11n
38
(K11n
38
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 10 4 1 11 6 9 8
Solving Sequence
1,8 4,7
3 2 5 6 11 9 10
c
7
c
3
c
2
c
4
c
6
c
11
c
8
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, 5u
4
+ u
3
+ 33u
2
+ 14a + 10u + 11, u
5
+ 6u
3
− u
2
− u − 1i
I
u
2
= hb + u, −u
3
− u
2
+ a − 3u − 2, u
4
+ u
3
+ 3u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 9 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, 5u
4
+ u
3
+ 33u
2
+ 14a + 10u + 11, u
5
+ 6u
3
− u
2
− u − 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
−
5
14
u
4
−
1
14
u
3
+ ··· −
5
7
u −
11
14
u
a
7
=
1
u
2
a
3
=
−
4
7
u
4
−
5
7
u
3
+ ··· −
15
7
u −
6
7
1
14
u
4
+
31
14
u
3
+ ··· +
1
7
u −
9
14
a
2
=
−
4
7
u
4
−
5
7
u
3
+ ··· −
15
7
u −
6
7
3
14
u
4
+
9
14
u
3
+ ··· +
10
7
u +
1
14
a
5
=
4
7
u
4
−
2
7
u
3
+ ··· +
22
7
u +
6
7
−0.357143u
4
+ 0.928571u
3
+ ··· − 0.714286u − 0.785714
a
6
=
−0.785714u
4
− 0.357143u
3
+ ··· − 2.57143u + 0.0714286
9
14
u
4
−
1
14
u
3
+ ··· +
2
7
u +
3
14
a
11
=
−u
u
a
9
=
u
2
+ 1
−u
2
a
10
=
−u
3
− 2u
u
3
+ u
a
10
=
−u
3
− 2u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
23
7
u
4
+
1
7
u
3
−
135
7
u
2
+
31
7
u +
18
7
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
5
− 5u
4
+ 20u
2
− u + 1
c
2
u
5
+ 25u
4
+ 198u
3
+ 390u
2
− 39u + 1
c
3
, c
6
u
5
+ 4u
4
+ 38u
3
+ 40u
2
− 40u + 16
c
5
, c
9
u
5
+ 2u
4
+ 2u
3
− u
2
− u − 1
c
7
, c
8
, c
10
c
11
u
5
+ 6u
3
+ u
2
− u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
5
− 25y
4
+ 198y
3
− 390y
2
− 39y −1
c
2
y
5
− 229y
4
+ 19626y
3
− 167594y
2
+ 741y −1
c
3
, c
6
y
5
+ 60y
4
+ 1044y
3
− 4768y
2
+ 320y −256
c
5
, c
9
y
5
+ 6y
3
− y
2
− y −1
c
7
, c
8
, c
10
c
11
y
5
+ 12y
4
+ 34y
3
− 13y
2
− y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.695222
a = −2.52902
b = 0.695222
−2.84858 −4.39070
u = −0.281458 + 0.392024I
a = −0.401414 + 0.226060I
b = −0.281458 + 0.392024I
0.206446 − 1.108910I 2.91822 + 5.88873I
u = −0.281458 − 0.392024I
a = −0.401414 − 0.226060I
b = −0.281458 − 0.392024I
0.206446 + 1.108910I 2.91822 − 5.88873I
u = −0.06615 + 2.48427I
a = 0.165924 − 1.354820I
b = −0.06615 + 2.48427I
10.26500 − 4.12490I −3.22285 + 1.83437I
u = −0.06615 − 2.48427I
a = 0.165924 + 1.354820I
b = −0.06615 − 2.48427I
10.26500 + 4.12490I −3.22285 − 1.83437I
5
II. I
u
2
= hb + u, −u
3
− u
2
+ a − 3u − 2, u
4
+ u
3
+ 3u
2
+ 2u + 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
u
3
+ u
2
+ 3u + 2
−u
a
7
=
1
u
2
a
3
=
u
3
+ u
2
+ 3u + 2
−u
a
2
=
u
3
+ u
2
+ 3u + 2
0
a
5
=
0
−u
a
6
=
1
u
2
a
11
=
−u
u
a
9
=
u
2
+ 1
−u
2
a
10
=
−u
3
− 2u
u
3
+ u
a
10
=
−u
3
− 2u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
3
+ 2u
2
+ 7u + 1
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
c
2
, c
4
(u + 1)
4
c
3
, c
6
u
4
c
5
u
4
+ u
3
+ u
2
+ 1
c
7
, c
8
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
9
u
4
− u
3
+ u
2
+ 1
c
10
, c
11
u
4
− u
3
+ 3u
2
− 2u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
6
y
4
c
5
, c
9
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
7
, c
8
, c
10
c
11
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.395123 + 0.506844I
a = 0.95668 + 1.22719I
b = 0.395123 − 0.506844I
−1.43393 − 1.41510I −1.48175 + 2.96122I
u = −0.395123 − 0.506844I
a = 0.95668 − 1.22719I
b = 0.395123 + 0.506844I
−1.43393 + 1.41510I −1.48175 − 2.96122I
u = −0.10488 + 1.55249I
a = 0.043315 + 0.641200I
b = 0.10488 − 1.55249I
−8.43568 − 3.16396I −3.01825 + 2.83489I
u = −0.10488 − 1.55249I
a = 0.043315 − 0.641200I
b = 0.10488 + 1.55249I
−8.43568 + 3.16396I −3.01825 − 2.83489I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
(u
5
− 5u
4
+ 20u
2
− u + 1)
c
2
(u + 1)
4
(u
5
+ 25u
4
+ 198u
3
+ 390u
2
− 39u + 1)
c
3
, c
6
u
4
(u
5
+ 4u
4
+ 38u
3
+ 40u
2
− 40u + 16)
c
4
(u + 1)
4
(u
5
− 5u
4
+ 20u
2
− u + 1)
c
5
(u
4
+ u
3
+ u
2
+ 1)(u
5
+ 2u
4
+ 2u
3
− u
2
− u − 1)
c
7
, c
8
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
5
+ 6u
3
+ u
2
− u + 1)
c
9
(u
4
− u
3
+ u
2
+ 1)(u
5
+ 2u
4
+ 2u
3
− u
2
− u − 1)
c
10
, c
11
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
5
+ 6u
3
+ u
2
− u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y −1)
4
(y
5
− 25y
4
+ 198y
3
− 390y
2
− 39y −1)
c
2
(y −1)
4
(y
5
− 229y
4
+ 19626y
3
− 167594y
2
+ 741y −1)
c
3
, c
6
y
4
(y
5
+ 60y
4
+ 1044y
3
− 4768y
2
+ 320y −256)
c
5
, c
9
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
5
+ 6y
3
− y
2
− y −1)
c
7
, c
8
, c
10
c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
5
+ 12y
4
+ 34y
3
− 13y
2
− y −1)
11
