b, a + 1, u − 1i
I
u
6
= hd + 1, c, b + 1, a + 1, u − 1i
I
u
7
= hda + d + 1, c, b + 1, u − 1i
I
v
1
= ha, d, c + 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 + 3u − 1, u
4
+ 2u
3
+ · · · + 2c + 1, b − u, −u
4
− u
3
+
3u
2
+ 2a + 3u, u
5
+ u
4
+ · · · + 3u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
1
2
u
4
+
1
2
u
3
−
3
2
u
2
−
3
2
u
u
a
7
=
1
2
u
4
+
1
2
u
3
−
3
2
u
2
−
5
2
u
u
a
9
=
−
1
2
u
4
− u
3
+ 2u
2
+ 4u −
1
2
1
2
u
3
−
1
2
u
2
−
3
2
u +
1
2
a
3
=
−u
2
+ 1
u
2
a
8
=
−
1
2
u
4
− u
3
+ 2u
2
+ 4u −
1
2
1
2
u
4
+
1
2
u
3
−
3
2
u
2
−
1
2
u
a
11
=
−
1
2
u
3
−
1
2
u
2
+
3
2
u +
1
2
−u
2
a
5
=
−u
−u
3
+ u
a
10
=
−
1
2
u
4
−
1
2
u
3
+
3
2
u
2
+
5
2
u
1
2
u
3
+
1
2
u
2
−
3
2
u +
1
2
a
10
=
−
1
2
u
4
−
1
2
u
3
+
3
2
u
2
+
5
2
u
1
2
u
3
+
1
2
u
2
−
3
2
u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −3u
4
− 6u
3
+ 10u
2
+ 22u − 13
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
, c
11
u
5
− u
4
− 4u
3
+ 4u
2
+ 3u + 1
c
2
u
5
+ 9u
4
+ 30u
3
+ 38u
2
+ u + 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
, c
11
y
5
− 9y
4
+ 30y
3
− 38y
2
+ y − 1
c
2
y
5
− 21y
4
+ 218y
3
− 1402y
2
− 75y − 1
c
3
, c
8
y
5
− 96y
2
− 64y − 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.287923 + 0.283171I
a = −0.471944 − 0.645049I
b = 0.287923 + 0.283171I
c = 0.71581 + 1.41065I
d = 0.044061 − 0.482429I
−0.341586 − 0.921914I −6.28644 + 7.57142I
u = 0.287923 − 0.283171I
a = −0.471944 + 0.645049I
b = 0.287923 − 0.283171I
c = 0.71581 − 1.41065I
d = 0.044061 + 0.482429I
−0.341586 + 0.921914I −6.28644 − 7.57142I
u = −1.72935 + 0.51571I
a = −1.26784 − 0.71317I
b = −1.72935 + 0.51571I
c = −0.297131 − 1.134290I
d = −0.16439 + 2.36316I
16.6614 + 10.9560I −13.7735 − 4.2698I
u = −1.72935 − 0.51571I
a = −1.26784 + 0.71317I
b = −1.72935 − 0.51571I
c = −0.297131 + 1.134290I
d = −0.16439 − 2.36316I
16.6614 − 10.9560I −13.7735 + 4.2698I
u = 1.88286
a = 1.47956
b = 1.88286
c = 1.16265
d = −0.759351
−17.8353 −13.8800
5
II. I
u
2
= h−u
4
+ 2u
2
+ d − 2u, u
4
+ u
3
− 2u
2
+ c + 2, b − u, u
3
+ 2u
2
+ a −
u − 1, u
5
+ 2u
4
+ · · · + 3u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
−u
3
− 2u
2
+ u + 1
u
a
7
=
−u
3
− 2u
2
+ 1
u
a
9
=
−u
4
− u
3
+ 2u
2
− 2
u
4
− 2u
2
+ 2u
a
3
=
−u
2
+ 1
u
2
a
8
=
−u
4
− u
3
+ 2u
2
− 2
−u
4
− u
3
+ 2u
2
− 1
a
11
=
u
4
+ 2u
3
− u
2
− u + 1
−u
2
a
5
=
−u
−u
3
+ u
a
10
=
u
3
+ 2u
2
− 1
−1
a
10
=
u
3
+ 2u
2
− 1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
4
+ 6u
3
− 8u
2
− 6u − 4
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
11
u
5
− 2u
4
− 2u
3
+ 3u
2
+ 3u − 1
c
2
u
5
+ 8u
4
+ 22u
3
+ 25u
2
+ 15u + 1
c
3
, c
8
u
5
− u
4
+ 5u
3
− u
2
+ 2u + 2
c
7
, c
9
, c
10
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
6
, c
11
y
5
− 8y
4
+ 22y
3
− 25y
2
+ 15y − 1
c
2
y
5
− 20y
4
+ 114y
3
+ 19y
2
+ 175y − 1
c
3
, c
8
y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y − 4
c
7
, c
9
, c
10
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 = 0.949895 + 0.441667I
a = 0.23423 − 2.34588I
b = 0.949895 + 0.441667I
c = −0.682871 − 0.618084I
d = 0.281458 + 0.392024I
−5.14125 − 1.10891I −14.3655 + 2.0411I
u = 0.949895 − 0.441667I
a = 0.23423 + 2.34588I
b = 0.949895 − 0.441667I
c = −0.682871 + 0.618084I
d = 0.281458 − 0.392024I
−5.14125 + 1.10891I −14.3655 − 2.0411I
u = −0.274898
a = 0.594739
b = −0.274898
c = −1.83380
d = −0.695222
−2.08622 −3.05700
u = −1.81245 + 0.17314I
a = −1.53160 − 0.27272I
b = −1.81245 + 0.17314I
c = 0.099771 + 1.129450I
d = 0.06615 − 2.48427I
−15.1998 + 4.1249I −13.10604 − 2.15443I
u = −1.81245 − 0.17314I
a = −1.53160 + 0.27272I
b = −1.81245 − 0.17314I
c = 0.099771 − 1.129450I
d = 0.06615 + 2.48427I
−15.1998 − 4.1249I −13.10604 + 2.15443I
9
III. I
u
3
= h−u
4
+ 2u
2
+ d − 2u, u
4
+ u
3
− 2u
2
+ c + 2, −u
4
− u
3
+ · · · + b −
1, −u
4
− 2u
3
+ · · · + a − 3, u
5
+ 2u
4
+ · · · + 3u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
u
4
+ 2u
3
− 2u
2
− 3u + 3
u
4
+ u
3
− 2u
2
− u + 1
a
7
=
u
3
− 2u + 2
u
4
+ u
3
− 2u
2
− u + 1
a
9
=
−u
4
− u
3
+ 2u
2
− 2
u
4
− 2u
2
+ 2u
a
3
=
−u
2
+ 1
u
2
a
8
=
−u
4
− u
3
+ 2u
2
− 2
−u
4
− u
3
+ 2u
2
− 1
a
11
=
−u
4
− u
3
+ 3u
2
+ u − 3
−u
4
+ 3u
2
− 2u − 2
a
5
=
−u
−u
3
+ u
a
10
=
−u
3
+ 2u − 2
−1
a
10
=
−u
3
+ 2u − 2
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
4
+ 6u
3
− 8u
2
− 6u − 4
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
7
c
9
, c
10
u
5
− 2u
4
− 2u
3
+ 3u
2
+ 3u − 1
c
2
u
5
+ 8u
4
+ 22u
3
+ 25u
2
+ 15u + 1
c
3
, c
8
u
5
− u
4
+ 5u
3
− u
2
+ 2u + 2
c
5
, c
6
, c
11
u
5
− u
3
− 3u
2
+ 4
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
c
9
, c
10
y
5
− 8y
4
+ 22y
3
− 25y
2
+ 15y − 1
c
2
y
5
− 20y
4
+ 114y
3
+ 19y
2
+ 175y − 1
c
3
, c
8
y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y − 4
c
5
, c
6
, c
11
y
5
− 2y
4
+ y
3
− 9y
2
+ 24y − 16
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.949895 + 0.441667I
a = −0.865610 + 0.402477I
b = −1.267020 + 0.176417I
c = −0.682871 − 0.618084I
d = 0.281458 + 0.392024I
−5.14125 − 1.10891I −14.3655 + 2.0411I
u = 0.949895 − 0.441667I
a = −0.865610 − 0.402477I
b = −1.267020 − 0.176417I
c = −0.682871 + 0.618084I
d = 0.281458 − 0.392024I
−5.14125 + 1.10891I −14.3655 − 2.0411I
u = −0.274898
a = 3.63772
b = 1.10870
c = −1.83380
d = −0.695222
−2.08622 −3.05700
u = −1.81245 + 0.17314I
a = 0.546751 + 0.052231I
b = 0.71268 − 1.30259I
c = 0.099771 + 1.129450I
d = 0.06615 − 2.48427I
−15.1998 + 4.1249I −13.10604 − 2.15443I
u = −1.81245 − 0.17314I
a = 0.546751 − 0.052231I
b = 0.71268 + 1.30259I
c = 0.099771 − 1.129450I
d = 0.06615 + 2.48427I
−15.1998 − 4.1249I −13.10604 + 2.15443I
13
IV. I
u
4
= h−5u
4
+ 6u
3
+ · · · + 4d + 14, 3u
4
− 2u
3
+ · · · + 8c − 10, u
4
− 2u
3
+
· · · + 4b − 2, u
4
− u
2
+ 4a + 3u, u
5
− u
3
+ 3u
2
− 4i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
−
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
7
=
−
1
2
u
3
+
1
2
u −
1
2
−
1
4
u
4
+
1
2
u
3
+ ··· −
5
4
u +
1
2
a
9
=
−
3
8
u
4
+
1
4
u
3
+ ··· −
3
8
u +
5
4
5
4
u
4
−
3
2
u
3
+ ··· +
9
4
u −
7
2
a
3
=
−u
2
+ 1
u
2
a
8
=
−
3
8
u
4
+
1
4
u
3
+ ··· −
3
8
u +
5
4
3
4
u
4
−
1
2
u
3
+ ··· +
3
4
u −
5
2
a
11
=
1
8
u
4
−
1
4
u
3
+ ··· +
5
8
u −
1
4
1
2
u
4
+
1
2
u
2
+
1
2
u − 2
a
5
=
−u
−u
3
+ u
a
10
=
1
2
u
3
−
1
2
u +
1
2
3
4
u
4
−
3
2
u
3
+ ··· +
7
4
u −
3
2
a
10
=
1
2
u
3
−
1
2
u +
1
2
3
4
u
4
−
3
2
u
3
+ ··· +
7
4
u −
3
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
3
+ 2u − 8
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
, c
11
u
5
− 2u
4
− 2u
3
+ 3u
2
+ 3u − 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
, c
11
y
5
− 8y
4
+ 22y
3
− 25y
2
+ 15y − 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.10870
a = −0.901960
b = −0.274898
c = 0.454684
d = −0.239061
−2.08622 −3.05700
u = −1.267020 + 0.176417I
a = 0.774241 + 0.107803I
b = 0.949895 + 0.441667I
c = 0.195051 + 0.728580I
d = 0.55136 − 2.96396I
−5.14125 − 1.10891I −14.3655 + 2.0411I
u = −1.267020 − 0.176417I
a = 0.774241 − 0.107803I
b = 0.949895 − 0.441667I
c = 0.195051 − 0.728580I
d = 0.55136 + 2.96396I
−5.14125 + 1.10891I −14.3655 − 2.0411I
u = 0.71268 + 1.30259I
a = −0.323261 + 0.590839I
b = −1.81245 − 0.17314I
c = 1.077610 + 0.878534I
d = −0.431826 − 0.856727I
−15.1998 − 4.1249I −13.10604 + 2.15443I
u = 0.71268 − 1.30259I
a = −0.323261 − 0.590839I
b = −1.81245 + 0.17314I
c = 1.077610 − 0.878534I
d = −0.431826 + 0.856727I
−15.1998 + 4.1249I −13.10604 − 2.15443I
17
V. I
u
5
= hd + 1, c, b, a + 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
6
=
−1
0
a
7
=
−1
0
a
9
=
0
−1
a
3
=
0
1
a
8
=
0
−1
a
11
=
1
0
a
5
=
−1
0
a
10
=
1
−1
a
10
=
1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u − 1
c
2
, c
4
, c
9
c
10
u + 1
c
3
, c
5
, c
6
c
8
, c
11
u
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
7
, c
9
, c
10
y − 1
c
3
, c
5
, c
6
c
8
, c
11
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 = 0
d = −1.00000
−3.28987 −12.0000
21
VI. I
u
6
= hd + 1, c, b + 1, a + 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
6
=
−1
−1
a
7
=
0
−1
a
9
=
0
−1
a
3
=
0
1
a
8
=
0
−1
a
11
=
0
−1
a
5
=
−1
0
a
10
=
0
−1
a
10
=
0
−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
5
, c
6
u − 1
c
2
, c
4
, c
11
u + 1
c
3
, c
7
, c
8
c
9
, c
10
u
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
11
y − 1
c
3
, c
7
, c
8
c
9
, c
10
y
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −1.00000
c = 0
d = −1.00000
−3.28987 −12.0000
25
VII. I
u
7
= hda + d + 1, c, b + 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
6
=
a
−1
a
7
=
a + 1
−1
a
9
=
0
d
a
3
=
0
1
a
8
=
0
d
a
11
=
a + 1
−1
a
5
=
−1
0
a
10
=
a + 1
d − 1
a
10
=
a + 1
d − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −d
2
− 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 −16.0570 + 0.6676I
27
VIII. I
v
1
= ha, d, c + 1, b − 1, v − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
1
0
a
2
=
1
0
a
6
=
0
1
a
7
=
−1
1
a
9
=
−1
0
a
3
=
1
0
a
8
=
−1
0
a
11
=
1
−1
a
5
=
1
0
a
10
=
0
−1
a
10
=
0
−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
6
, c
9
c
10
u + 1
c
7
, c
11
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 = 0
−3.28987 −12.0000
31
IX. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
u(u − 1)
2
(u
5
− u
3
− 3u
2
+ 4)(u
5
− 2u
4
− 2u
3
+ 3u
2
+ 3u − 1)
2
· (u
5
− u
4
− 4u
3
+ 4u
2
+ 3u + 1)
c
2
u(u + 1)
2
(u
5
+ 2u
4
+ u
3
+ 9u
2
+ 24u + 16)
· ((u
5
+ 8u
4
+ 22u
3
+ 25u
2
+ 15u + 1)
2
)(u
5
+ 9u
4
+ 30u
3
+ 38u
2
+ u + 1)
c
3
, c
8
u
3
(u
5
− u
4
+ 5u
3
− u
2
+ 2u + 2)
3
(u
5
+ 4u
4
+ 8u
3
+ 8u
2
+ 4)
c
4
, c
9
, c
10
u(u + 1)
2
(u
5
− u
3
− 3u
2
+ 4)(u
5
− 2u
4
− 2u
3
+ 3u
2
+ 3u − 1)
2
· (u
5
− u
4
− 4u
3
+ 4u
2
+ 3u + 1)
c
5
, c
6
, c
11
u(u − 1)(u + 1)(u
5
− u
3
− 3u
2
+ 4)(u
5
− 2u
4
+ ··· + 3u − 1)
2
· (u
5
− u
4
− 4u
3
+ 4u
2
+ 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
, c
11
y(y − 1)
2
(y
5
− 9y
4
+ 30y
3
− 38y
2
+ y − 1)
· ((y
5
− 8y
4
+ 22y
3
− 25y
2
+ 15y − 1)
2
)(y
5
− 2y
4
+ y
3
− 9y
2
+ 24y − 16)
c
2
y(y − 1)
2
(y
5
− 21y
4
+ 218y
3
− 1402y
2
− 75y − 1)
· (y
5
− 20y
4
+ 114y
3
+ 19y
2
+ 175y − 1)
2
· (y
5
− 2y
4
+ 13y
3
− 97y
2
+ 288y − 256)
c
3
, c
8
y
3
(y
5
− 96y
2
− 64y − 16)(y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y − 4)
3
33
