11n
91
(K11n
91
)
A knot diagram
1
Linearized knot diagam
6 1 9 10 2 3 11 4 3 7 10
Solving Sequence
7,10
11 8
1,3
2 6 5 9 4
c
10
c
7
c
11
c
2
c
6
c
5
c
9
c
3
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−10091257321u
20
+ 17716980292u
19
+ ··· + 366196956382b + 49796914345,
591910942537u
20
− 2794488596145u
19
+ ··· + 4394363476584a − 481484753828,
u
21
− 3u
20
+ ··· − 2u − 3i
I
u
2
= hb
2
+ 2, a
2
− a + 1, u + 1i
I
u
3
= hb, a
2
− a + 1, u − 1i
* 3 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−1.01×10
10
u
20
+1.77×10
10
u
19
+· · ·+3.66×10
11
b+4.98×10
10
, 5.92×
10
11
u
20
−2.79×10
12
u
19
+· · ·+4.39×10
12
a−4.81×10
11
, u
21
−3u
20
+· · ·−2u−3i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
8
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
2
a
3
=
−0.134698u
20
+ 0.635926u
19
+ ··· − 0.0316545u + 0.109569
0.0275569u
20
− 0.0483810u
19
+ ··· − 1.11621u − 0.135984
a
2
=
−0.0599090u
20
+ 0.413165u
19
+ ··· − 1.26842u + 0.402548
0.129090u
20
− 0.405931u
19
+ ··· − 1.14509u − 0.299649
a
6
=
−0.0944207u
20
+ 0.398371u
19
+ ··· + 0.793023u − 0.448119
0.0612997u
20
− 0.138407u
19
+ ··· + 0.143901u − 0.565085
a
5
=
0.340816u
20
− 1.13692u
19
+ ··· − 0.307450u − 1.55518
0.154312u
20
− 0.484554u
19
+ ··· − 0.533633u − 0.125533
a
9
=
−0.188362u
20
+ 0.626385u
19
+ ··· + 2.11578u + 0.520624
0.0158081u
20
+ 0.0256218u
19
+ ··· + 0.586386u − 0.748984
a
4
=
−0.186504u
20
+ 0.652367u
19
+ ··· − 0.226183u + 1.42965
−0.154312u
20
+ 0.484554u
19
+ ··· + 0.533633u + 0.125533
a
4
=
−0.186504u
20
+ 0.652367u
19
+ ··· − 0.226183u + 1.42965
−0.154312u
20
+ 0.484554u
19
+ ··· + 0.533633u + 0.125533
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
57646172449
732393912764
u
20
+
387941164973
732393912764
u
19
+ ··· +
4279700760483
732393912764
u +
1134852296901
183098478191
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
21
− 2u
20
+ ··· + 7u − 3
c
2
u
21
+ 14u
20
+ ··· + 49u − 9
c
3
, c
8
, c
9
u
21
+ u
20
+ ··· + 8u − 4
c
4
u
21
− u
20
+ ··· − 64u − 548
c
6
u
21
+ 2u
20
+ ··· + 19u − 3
c
7
, c
10
u
21
− 3u
20
+ ··· − 2u − 3
c
11
u
21
− 3u
20
+ ··· + 22u − 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
21
+ 14y
20
+ ··· + 49y −9
c
2
y
21
− 10y
20
+ ··· + 7117y −81
c
3
, c
8
, c
9
y
21
+ 31y
20
+ ··· − 160y −16
c
4
y
21
+ 91y
20
+ ··· − 4121248y −300304
c
6
y
21
− 34y
20
+ ··· + 193y −9
c
7
, c
10
y
21
− 3y
20
+ ··· + 22y −9
c
11
y
21
+ 37y
20
+ ··· + 2158y −81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.678453 + 0.688147I
a = 0.845365 + 0.796760I
b = −0.271568 + 1.062540I
−2.57223 − 2.30104I 5.12347 + 3.68698I
u = −0.678453 − 0.688147I
a = 0.845365 − 0.796760I
b = −0.271568 − 1.062540I
−2.57223 + 2.30104I 5.12347 − 3.68698I
u = 0.316999 + 0.813917I
a = −0.403688 + 0.492380I
b = 0.713808 + 0.270061I
−2.21076 + 2.09468I 3.00496 − 3.91489I
u = 0.316999 − 0.813917I
a = −0.403688 − 0.492380I
b = 0.713808 − 0.270061I
−2.21076 − 2.09468I 3.00496 + 3.91489I
u = 1.145710 + 0.140166I
a = 0.239466 − 0.410115I
b = −0.264392 + 0.442489I
1.11524 + 1.37460I 4.04933 + 2.02582I
u = 1.145710 − 0.140166I
a = 0.239466 + 0.410115I
b = −0.264392 − 0.442489I
1.11524 − 1.37460I 4.04933 − 2.02582I
u = −1.134400 + 0.465760I
a = −0.130701 − 0.200335I
b = −0.21844 − 1.42669I
−5.12357 + 0.62763I 1.407681 + 0.031302I
u = −1.134400 − 0.465760I
a = −0.130701 + 0.200335I
b = −0.21844 + 1.42669I
−5.12357 − 0.62763I 1.407681 − 0.031302I
u = −0.831921 + 0.976886I
a = −0.814769 − 0.824529I
b = 0.567067 − 1.129180I
−6.41753 − 6.55364I 2.05255 + 5.68240I
u = −0.831921 − 0.976886I
a = −0.814769 + 0.824529I
b = 0.567067 + 1.129180I
−6.41753 + 6.55364I 2.05255 − 5.68240I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.684877 + 0.126673I
a = 0.28002 + 1.72236I
b = −0.033686 + 0.472962I
0.91652 − 2.35539I 2.06528 + 5.30676I
u = −0.684877 − 0.126673I
a = 0.28002 − 1.72236I
b = −0.033686 − 0.472962I
0.91652 + 2.35539I 2.06528 − 5.30676I
u = −0.153992 + 0.545893I
a = −1.93201 − 1.20990I
b = 0.130919 − 1.391490I
−5.17740 + 1.74265I 0.53526 − 2.03708I
u = −0.153992 − 0.545893I
a = −1.93201 + 1.20990I
b = 0.130919 + 1.391490I
−5.17740 − 1.74265I 0.53526 + 2.03708I
u = 0.515961
a = 0.725450
b = −0.417301
0.754310 13.3550
u = 1.06707 + 1.07605I
a = 0.80534 − 1.32502I
b = −0.08814 − 1.77927I
−12.95410 + 3.94853I 4.46142 − 1.91994I
u = 1.06707 − 1.07605I
a = 0.80534 + 1.32502I
b = −0.08814 + 1.77927I
−12.95410 − 3.94853I 4.46142 + 1.91994I
u = 0.92823 + 1.31603I
a = −0.465176 + 1.276900I
b = −0.01070 + 1.85472I
−17.8929 − 1.4476I 1.24285 + 0.69389I
u = 0.92823 − 1.31603I
a = −0.465176 − 1.276900I
b = −0.01070 − 1.85472I
−17.8929 + 1.4476I 1.24285 − 0.69389I
u = 1.26765 + 1.02698I
a = −0.95325 + 1.07681I
b = 0.18378 + 1.78761I
−16.6802 + 9.9035I 2.37944 − 4.78800I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.26765 − 1.02698I
a = −0.95325 − 1.07681I
b = 0.18378 − 1.78761I
−16.6802 − 9.9035I 2.37944 + 4.78800I
7
II. I
u
2
= hb
2
+ 2, a
2
− a + 1, u + 1i
(i) Arc colorings
a
7
=
0
−1
a
10
=
1
0
a
11
=
1
−1
a
8
=
−1
0
a
1
=
0
−1
a
3
=
a
b
a
2
=
a
b − a
a
6
=
−a + 1
−ba − 1
a
5
=
−a
−b
a
9
=
ba + 1
−2
a
4
=
b − a
−b
a
4
=
b − a
−b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a + 4
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
2
− u + 1)
2
c
2
, c
5
(u
2
+ u + 1)
2
c
3
, c
4
, c
8
c
9
(u
2
+ 2)
2
c
7
, c
11
(u − 1)
4
c
10
(u + 1)
4
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
(y
2
+ y + 1)
2
c
3
, c
4
, c
8
c
9
(y + 2)
4
c
7
, c
10
, c
11
(y −1)
4
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.500000 + 0.866025I
b = 1.414210I
−3.28987 − 2.02988I 6.00000 + 3.46410I
u = −1.00000
a = 0.500000 + 0.866025I
b = − 1.414210I
−3.28987 − 2.02988I 6.00000 + 3.46410I
u = −1.00000
a = 0.500000 − 0.866025I
b = 1.414210I
−3.28987 + 2.02988I 6.00000 − 3.46410I
u = −1.00000
a = 0.500000 − 0.866025I
b = − 1.414210I
−3.28987 + 2.02988I 6.00000 − 3.46410I
11
III. I
u
3
= hb, a
2
− a + 1, u − 1i
(i) Arc colorings
a
7
=
0
1
a
10
=
1
0
a
11
=
1
−1
a
8
=
1
0
a
1
=
0
−1
a
3
=
a
0
a
2
=
a
−a
a
6
=
a − 1
1
a
5
=
a
0
a
9
=
1
0
a
4
=
a
0
a
4
=
a
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a + 10
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
2
+ u + 1
c
3
, c
4
, c
8
c
9
u
2
c
5
u
2
− u + 1
c
7
(u + 1)
2
c
10
, c
11
(u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
y
2
+ y + 1
c
3
, c
4
, c
8
c
9
y
2
c
7
, c
10
, c
11
(y −1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.500000 + 0.866025I
b = 0
1.64493 − 2.02988I 12.00000 + 3.46410I
u = 1.00000
a = 0.500000 − 0.866025I
b = 0
1.64493 + 2.02988I 12.00000 − 3.46410I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
2
)(u
2
+ u + 1)(u
21
− 2u
20
+ ··· + 7u − 3)
c
2
((u
2
+ u + 1)
3
)(u
21
+ 14u
20
+ ··· + 49u − 9)
c
3
, c
8
, c
9
u
2
(u
2
+ 2)
2
(u
21
+ u
20
+ ··· + 8u − 4)
c
4
u
2
(u
2
+ 2)
2
(u
21
− u
20
+ ··· − 64u − 548)
c
5
(u
2
− u + 1)(u
2
+ u + 1)
2
(u
21
− 2u
20
+ ··· + 7u − 3)
c
6
((u
2
− u + 1)
2
)(u
2
+ u + 1)(u
21
+ 2u
20
+ ··· + 19u − 3)
c
7
((u − 1)
4
)(u + 1)
2
(u
21
− 3u
20
+ ··· − 2u − 3)
c
10
((u − 1)
2
)(u + 1)
4
(u
21
− 3u
20
+ ··· − 2u − 3)
c
11
((u − 1)
6
)(u
21
− 3u
20
+ ··· + 22u − 9)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
((y
2
+ y + 1)
3
)(y
21
+ 14y
20
+ ··· + 49y −9)
c
2
((y
2
+ y + 1)
3
)(y
21
− 10y
20
+ ··· + 7117y −81)
c
3
, c
8
, c
9
y
2
(y + 2)
4
(y
21
+ 31y
20
+ ··· − 160y −16)
c
4
y
2
(y + 2)
4
(y
21
+ 91y
20
+ ··· − 4121248y −300304)
c
6
((y
2
+ y + 1)
3
)(y
21
− 34y
20
+ ··· + 193y −9)
c
7
, c
10
((y −1)
6
)(y
21
− 3y
20
+ ··· + 22y −9)
c
11
((y −1)
6
)(y
21
+ 37y
20
+ ··· + 2158y −81)
17
