10
130
(K10n
20
)
A knot diagram
1
Linearized knot diagam
4 8 5 2 10 9 5 2 6 7
Solving Sequence
5,10 2,6
4 1 3 9 7 8
c
5
c
4
c
1
c
3
c
9
c
6
c
8
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
7
+ u
6
− 4u
5
+ 3u
4
− 4u
3
+ 2u
2
+ b − 1,
− u
10
+ 2u
9
− 8u
8
+ 11u
7
− 20u
6
+ 19u
5
− 17u
4
+ 8u
3
− u
2
+ a − 5u + 1,
u
11
− 2u
10
+ 8u
9
− 12u
8
+ 22u
7
− 24u
6
+ 24u
5
− 15u
4
+ 7u
3
+ 3u
2
− 2u + 1i
I
u
2
= hb + 1, −u
2
+ a − u − 1, u
3
+ u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 14 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
− 4u
5
+ 3u
4
− 4u
3
+ 2u
2
+ b − 1, −u
10
+ 2u
9
+ · · · + a +
1, u
11
− 2u
10
+ · · · − 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
u
10
− 2u
9
+ 8u
8
− 11u
7
+ 20u
6
− 19u
5
+ 17u
4
− 8u
3
+ u
2
+ 5u − 1
u
7
− u
6
+ 4u
5
− 3u
4
+ 4u
3
− 2u
2
+ 1
a
6
=
1
−u
2
a
4
=
−u
10
+ 2u
9
− 7u
8
+ 10u
7
− 16u
6
+ 15u
5
− 13u
4
+ 4u
3
− u
2
− 4u + 1
u
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ u − 1
a
1
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
3
=
−u
10
+ 2u
9
− 6u
8
+ 9u
7
− 11u
6
+ 11u
5
− 6u
4
+ u
2
− 3u
u
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ u − 1
a
9
=
−u
u
3
+ u
a
7
=
u
2
+ 1
−u
4
− 2u
2
a
8
=
−u
4
− u
2
+ 1
−u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −u
10
+ 2u
9
− 7u
8
+ 8u
7
− 12u
6
+ 3u
5
+ 3u
4
− 16u
3
+ 15u
2
− 13u + 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
11
− 4u
10
− u
9
+ 17u
8
+ u
7
− 40u
6
+ 3u
5
+ 37u
4
− 3u
3
− 9u
2
+ 7u − 1
c
2
, c
8
u
11
− u
10
+ ··· − 4u − 8
c
3
u
11
+ 18u
10
+ ··· + 31u + 1
c
5
, c
6
, c
9
u
11
+ 2u
10
+ ··· − 2u − 1
c
7
u
11
+ 12u
9
+ 36u
7
− 2u
6
+ 2u
5
− 13u
4
+ 13u
3
− u
2
− 1
c
10
u
11
− 2u
10
+ ··· − 6u − 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
11
− 18y
10
+ ··· + 31y −1
c
2
, c
8
y
11
+ 21y
10
+ ··· + 336y −64
c
3
y
11
− 46y
10
+ ··· + 863y −1
c
5
, c
6
, c
9
y
11
+ 12y
10
+ ··· − 2y −1
c
7
y
11
+ 24y
10
+ ··· − 2y −1
c
10
y
11
+ 12y
10
+ ··· − 594y −81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.816018 + 0.563764I
a = −0.368670 − 1.053910I
b = −1.86528 + 0.08844I
−12.35850 + 2.70718I 0.47291 − 2.44627I
u = 0.816018 − 0.563764I
a = −0.368670 + 1.053910I
b = −1.86528 − 0.08844I
−12.35850 − 2.70718I 0.47291 + 2.44627I
u = −0.157733 + 1.338590I
a = −0.577850 + 0.189675I
b = −0.283200 + 0.366521I
−3.43504 − 2.25109I 3.70368 + 2.34373I
u = −0.157733 − 1.338590I
a = −0.577850 − 0.189675I
b = −0.283200 − 0.366521I
−3.43504 + 2.25109I 3.70368 − 2.34373I
u = 0.05807 + 1.49843I
a = 1.69315 + 0.17490I
b = 1.26769 − 0.68760I
−8.01785 + 1.82060I −2.54374 − 1.21714I
u = 0.05807 − 1.49843I
a = 1.69315 − 0.17490I
b = 1.26769 + 0.68760I
−8.01785 − 1.82060I −2.54374 + 1.21714I
u = −0.480017
a = −0.562904
b = −0.182568
0.824865 12.3320
u = 0.238107 + 0.385438I
a = 0.41631 + 1.75871I
b = 0.911055 − 0.299346I
−1.69473 + 0.83621I −2.12521 − 2.51411I
u = 0.238107 − 0.385438I
a = 0.41631 − 1.75871I
b = 0.911055 + 0.299346I
−1.69473 − 0.83621I −2.12521 + 2.51411I
u = 0.28555 + 1.56335I
a = −1.88149 − 0.96849I
b = −1.93898 + 0.26128I
−19.3195 + 6.7782I −2.17368 − 2.81310I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.28555 − 1.56335I
a = −1.88149 + 0.96849I
b = −1.93898 − 0.26128I
−19.3195 − 6.7782I −2.17368 + 2.81310I
6
II. I
u
2
= hb + 1, −u
2
+ a − u − 1, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
u
2
+ u + 1
−1
a
6
=
1
−u
2
a
4
=
u
2
+ u + 2
−1
a
1
=
−1
0
a
3
=
u
2
+ u + 1
−1
a
9
=
−u
−u
2
− u − 1
a
7
=
u
2
+ 1
−u
2
− u − 1
a
8
=
−u
−u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
2
+ 4u + 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u − 1)
3
c
2
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
6
u
3
+ u
2
+ 2u + 1
c
7
, c
10
u
3
+ u
2
− 1
c
9
u
3
− u
2
+ 2u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
(y −1)
3
c
2
, c
8
y
3
c
5
, c
6
, c
9
y
3
+ 3y
2
+ 2y −1
c
7
, c
10
y
3
− y
2
+ 2y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −0.877439 + 0.744862I
b = −1.00000
−4.66906 − 2.82812I −1.84740 + 3.54173I
u = −0.215080 − 1.307140I
a = −0.877439 − 0.744862I
b = −1.00000
−4.66906 + 2.82812I −1.84740 − 3.54173I
u = −0.569840
a = 0.754878
b = −1.00000
−0.531480 2.69480
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
3
· (u
11
− 4u
10
− u
9
+ 17u
8
+ u
7
− 40u
6
+ 3u
5
+ 37u
4
− 3u
3
− 9u
2
+ 7u − 1)
c
2
, c
8
u
3
(u
11
− u
10
+ ··· − 4u − 8)
c
3
((u − 1)
3
)(u
11
+ 18u
10
+ ··· + 31u + 1)
c
4
(u + 1)
3
· (u
11
− 4u
10
− u
9
+ 17u
8
+ u
7
− 40u
6
+ 3u
5
+ 37u
4
− 3u
3
− 9u
2
+ 7u − 1)
c
5
, c
6
(u
3
+ u
2
+ 2u + 1)(u
11
+ 2u
10
+ ··· − 2u − 1)
c
7
(u
3
+ u
2
− 1)(u
11
+ 12u
9
+ 36u
7
− 2u
6
+ 2u
5
− 13u
4
+ 13u
3
− u
2
− 1)
c
9
(u
3
− u
2
+ 2u − 1)(u
11
+ 2u
10
+ ··· − 2u − 1)
c
10
(u
3
+ u
2
− 1)(u
11
− 2u
10
+ ··· − 6u − 9)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
3
)(y
11
− 18y
10
+ ··· + 31y −1)
c
2
, c
8
y
3
(y
11
+ 21y
10
+ ··· + 336y −64)
c
3
((y −1)
3
)(y
11
− 46y
10
+ ··· + 863y −1)
c
5
, c
6
, c
9
(y
3
+ 3y
2
+ 2y −1)(y
11
+ 12y
10
+ ··· − 2y −1)
c
7
(y
3
− y
2
+ 2y −1)(y
11
+ 24y
10
+ ··· − 2y −1)
c
10
(y
3
− y
2
+ 2y −1)(y
11
+ 12y
10
+ ··· − 594y −81)
12
