10
132
(K10n
13
)
A knot diagram
1
Linearized knot diagam
5 10 7 6 2 4 10 7 3 8
Solving Sequence
3,7 4,10
8 2 6 5 1 9
c
3
c
7
c
2
c
6
c
5
c
1
c
9
c
4
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
4
− 5u
3
− 11u
2
+ 9b − 14u − 1, −4u
4
− u
3
− 22u
2
+ 9a − u + 7, u
5
+ 6u
3
+ u + 1i
I
u
2
= hb, −u
2
+ a − u − 2, u
3
+ u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 8 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−2u
4
−5u
3
−11u
2
+9b−14u−1, −4u
4
−u
3
−22u
2
+9a−u+7, u
5
+6u
3
+u+1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u
2
a
10
=
4
9
u
4
+
1
9
u
3
+ ··· +
1
9
u −
7
9
2
9
u
4
+
5
9
u
3
+ ··· +
14
9
u +
1
9
a
8
=
2
9
u
4
−
4
9
u
3
+ ··· −
13
9
u −
8
9
1
3
u
4
−
2
3
u
3
+ ··· +
4
3
u −
1
3
a
2
=
−
1
3
u
4
−
1
3
u
3
+ ··· −
7
3
u +
1
3
−
1
9
u
4
+
2
9
u
3
+ ··· −
7
9
u −
5
9
a
6
=
−u
u
3
+ u
a
5
=
u
2
+ 1
−u
4
− 2u
2
a
1
=
−
5
9
u
4
+
1
9
u
3
+ ··· −
17
9
u +
2
9
5
9
u
4
+
17
9
u
3
+ ··· −
10
9
u −
2
9
a
9
=
2
9
u
4
−
4
9
u
3
+ ··· −
13
9
u −
8
9
2
9
u
4
+
5
9
u
3
+ ··· +
14
9
u +
1
9
(ii) Obstruction class = −1
(iii) Cusp Shapes =
10
3
u
4
−
2
3
u
3
+
61
3
u
2
−
11
3
u +
17
3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
5
− 2u
4
+ 2u
3
+ u − 1
c
2
, c
9
u
5
+ u
4
+ 17u
3
− 4u
2
+ 20u − 8
c
3
, c
4
, c
6
u
5
+ 6u
3
+ u − 1
c
7
, c
10
u
5
− 4u
4
+ u
3
+ 5u
2
+ 6u − 1
c
8
u
5
+ 14u
4
+ 53u
3
+ 21u
2
+ 46u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
5
+ 6y
3
+ y −1
c
2
, c
9
y
5
+ 33y
4
+ 337y
3
+ 680y
2
+ 336y −64
c
3
, c
4
, c
6
y
5
+ 12y
4
+ 38y
3
+ 12y
2
+ y −1
c
7
, c
10
y
5
− 14y
4
+ 53y
3
− 21y
2
+ 46y −1
c
8
y
5
− 90y
4
+ 2313y
3
+ 4407y
2
+ 2074y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.238576 + 0.571771I
a = −1.43645 + 0.65503I
b = 0.029437 + 1.140530I
−1.70245 − 1.37362I −0.55634 + 3.01933I
u = 0.238576 − 0.571771I
a = −1.43645 − 0.65503I
b = 0.029437 − 1.140530I
−1.70245 + 1.37362I −0.55634 − 3.01933I
u = −0.446847
a = −0.331534
b = −0.380649
0.907840 11.5570
u = −0.01515 + 2.41455I
a = 0.102214 − 1.095320I
b = 0.66089 − 3.96349I
16.0529 − 4.0569I 0.27760 + 1.88627I
u = −0.01515 − 2.41455I
a = 0.102214 + 1.095320I
b = 0.66089 + 3.96349I
16.0529 + 4.0569I 0.27760 − 1.88627I
5
II. I
u
2
= hb, −u
2
+ a − u − 2, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u
2
a
10
=
u
2
+ u + 2
0
a
8
=
u
2
+ u + 2
u
a
2
=
1
0
a
6
=
−u
−u
2
− u − 1
a
5
=
u
2
+ 1
−u
2
− u − 1
a
1
=
0
−u
a
9
=
u
2
+ u + 2
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
+ 3u + 3
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
− u
2
+ 1
c
2
, c
9
u
3
c
3
, c
4
u
3
+ u
2
+ 2u + 1
c
5
u
3
+ u
2
− 1
c
6
u
3
− u
2
+ 2u − 1
c
7
(u − 1)
3
c
8
, c
10
(u + 1)
3
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
3
− y
2
+ 2y −1
c
2
, c
9
y
3
c
3
, c
4
, c
6
y
3
+ 3y
2
+ 2y −1
c
7
, c
8
, c
10
(y −1)
3
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = 0.122561 + 0.744862I
b = 0
−4.66906 − 2.82812I 0.69240 + 3.35914I
u = −0.215080 − 1.307140I
a = 0.122561 − 0.744862I
b = 0
−4.66906 + 2.82812I 0.69240 − 3.35914I
u = −0.569840
a = 1.75488
b = 0
−0.531480 1.61520
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 1)(u
5
− 2u
4
+ 2u
3
+ u − 1)
c
2
, c
9
u
3
(u
5
+ u
4
+ 17u
3
− 4u
2
+ 20u − 8)
c
3
, c
4
(u
3
+ u
2
+ 2u + 1)(u
5
+ 6u
3
+ u − 1)
c
5
(u
3
+ u
2
− 1)(u
5
− 2u
4
+ 2u
3
+ u − 1)
c
6
(u
3
− u
2
+ 2u − 1)(u
5
+ 6u
3
+ u − 1)
c
7
(u − 1)
3
(u
5
− 4u
4
+ u
3
+ 5u
2
+ 6u − 1)
c
8
(u + 1)
3
(u
5
+ 14u
4
+ 53u
3
+ 21u
2
+ 46u + 1)
c
10
(u + 1)
3
(u
5
− 4u
4
+ u
3
+ 5u
2
+ 6u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
3
− y
2
+ 2y −1)(y
5
+ 6y
3
+ y −1)
c
2
, c
9
y
3
(y
5
+ 33y
4
+ 337y
3
+ 680y
2
+ 336y −64)
c
3
, c
4
, c
6
(y
3
+ 3y
2
+ 2y −1)(y
5
+ 12y
4
+ 38y
3
+ 12y
2
+ y −1)
c
7
, c
10
(y −1)
3
(y
5
− 14y
4
+ 53y
3
− 21y
2
+ 46y −1)
c
8
(y −1)
3
(y
5
− 90y
4
+ 2313y
3
+ 4407y
2
+ 2074y −1)
11
