10
139
(K10n
27
)
A knot diagram
1
Linearized knot diagam
6 9 7 8 1 2 4 2 8 5
Solving Sequence
1,5
6
2,8
4 7 10 9 3
c
5
c
1
c
4
c
7
c
10
c
9
c
2
c
3
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
3
− 3u
2
+ b − 2u + 1, −u
3
− 2u
2
+ 2a − 2u, u
4
+ 4u
3
+ 6u
2
+ 2u − 2i
I
u
2
= hb + 1, 2a − u + 2, u
2
− 2i
I
v
1
= ha, b − 1, v + 1i
* 3 irreducible components of dim
C
= 0, with total 7 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
3
− 3u
2
+ b − 2u + 1, −u
3
− 2u
2
+ 2a − 2u, u
4
+ 4u
3
+ 6u
2
+ 2u − 2i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
2
=
−u
−u
3
+ u
a
8
=
1
2
u
3
+ u
2
+ u
u
3
+ 3u
2
+ 2u − 1
a
4
=
−
1
2
u
3
− u
2
+ 1
−2u
2
− 2u + 1
a
7
=
−u
2
+ 1
4u
3
+ 8u
2
+ 2u − 2
a
10
=
u
u
a
9
=
3
2
u
3
+ 5u
2
+ 4u − 2
−3u
3
− 5u
2
+ u + 1
a
3
=
3
2
u
3
+ u
2
− 2u
8u
3
+ 28u
2
+ 16u − 11
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u − 16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
10
u
4
− 4u
3
+ 6u
2
− 2u − 2
c
2
, c
3
, c
4
c
7
, c
8
u
4
+ 2u
3
+ 4u
2
− 2u − 1
c
9
u
4
− 4u
3
+ 22u
2
+ 12u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
10
y
4
− 4y
3
+ 16y
2
− 28y + 4
c
2
, c
3
, c
4
c
7
, c
8
y
4
+ 4y
3
+ 22y
2
− 12y + 1
c
9
y
4
+ 28y
3
+ 582y
2
− 100y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.47463
a = −0.903408
b = −0.632293
−6.80412 −13.0510
u = 0.395337
a = 0.582522
b = 0.321336
−0.588647 −16.7910
u = −1.46036 + 1.13932I
a = 0.660443 + 0.716885I
b = 1.15548 − 1.89385I
4.51885 + 4.85117I −13.07929 − 2.27864I
u = −1.46036 − 1.13932I
a = 0.660443 − 0.716885I
b = 1.15548 + 1.89385I
4.51885 − 4.85117I −13.07929 + 2.27864I
5
II. I
u
2
= hb + 1, 2a − u + 2, u
2
− 2i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
6
=
1
2
a
2
=
−u
−u
a
8
=
1
2
u − 1
−1
a
4
=
1
2
u
−1
a
7
=
−1
0
a
10
=
u
u
a
9
=
3
2
u − 1
u − 1
a
3
=
1
2
u − 1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −20
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
10
u
2
− 2
c
2
, c
7
(u − 1)
2
c
3
, c
4
, c
8
c
9
(u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
10
(y −2)
2
c
2
, c
3
, c
4
c
7
, c
8
, c
9
(y −1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = −0.292893
b = −1.00000
−8.22467 −20.0000
u = −1.41421
a = −1.70711
b = −1.00000
−8.22467 −20.0000
9
III. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
1
=
−1
0
a
5
=
1
0
a
6
=
1
0
a
2
=
−1
0
a
8
=
0
1
a
4
=
1
−1
a
7
=
1
0
a
10
=
−1
0
a
9
=
−1
1
a
3
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
10
u
c
2
, c
7
, c
9
u + 1
c
3
, c
4
, c
8
u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
10
y
c
2
, c
3
, c
4
c
7
, c
8
, c
9
y − 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
10
u(u
2
− 2)(u
4
− 4u
3
+ 6u
2
− 2u − 2)
c
2
, c
7
(u − 1)
2
(u + 1)(u
4
+ 2u
3
+ 4u
2
− 2u − 1)
c
3
, c
4
, c
8
(u − 1)(u + 1)
2
(u
4
+ 2u
3
+ 4u
2
− 2u − 1)
c
9
(u + 1)
3
(u
4
− 4u
3
+ 22u
2
+ 12u + 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
10
y(y − 2)
2
(y
4
− 4y
3
+ 16y
2
− 28y + 4)
c
2
, c
3
, c
4
c
7
, c
8
(y − 1)
3
(y
4
+ 4y
3
+ 22y
2
− 12y + 1)
c
9
(y − 1)
3
(y
4
+ 28y
3
+ 582y
2
− 100y + 1)
15
