11n
141
(K11n
141
)
A knot diagram
1
Linearized knot diagam
8 1 9 8 11 10 2 4 1 6 5
Solving Sequence
5,8
4 9
1,3
2 11 6 10 7
c
4
c
8
c
3
c
2
c
11
c
5
c
10
c
6
c
1
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
7
− u
6
− 6u
5
− 7u
4
− 9u
3
− 15u
2
+ 4b − 1, a − 1, u
8
+ 7u
6
+ u
5
+ 14u
4
+ 4u
3
+ 5u
2
− u + 1i
I
u
2
= h−u
5
+ 2u
4
− u
3
+ 5u
2
+ 5b − u, u
5
+ 2u
3
+ 2u
2
+ 5a + u + 7, u
6
− u
5
+ 4u
4
− 4u
3
+ 6u
2
− 4u + 5i
I
u
3
= hb
2
+ bu + 1, a + 1, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 18 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
7
− u
6
− 6u
5
− 7u
4
− 9u
3
− 15u
2
+ 4b − 1, a − 1, u
8
+ 7u
6
+ u
5
+
14u
4
+ 4u
3
+ 5u
2
− u + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
1
=
1
1
4
u
7
+
1
4
u
6
+ ··· +
15
4
u
2
+
1
4
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
1
1
4
u
7
+
1
4
u
6
+ ··· +
11
4
u
2
+
1
4
a
11
=
1
4
u
7
+
1
4
u
6
+ ··· +
15
4
u
2
+
5
4
1
4
u
7
+
1
4
u
6
+ ··· +
15
4
u
2
+
1
4
a
6
=
−
1
4
u
7
+
1
4
u
6
+ ··· −
1
2
u +
3
4
−
1
2
u
7
−
7
2
u
5
+ ··· −
1
2
u −
1
2
a
10
=
−
1
4
u
7
+
1
4
u
6
+ ··· +
3
2
u +
1
4
1
2
u
5
−
1
2
u
4
+ ··· + 2u −
1
2
a
7
=
u
1
4
u
7
−
1
4
u
6
+ ··· −
1
2
u −
1
4
a
7
=
u
1
4
u
7
−
1
4
u
6
+ ··· −
1
2
u −
1
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
7
− 2u
6
− 13u
5
− 15u
4
− 25u
3
− 30u
2
− 12u + 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
u
8
+ 7u
6
+ u
5
+ 14u
4
+ 4u
3
+ 5u
2
− u + 1
c
2
u
8
+ 14u
7
+ 77u
6
+ 205u
5
+ 260u
4
+ 140u
3
+ 61u
2
+ 9u + 1
c
5
, c
6
, c
10
c
11
u
8
+ 3u
7
+ 9u
6
+ 16u
5
+ 23u
4
+ 24u
3
+ 18u
2
+ 7u + 2
c
9
u
8
+ u
7
+ 21u
6
+ 24u
5
+ 109u
4
+ 142u
3
− 10u
2
− 23u + 24
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
y
8
+ 14y
7
+ 77y
6
+ 205y
5
+ 260y
4
+ 140y
3
+ 61y
2
+ 9y + 1
c
2
y
8
− 42y
7
+ ··· + 41y + 1
c
5
, c
6
, c
10
c
11
y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 47y
4
+ 64y
3
+ 80y
2
+ 23y + 4
c
9
y
8
+ 41y
7
+ ··· − 1009y + 576
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.405950 + 0.590547I
a = 1.00000
b = −0.00585 − 1.54991I
6.99421 − 1.46497I 5.63406 + 4.72165I
u = −0.405950 − 0.590547I
a = 1.00000
b = −0.00585 + 1.54991I
6.99421 + 1.46497I 5.63406 − 4.72165I
u = 0.195934 + 0.349055I
a = 1.00000
b = −0.218002 + 0.455338I
0.133570 + 0.902562I 2.84755 − 7.78366I
u = 0.195934 − 0.349055I
a = 1.00000
b = −0.218002 − 0.455338I
0.133570 − 0.902562I 2.84755 + 7.78366I
u = 0.33222 + 1.78481I
a = 1.00000
b = −0.34865 + 1.60107I
−9.04281 + 7.80349I 0.02756 − 3.21559I
u = 0.33222 − 1.78481I
a = 1.00000
b = −0.34865 − 1.60107I
−9.04281 − 7.80349I 0.02756 + 3.21559I
u = −0.12220 + 1.91634I
a = 1.00000
b = −0.927504 − 0.597003I
−16.1792 − 3.0379I −2.50917 + 2.22003I
u = −0.12220 − 1.91634I
a = 1.00000
b = −0.927504 + 0.597003I
−16.1792 + 3.0379I −2.50917 − 2.22003I
5
II. I
u
2
= h−u
5
+ 2u
4
− u
3
+ 5u
2
+ 5b − u, u
5
+ 2u
3
+ 2u
2
+ 5a + u + 7, u
6
−
u
5
+ 4u
4
− 4u
3
+ 6u
2
− 4u + 5i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
1
=
−
1
5
u
5
−
2
5
u
3
+ ··· −
1
5
u −
7
5
1
5
u
5
−
2
5
u
4
+
1
5
u
3
− u
2
+
1
5
u
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
−
1
5
u
5
−
2
5
u
3
+ ··· −
1
5
u −
7
5
−1
a
11
=
−
2
5
u
4
−
1
5
u
3
−
7
5
u
2
−
7
5
1
5
u
5
−
2
5
u
4
+
1
5
u
3
− u
2
+
1
5
u
a
6
=
2
5
u
5
−
2
5
u
4
+ ··· +
7
5
u −
8
5
2
5
u
5
−
4
5
u
4
+ ··· +
7
5
u − 2
a
10
=
1
5
u
5
−
1
5
u
4
+ ··· +
6
5
u −
4
5
1
5
u
5
−
2
5
u
4
+ ··· +
6
5
u − 1
a
7
=
2
5
u
5
−
3
5
u
4
+ ··· +
12
5
u −
9
5
1
5
u
5
−
2
5
u
4
+ ··· +
6
5
u − 1
a
7
=
2
5
u
5
−
3
5
u
4
+ ··· +
12
5
u −
9
5
1
5
u
5
−
2
5
u
4
+ ··· +
6
5
u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
4
5
u
5
+
8
5
u
4
−
24
5
u
3
+ 4u
2
−
24
5
u + 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
u
6
− u
5
+ 4u
4
− 4u
3
+ 6u
2
− 4u + 5
c
2
u
6
+ 7u
5
+ 20u
4
+ 34u
3
+ 44u
2
+ 44u + 25
c
5
, c
6
, c
10
c
11
(u
3
− u
2
+ 2u − 1)
2
c
9
(u
3
− u
2
+ 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
y
6
+ 7y
5
+ 20y
4
+ 34y
3
+ 44y
2
+ 44y + 25
c
2
y
6
− 9y
5
+ 12y
4
+ 38y
3
− 56y
2
+ 264y + 625
c
5
, c
6
, c
10
c
11
(y
3
+ 3y
2
+ 2y − 1)
2
c
9
(y
3
− y
2
+ 2y − 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.862082 + 0.785389I
a = −0.873959 − 0.978854I
b = 0.215080 − 1.307140I
−0.26574 + 2.82812I 1.50976 − 2.97945I
u = 0.862082 − 0.785389I
a = −0.873959 + 0.978854I
b = 0.215080 + 1.307140I
−0.26574 − 2.82812I 1.50976 + 2.97945I
u = −0.377439 + 1.194730I
a = −0.818504 + 0.574501I
b = 0.569840
−4.40332 −5.01951 + 0.I
u = −0.377439 − 1.194730I
a = −0.818504 − 0.574501I
b = 0.569840
−4.40332 −5.01951 + 0.I
u = 0.01536 + 1.53025I
a = −0.507537 − 0.568454I
b = 0.215080 + 1.307140I
−0.26574 − 2.82812I 1.50976 + 2.97945I
u = 0.01536 − 1.53025I
a = −0.507537 + 0.568454I
b = 0.215080 − 1.307140I
−0.26574 + 2.82812I 1.50976 − 2.97945I
9
III. I
u
3
= hb
2
+ bu + 1, a + 1, u
2
+ 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
−1
a
9
=
u
0
a
1
=
−1
b
a
3
=
0
−1
a
2
=
−1
b − 1
a
11
=
b − 1
b
a
6
=
−bu − b
−bu − 1
a
10
=
bu + u
−b + u
a
7
=
−u
bu
a
7
=
−u
bu
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
2
c
2
(u + 1)
4
c
5
, c
6
, c
10
c
11
u
4
+ 3u
2
+ 1
c
9
(u
2
− u − 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(y + 1)
4
c
2
(y − 1)
4
c
5
, c
6
, c
10
c
11
(y
2
+ 3y + 1)
2
c
9
(y
2
− 3y + 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −1.00000
b = 0.618034I
−2.30291 0
u = 1.000000I
a = −1.00000
b = − 1.61803I
5.59278 0
u = − 1.000000I
a = −1.00000
b = − 0.618034I
−2.30291 0
u = − 1.000000I
a = −1.00000
b = 1.61803I
5.59278 0
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
2
(u
6
− u
5
+ 4u
4
− 4u
3
+ 6u
2
− 4u + 5)
· (u
8
+ 7u
6
+ u
5
+ 14u
4
+ 4u
3
+ 5u
2
− u + 1)
c
2
(u + 1)
4
(u
6
+ 7u
5
+ 20u
4
+ 34u
3
+ 44u
2
+ 44u + 25)
· (u
8
+ 14u
7
+ 77u
6
+ 205u
5
+ 260u
4
+ 140u
3
+ 61u
2
+ 9u + 1)
c
5
, c
6
, c
10
c
11
(u
3
− u
2
+ 2u − 1)
2
(u
4
+ 3u
2
+ 1)
· (u
8
+ 3u
7
+ 9u
6
+ 16u
5
+ 23u
4
+ 24u
3
+ 18u
2
+ 7u + 2)
c
9
(u
2
− u − 1)
2
(u
3
− u
2
+ 1)
2
· (u
8
+ u
7
+ 21u
6
+ 24u
5
+ 109u
4
+ 142u
3
− 10u
2
− 23u + 24)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(y + 1)
4
(y
6
+ 7y
5
+ 20y
4
+ 34y
3
+ 44y
2
+ 44y + 25)
· (y
8
+ 14y
7
+ 77y
6
+ 205y
5
+ 260y
4
+ 140y
3
+ 61y
2
+ 9y + 1)
c
2
(y − 1)
4
(y
6
− 9y
5
+ 12y
4
+ 38y
3
− 56y
2
+ 264y + 625)
· (y
8
− 42y
7
+ ··· + 41y + 1)
c
5
, c
6
, c
10
c
11
(y
2
+ 3y + 1)
2
(y
3
+ 3y
2
+ 2y − 1)
2
· (y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 47y
4
+ 64y
3
+ 80y
2
+ 23y + 4)
c
9
((y
2
− 3y + 1)
2
)(y
3
− y
2
+ 2y − 1)
2
(y
8
+ 41y
7
+ ··· − 1009y + 576)
15
