10
161
(K10n
31
)
A knot diagram
1
Linearized knot diagam
7 9 7 2 9 1 3 6 4 2
Solving Sequence
2,9 3,7
4 1 6 5 8 10
c
2
c
3
c
1
c
6
c
5
c
8
c
10
c
4
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−29u
5
+ 62u
4
− 233u
3
− 17u
2
+ 73b − 178u − 27, −15u
5
+ 22u
4
− 118u
3
− 39u
2
+ 73a − 228u − 19,
u
6
− 2u
5
+ 8u
4
+ u
3
+ 7u
2
− u − 1i
I
u
2
= hu
2
+ b + u, u
3
+ u
2
+ a − u, u
4
+ u
3
− 1i
* 2 irreducible components of dim
C
= 0, with total 10 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−29u
5
+ 62u
4
+ · · · + 73b − 27, −15u
5
+ 22u
4
+ · · · + 73a − 19, u
6
−
2u
5
+ 8u
4
+ u
3
+ 7u
2
− u − 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
7
=
0.205479u
5
− 0.301370u
4
+ ··· + 3.12329u + 0.260274
0.397260u
5
− 0.849315u
4
+ ··· + 2.43836u + 0.369863
a
4
=
0.260274u
5
− 0.315068u
4
+ ··· + 2.35616u + 1.86301
0.260274u
5
− 0.315068u
4
+ ··· + 2.35616u + 0.863014
a
1
=
−0.205479u
5
+ 0.301370u
4
+ ··· − 3.12329u − 0.260274
−0.452055u
5
+ 0.863014u
4
+ ··· − 2.67123u − 0.972603
a
6
=
−1
−0.260274u
5
+ 0.315068u
4
+ ··· − 2.35616u − 0.863014
a
5
=
−1
−0.260274u
5
+ 0.315068u
4
+ ··· − 2.35616u − 0.863014
a
8
=
u
0.205479u
5
− 0.301370u
4
+ ··· + 2.12329u + 0.260274
a
10
=
−0.657534u
5
+ 1.16438u
4
+ ··· − 5.79452u − 1.23288
−0.452055u
5
+ 0.863014u
4
+ ··· − 2.67123u − 0.972603
(ii) Obstruction class = −1
(iii) Cusp Shapes =
145
73
u
5
−
310
73
u
4
+
1165
73
u
3
+
85
73
u
2
+
744
73
u −
887
73
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
6
− 6u
5
+ 16u
4
− 21u
3
+ 11u
2
+ 2u − 4
c
2
, c
5
, c
8
u
6
− 2u
5
+ 8u
4
+ u
3
+ 7u
2
− u − 1
c
3
, c
7
, c
9
u
6
+ u
5
+ 9u
4
− 11u
3
− 4u
2
− 2u − 1
c
4
u
6
− 3u
5
− 3u
4
+ 15u
3
− 10u
2
+ 1
c
10
u
6
+ 4u
5
+ 26u
4
+ 73u
3
+ 77u
2
+ 92u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
6
− 4y
5
+ 26y
4
− 73y
3
+ 77y
2
− 92y + 16
c
2
, c
5
, c
8
y
6
+ 12y
5
+ 82y
4
+ 105y
3
+ 35y
2
− 15y + 1
c
3
, c
7
, c
9
y
6
+ 17y
5
+ 95y
4
− 191y
3
− 46y
2
+ 4y + 1
c
4
y
6
− 15y
5
+ 79y
4
− 163y
3
+ 94y
2
− 20y + 1
c
10
y
6
+ 36y
5
+ 246y
4
− 2029y
3
− 6671y
2
− 6000y + 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.244201 + 0.971888I
a = −0.24405 + 1.39509I
b = 0.878332 − 0.695514I
2.08576 + 2.67800I −9.11994 − 5.42135I
u = −0.244201 − 0.971888I
a = −0.24405 − 1.39509I
b = 0.878332 + 0.695514I
2.08576 − 2.67800I −9.11994 + 5.42135I
u = 0.403945
a = 1.70981
b = 1.58486
−7.78420 −6.88360
u = −0.304480
a = −0.689933
b = −0.449415
−0.637429 −15.6380
u = 1.19447 + 2.58259I
a = 0.234107 − 0.606474I
b = 1.55395 + 1.43504I
13.6396 − 5.6388I −8.61921 + 2.01004I
u = 1.19447 − 2.58259I
a = 0.234107 + 0.606474I
b = 1.55395 − 1.43504I
13.6396 + 5.6388I −8.61921 − 2.01004I
5
II. I
u
2
= hu
2
+ b + u, u
3
+ u
2
+ a − u, u
4
+ u
3
− 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
7
=
−u
3
− u
2
+ u
−u
2
− u
a
4
=
u
2
+ u
u
2
+ u + 1
a
1
=
u
3
+ u
2
− u
−u
3
− u
2
− 1
a
6
=
−1
u + 1
a
5
=
−1
u
2
+ u + 1
a
8
=
u
−u
2
a
10
=
−u − 1
−u
3
− u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
3
− 7u
2
− 7u − 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
− 2u
2
+ u + 1
c
2
, c
8
u
4
+ u
3
− 1
c
3
, c
9
u
4
+ u − 1
c
4
u
4
+ 4u
3
+ 4u
2
+ u + 1
c
5
u
4
− u
3
− 1
c
6
u
4
− 2u
2
− u + 1
c
7
u
4
− u − 1
c
10
u
4
− 4u
3
+ 6u
2
− 5u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
4
− 4y
3
+ 6y
2
− 5y + 1
c
2
, c
5
, c
8
y
4
− y
3
− 2y
2
+ 1
c
3
, c
7
, c
9
y
4
− 2y
2
− y + 1
c
4
y
4
− 8y
3
+ 10y
2
+ 7y + 1
c
10
y
4
− 4y
3
− 2y
2
− 13y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.219447 + 0.914474I
a = 0.02868 + 1.94846I
b = 1.007550 − 0.513116I
3.04135 + 1.96274I −4.02709 − 2.32656I
u = −0.219447 − 0.914474I
a = 0.02868 − 1.94846I
b = 1.007550 + 0.513116I
3.04135 − 1.96274I −4.02709 + 2.32656I
u = 0.819173
a = −0.401572
b = −1.49022
−8.36260 −21.5310
u = −1.38028
a = −0.655786
b = −0.524889
−4.29983 −8.41490
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
4
− 2u
2
+ u + 1)(u
6
− 6u
5
+ 16u
4
− 21u
3
+ 11u
2
+ 2u − 4)
c
2
, c
8
(u
4
+ u
3
− 1)(u
6
− 2u
5
+ 8u
4
+ u
3
+ 7u
2
− u − 1)
c
3
, c
9
(u
4
+ u − 1)(u
6
+ u
5
+ 9u
4
− 11u
3
− 4u
2
− 2u − 1)
c
4
(u
4
+ 4u
3
+ 4u
2
+ u + 1)(u
6
− 3u
5
− 3u
4
+ 15u
3
− 10u
2
+ 1)
c
5
(u
4
− u
3
− 1)(u
6
− 2u
5
+ 8u
4
+ u
3
+ 7u
2
− u − 1)
c
6
(u
4
− 2u
2
− u + 1)(u
6
− 6u
5
+ 16u
4
− 21u
3
+ 11u
2
+ 2u − 4)
c
7
(u
4
− u − 1)(u
6
+ u
5
+ 9u
4
− 11u
3
− 4u
2
− 2u − 1)
c
10
(u
4
− 4u
3
+ 6u
2
− 5u + 1)(u
6
+ 4u
5
+ ··· + 92u + 16)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
4
− 4y
3
+ 6y
2
− 5y + 1)(y
6
− 4y
5
+ ··· − 92y + 16)
c
2
, c
5
, c
8
(y
4
− y
3
− 2y
2
+ 1)(y
6
+ 12y
5
+ 82y
4
+ 105y
3
+ 35y
2
− 15y + 1)
c
3
, c
7
, c
9
(y
4
− 2y
2
− y + 1)(y
6
+ 17y
5
+ 95y
4
− 191y
3
− 46y
2
+ 4y + 1)
c
4
(y
4
− 8y
3
+ 10y
2
+ 7y + 1)(y
6
− 15y
5
+ ··· − 20y + 1)
c
10
(y
4
− 4y
3
− 2y
2
− 13y + 1)
· (y
6
+ 36y
5
+ 246y
4
− 2029y
3
− 6671y
2
− 6000y + 256)
11
