10
150
(K10n
9
)
A knot diagram
1
Linearized knot diagam
9 4 1 7 4 10 5 2 7 3
Solving Sequence
1,4
3 2
7,10
6 5 8 9
c
3
c
2
c
10
c
6
c
5
c
7
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h5218u
16
− 13845u
15
+ ··· + 24209b − 23873, 14691u
16
− 23006u
15
+ ··· + 24209a − 62170,
u
17
− 2u
16
+ ··· − 3u − 1i
I
u
2
= hb − 1, −u
2
+ a + u − 1, u
3
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 20 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
= h5218u
16
− 13845u
15
+ · · · + 24209b − 23873, 14691u
16
− 23006u
15
+
· · · + 24209a − 62170, u
17
− 2u
16
+ · · · − 3u − 1i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
3
=
1
−u
2
a
2
=
−u
2
+ 1
−u
2
a
7
=
−0.606840u
16
+ 0.950308u
15
+ ··· − 2.86026u + 2.56805
−0.215540u
16
+ 0.571895u
15
+ ··· − 0.628196u + 0.986121
a
10
=
u
−u
3
+ u
a
6
=
−0.650337u
16
+ 1.05824u
15
+ ··· − 2.69995u + 2.65839
−0.137841u
16
+ 0.712421u
15
+ ··· − 0.487215u + 1.05552
a
5
=
−0.788178u
16
+ 1.77066u
15
+ ··· − 3.18716u + 3.71391
−0.137841u
16
+ 0.712421u
15
+ ··· − 0.487215u + 1.05552
a
8
=
0.421785u
16
− 1.78739u
15
+ ··· − 0.282003u − 1.72217
−0.844603u
16
+ 0.281053u
15
+ ··· − 2.71804u − 0.861209
a
9
=
−0.396877u
16
+ 1.39225u
15
+ ··· − 0.955099u + 1.13198
0.271015u
16
+ 0.327482u
15
+ ··· + 1.53959u + 0.667892
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
76049
24209
u
16
+
104431
24209
u
15
+ ··· −
330360
24209
u −
115800
24209
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
17
− 2u
16
+ ··· + u − 1
c
2
u
17
+ 12u
16
+ ··· + 7u + 1
c
3
, c
10
u
17
− 2u
16
+ ··· − 3u − 1
c
4
, c
7
u
17
− 4u
16
+ ··· + 16u − 1
c
5
u
17
+ 22u
16
+ ··· + 256u + 1
c
6
, c
9
u
17
+ 3u
16
+ ··· + 20u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
17
+ 18y
15
+ ··· + 7y −1
c
2
y
17
− 12y
16
+ ··· + 155y −1
c
3
, c
10
y
17
− 12y
16
+ ··· + 7y −1
c
4
, c
7
y
17
− 22y
16
+ ··· + 256y −1
c
5
y
17
− 50y
16
+ ··· + 60796y −1
c
6
, c
9
y
17
+ 21y
16
+ ··· + 976y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.876782 + 0.644726I
a = 0.092257 − 0.124101I
b = −0.568271 + 0.184291I
2.13008 − 2.53959I 0.76560 + 1.98769I
u = 0.876782 − 0.644726I
a = 0.092257 + 0.124101I
b = −0.568271 − 0.184291I
2.13008 + 2.53959I 0.76560 − 1.98769I
u = −1.089060 + 0.132960I
a = −0.02578 − 2.03485I
b = 0.834229 − 0.235726I
−3.18058 + 0.67411I −10.63151 + 5.49435I
u = −1.089060 − 0.132960I
a = −0.02578 + 2.03485I
b = 0.834229 + 0.235726I
−3.18058 − 0.67411I −10.63151 − 5.49435I
u = −0.026050 + 1.128120I
a = −1.354380 + 0.277932I
b = −1.63657 + 0.18009I
−8.13487 + 4.20505I −7.98094 − 2.47792I
u = −0.026050 − 1.128120I
a = −1.354380 − 0.277932I
b = −1.63657 − 0.18009I
−8.13487 − 4.20505I −7.98094 + 2.47792I
u = −0.819663
a = 0.742247
b = −0.0636841
−1.19406 −8.42610
u = 1.229710 + 0.222583I
a = −0.189457 + 1.004150I
b = 0.83094 + 1.19370I
−4.39628 − 4.11745I −11.29745 + 5.99012I
u = 1.229710 − 0.222583I
a = −0.189457 − 1.004150I
b = 0.83094 − 1.19370I
−4.39628 + 4.11745I −11.29745 − 5.99012I
u = 1.26347
a = −0.266454
b = 1.87117
−6.78936 −15.0240
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.39748 + 0.52974I
a = −0.069718 − 1.260110I
b = −1.72864 − 0.39180I
−12.6337 − 10.0814I −9.96961 + 5.13034I
u = 1.39748 − 0.52974I
a = −0.069718 + 1.260110I
b = −1.72864 + 0.39180I
−12.6337 + 10.0814I −9.96961 − 5.13034I
u = −1.39973 + 0.55866I
a = −0.294421 + 0.977752I
b = −1.71162 + 0.05597I
−12.44690 + 1.83083I −10.41430 − 0.85064I
u = −1.39973 − 0.55866I
a = −0.294421 − 0.977752I
b = −1.71162 − 0.05597I
−12.44690 − 1.83083I −10.41430 + 0.85064I
u = −0.057966 + 0.464686I
a = 1.90019 − 0.95414I
b = 0.504075 − 0.513259I
−0.61170 + 1.48793I −4.64409 − 4.66231I
u = −0.057966 − 0.464686I
a = 1.90019 + 0.95414I
b = 0.504075 + 0.513259I
−0.61170 − 1.48793I −4.64409 + 4.66231I
u = −0.306131
a = 3.40681
b = 1.14424
−2.29521 −1.20570
6
II. I
u
2
= hb − 1, −u
2
+ a + u − 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
3
=
1
−u
2
a
2
=
−u
2
+ 1
−u
2
a
7
=
u
2
− u + 1
1
a
10
=
u
−u
2
+ u + 1
a
6
=
u
2
− u + 1
1
a
5
=
u
2
− u + 2
1
a
8
=
−1
0
a
9
=
u
−u
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
+ 8u − 16
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ u
2
+ 2u + 1
c
2
, c
8
u
3
− u
2
+ 2u − 1
c
3
u
3
− u
2
+ 1
c
4
(u − 1)
3
c
5
, c
7
(u + 1)
3
c
6
, c
9
u
3
c
10
u
3
+ u
2
− 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
8
y
3
+ 3y
2
+ 2y −1
c
3
, c
10
y
3
− y
2
+ 2y −1
c
4
, c
5
, c
7
(y −1)
3
c
6
, c
9
y
3
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.337641 + 0.562280I
b = 1.00000
1.37919 − 2.82812I −9.19557 + 4.65175I
u = 0.877439 − 0.744862I
a = 0.337641 − 0.562280I
b = 1.00000
1.37919 + 2.82812I −9.19557 − 4.65175I
u = −0.754878
a = 2.32472
b = 1.00000
−2.75839 −22.6090
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
+ u
2
+ 2u + 1)(u
17
− 2u
16
+ ··· + u − 1)
c
2
(u
3
− u
2
+ 2u − 1)(u
17
+ 12u
16
+ ··· + 7u + 1)
c
3
(u
3
− u
2
+ 1)(u
17
− 2u
16
+ ··· − 3u − 1)
c
4
((u − 1)
3
)(u
17
− 4u
16
+ ··· + 16u − 1)
c
5
((u + 1)
3
)(u
17
+ 22u
16
+ ··· + 256u + 1)
c
6
, c
9
u
3
(u
17
+ 3u
16
+ ··· + 20u + 8)
c
7
((u + 1)
3
)(u
17
− 4u
16
+ ··· + 16u − 1)
c
8
(u
3
− u
2
+ 2u − 1)(u
17
− 2u
16
+ ··· + u − 1)
c
10
(u
3
+ u
2
− 1)(u
17
− 2u
16
+ ··· − 3u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y
3
+ 3y
2
+ 2y −1)(y
17
+ 18y
15
+ ··· + 7y −1)
c
2
(y
3
+ 3y
2
+ 2y −1)(y
17
− 12y
16
+ ··· + 155y −1)
c
3
, c
10
(y
3
− y
2
+ 2y −1)(y
17
− 12y
16
+ ··· + 7y −1)
c
4
, c
7
((y −1)
3
)(y
17
− 22y
16
+ ··· + 256y −1)
c
5
((y −1)
3
)(y
17
− 50y
16
+ ··· + 60796y −1)
c
6
, c
9
y
3
(y
17
+ 21y
16
+ ··· + 976y −64)
12
