11n
104
(K11n
104
)
A knot diagram
1
Linearized knot diagam
5 1 8 9 2 11 1 10 4 5 7
Solving Sequence
1,5
2
3,7
8 11 6 10 9 4
c
1
c
2
c
7
c
11
c
6
c
10
c
8
c
4
c
3
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, u
6
− 2u
5
− 9u
4
+ 28u
3
− u
2
+ 8a + 14u + 1, u
8
− 2u
7
− 10u
6
+ 30u
5
+ 8u
4
− 14u
3
+ 2u
2
+ 2u − 1i
I
u
2
= hb − 1, a
4
− 4a
3
+ 4a
2
+ 1, u + 1i
I
u
3
= hb + 1, a
3
+ 3a
2
+ 3a + 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 15 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
= hb −u, u
6
− 2u
5
− 9u
4
+ 28u
3
− u
2
+ 8a + 14u + 1, u
8
− 2u
7
+ · · · + 2u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
−u
2
+ 1
u
2
a
7
=
−
1
8
u
6
+
1
4
u
5
+ ··· −
7
4
u −
1
8
u
a
8
=
−
1
8
u
6
+
1
4
u
5
+ ··· −
11
4
u −
1
8
u
a
11
=
1
8
u
7
−
1
4
u
6
+ ··· +
1
8
u + 1
−u
2
a
6
=
−u
−u
3
+ u
a
10
=
1
8
u
7
−
1
4
u
6
+ ··· +
1
8
u + 1
−
1
8
u
7
+
1
4
u
6
+ ··· −
7
4
u
2
−
1
8
u
a
9
=
−
5
8
u
7
+
15
8
u
6
+ ··· −
65
8
u +
13
8
1
2
u
7
−
11
8
u
6
+ ··· + 4u −
9
8
a
4
=
−
5
8
u
7
+
11
8
u
6
+ ··· −
9
8
u +
9
8
1
8
u
7
−
1
4
u
6
+ ··· +
7
4
u
2
+
1
8
u
a
4
=
−
5
8
u
7
+
11
8
u
6
+ ··· −
9
8
u +
9
8
1
8
u
7
−
1
4
u
6
+ ··· +
7
4
u
2
+
1
8
u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1
2
u
7
−
11
4
u
6
−
1
2
u
5
+
121
4
u
4
−
115
2
u
3
+
43
4
u
2
+
43
2
u −
65
4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
u
8
+ 2u
7
− 10u
6
− 30u
5
+ 8u
4
+ 14u
3
+ 2u
2
− 2u − 1
c
2
u
8
+ 24u
7
+ 236u
6
+ 1112u
5
+ 870u
4
+ 264u
3
+ 44u
2
+ 8u + 1
c
3
, c
10
u
8
− 4u
7
− 12u
6
+ 70u
5
− 54u
4
− 46u
3
+ 38u
2
+ 10u + 10
c
4
, c
9
u
8
+ 4u
7
+ 10u
6
+ 16u
5
+ 18u
4
+ 16u
3
+ 10u
2
+ 6u + 2
c
8
u
8
+ 4u
7
+ 8u
6
− 4u
5
− 32u
4
− 48u
3
− 20u
2
+ 4u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
y
8
− 24y
7
+ 236y
6
− 1112y
5
+ 870y
4
− 264y
3
+ 44y
2
− 8y + 1
c
2
y
8
− 104y
7
+ ··· + 24y + 1
c
3
, c
10
y
8
− 40y
7
+ ··· + 660y + 100
c
4
, c
9
y
8
+ 4y
7
+ 8y
6
− 4y
5
− 32y
4
− 48y
3
− 20y
2
+ 4y + 4
c
8
y
8
+ 32y
6
− 184y
5
+ 296y
4
− 928y
3
+ 528y
2
− 176y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.594812 + 0.065631I
a = 1.77892 − 0.41529I
b = −0.594812 + 0.065631I
−4.20158 + 3.92770I −10.18918 − 5.00146I
u = −0.594812 − 0.065631I
a = 1.77892 + 0.41529I
b = −0.594812 − 0.065631I
−4.20158 − 3.92770I −10.18918 + 5.00146I
u = 0.495898
a = −1.31523
b = 0.495898
−1.19322 −8.17950
u = 0.279091 + 0.329009I
a = −0.414734 − 0.712553I
b = 0.279091 + 0.329009I
−0.535301 − 1.039080I −7.61110 + 6.36007I
u = 0.279091 − 0.329009I
a = −0.414734 + 0.712553I
b = 0.279091 − 0.329009I
−0.535301 + 1.039080I −7.61110 − 6.36007I
u = 2.73980 + 1.24096I
a = 0.621831 − 0.351657I
b = 2.73980 + 1.24096I
11.45110 − 7.34942I −11.27453 + 2.75920I
u = 2.73980 − 1.24096I
a = 0.621831 + 0.351657I
b = 2.73980 − 1.24096I
11.45110 + 7.34942I −11.27453 − 2.75920I
u = −3.34406
a = −0.656809
b = −3.34406
15.7287 −9.67090
5
II. I
u
2
= hb − 1, a
4
− 4a
3
+ 4a
2
+ 1, u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
−1
a
2
=
1
1
a
3
=
0
1
a
7
=
a
1
a
8
=
a − 1
1
a
11
=
−a + 1
−1
a
6
=
1
0
a
10
=
−a + 1
a − 2
a
9
=
−a
3
+ 3a
2
− 2a
a
3
− 4a
2
+ 5a − 1
a
4
=
−a
2
+ 2a − 1
−a + 2
a
4
=
−a
2
+ 2a − 1
−a + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a
2
− 8a − 16
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
(u + 1)
4
c
3
, c
10
u
4
− 2u
2
+ 2
c
4
, c
9
u
4
+ 2u
2
+ 2
c
5
, c
11
(u − 1)
4
c
8
(u
2
− 2u + 2)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
11
(y −1)
4
c
3
, c
10
(y
2
− 2y + 2)
2
c
4
, c
9
(y
2
+ 2y + 2)
2
c
8
(y
2
+ 4)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.098684 + 0.455090I
b = 1.00000
−5.75727 + 3.66386I −16.0000 − 4.0000I
u = −1.00000
a = −0.098684 − 0.455090I
b = 1.00000
−5.75727 − 3.66386I −16.0000 + 4.0000I
u = −1.00000
a = 2.09868 + 0.45509I
b = 1.00000
−5.75727 − 3.66386I −16.0000 + 4.0000I
u = −1.00000
a = 2.09868 − 0.45509I
b = 1.00000
−5.75727 + 3.66386I −16.0000 − 4.0000I
9
III. I
u
3
= hb + 1, a
3
+ 3a
2
+ 3a + 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
7
=
a
−1
a
8
=
a + 1
−1
a
11
=
a + 1
−1
a
6
=
−1
0
a
10
=
a + 1
−a − 2
a
9
=
a + 1
a
2
+ 2a
a
4
=
−a
2
− 2a − 1
a + 2
a
4
=
−a
2
− 2a − 1
a + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a
2
+ 8a − 8
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
(u − 1)
3
c
2
, c
5
, c
11
(u + 1)
3
c
3
, c
4
, c
8
c
9
, c
10
u
3
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
11
(y −1)
3
c
3
, c
4
, c
8
c
9
, c
10
y
3
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −1.00000
−3.28987 −12.0000
u = 1.00000
a = −1.00000
b = −1.00000
−3.28987 −12.0000
u = 1.00000
a = −1.00000
b = −1.00000
−3.28987 −12.0000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
((u − 1)
3
)(u + 1)
4
(u
8
+ 2u
7
+ ··· − 2u − 1)
c
2
(u + 1)
7
· (u
8
+ 24u
7
+ 236u
6
+ 1112u
5
+ 870u
4
+ 264u
3
+ 44u
2
+ 8u + 1)
c
3
, c
10
u
3
(u
4
− 2u
2
+ 2)
· (u
8
− 4u
7
− 12u
6
+ 70u
5
− 54u
4
− 46u
3
+ 38u
2
+ 10u + 10)
c
4
, c
9
u
3
(u
4
+ 2u
2
+ 2)(u
8
+ 4u
7
+ ··· + 6u + 2)
c
5
, c
11
((u − 1)
4
)(u + 1)
3
(u
8
+ 2u
7
+ ··· − 2u − 1)
c
8
u
3
(u
2
− 2u + 2)
2
(u
8
+ 4u
7
+ ··· + 4u + 4)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
(y −1)
7
· (y
8
− 24y
7
+ 236y
6
− 1112y
5
+ 870y
4
− 264y
3
+ 44y
2
− 8y + 1)
c
2
((y −1)
7
)(y
8
− 104y
7
+ ··· + 24y + 1)
c
3
, c
10
y
3
(y
2
− 2y + 2)
2
(y
8
− 40y
7
+ ··· + 660y + 100)
c
4
, c
9
y
3
(y
2
+ 2y + 2)
2
(y
8
+ 4y
7
+ ··· + 4y + 4)
c
8
y
3
(y
2
+ 4)
2
(y
8
+ 32y
6
+ ··· − 176y + 16)
15
