10
56
(K10a
28
)
A knot diagram
1
Linearized knot diagam
8 5 6 9 3 10 1 7 4 2
Solving Sequence
1,8 2,5
3 7 9 4 10 6
c
1
c
2
c
7
c
8
c
4
c
10
c
6
c
3
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
34
+ 4u
33
+ ··· + b − 2, 2u
34
+ 2u
33
+ ··· + a − 2, u
35
+ 2u
34
+ ··· − 2u − 1i
I
u
2
= h−u
2
+ b, a − u, u
3
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 38 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
= h2u
34
+4u
33
+· · ·+b−2, 2u
34
+2u
33
+· · ·+a−2, u
35
+2u
34
+· · ·−2u−1i
(i) Arc colorings
a
1
=
1
0
a
8
=
0
u
a
2
=
1
u
2
a
5
=
−2u
34
− 2u
33
+ ··· + 6u + 2
−2u
34
− 4u
33
+ ··· + 2u + 2
a
3
=
u
34
+ u
33
+ ··· − 4u − 1
u
34
+ 2u
33
+ ··· + 9u
2
− 1
a
7
=
u
u
a
9
=
−u
3
−u
3
+ u
a
4
=
u
32
− 5u
30
+ ··· + 4u + 1
−u
33
+ 5u
31
+ ··· − 4u
2
− u
a
10
=
−u
2
+ 1
−u
4
a
6
=
−u
7
+ 2u
5
− 2u
3
+ 2u
−u
9
+ u
7
− u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 7u
34
+ 10u
33
− 33u
32
− 62u
31
+ 104u
30
+ 233u
29
− 217u
28
−
640u
27
+ 303u
26
+ 1349u
25
−205u
24
−2310u
23
−276u
22
+ 3206u
21
+ 1184u
20
−3622u
19
−
2289u
18
+ 3265u
17
+ 3165u
16
− 2158u
15
− 3352u
14
+ 824u
13
+ 2806u
12
+ 244u
11
−
1824u
10
− 716u
9
+ 828u
8
+ 636u
7
− 230u
6
− 376u
5
− 32u
4
+ 126u
3
+ 61u
2
− 5u − 13
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
35
+ 2u
34
+ ··· − 2u − 1
c
2
, c
3
, c
5
u
35
− 4u
34
+ ··· + 3u − 1
c
4
, c
9
u
35
− u
34
+ ··· − 28u − 8
c
6
u
35
− 2u
34
+ ··· + 36u − 36
c
8
, c
10
u
35
+ 12u
34
+ ··· + 10u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
35
− 12y
34
+ ··· + 10y −1
c
2
, c
3
, c
5
y
35
− 34y
34
+ ··· + 19y −1
c
4
, c
9
y
35
+ 21y
34
+ ··· + 16y −64
c
6
y
35
− 12y
34
+ ··· + 22392y −1296
c
8
, c
10
y
35
+ 24y
34
+ ··· + 10y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.664256 + 0.761558I
a = −0.45768 − 1.47251I
b = −1.76114 − 0.11415I
1.24148 − 2.67684I −4.78426 + 2.93641I
u = −0.664256 − 0.761558I
a = −0.45768 + 1.47251I
b = −1.76114 + 0.11415I
1.24148 + 2.67684I −4.78426 − 2.93641I
u = −0.741471 + 0.622830I
a = −0.57819 + 1.67504I
b = 0.590055 + 1.240710I
−0.35079 + 1.76625I −8.73044 − 2.55261I
u = −0.741471 − 0.622830I
a = −0.57819 − 1.67504I
b = 0.590055 − 1.240710I
−0.35079 − 1.76625I −8.73044 + 2.55261I
u = 1.037620 + 0.057613I
a = 0.554534 − 0.308977I
b = 0.096321 − 0.988163I
−4.46867 − 2.44036I −13.20394 + 3.90896I
u = 1.037620 − 0.057613I
a = 0.554534 + 0.308977I
b = 0.096321 + 0.988163I
−4.46867 + 2.44036I −13.20394 − 3.90896I
u = −1.04680
a = 2.40428
b = 1.22975
−6.56245 −13.9210
u = 0.647381 + 0.692758I
a = −0.044489 + 0.551561I
b = −1.82365 + 0.07795I
−1.45204 + 0.58793I −6.80279 + 0.37603I
u = 0.647381 − 0.692758I
a = −0.044489 − 0.551561I
b = −1.82365 − 0.07795I
−1.45204 − 0.58793I −6.80279 − 0.37603I
u = −0.636751 + 0.841462I
a = 0.997900 + 0.837792I
b = 2.05544 − 1.02610I
−4.54695 − 6.58963I −8.16646 + 3.21535I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.636751 − 0.841462I
a = 0.997900 − 0.837792I
b = 2.05544 + 1.02610I
−4.54695 + 6.58963I −8.16646 − 3.21535I
u = 0.799224 + 0.732897I
a = −0.137052 − 0.887085I
b = 1.192050 − 0.720803I
3.23509 − 1.72545I −0.63608 + 2.52233I
u = 0.799224 − 0.732897I
a = −0.137052 + 0.887085I
b = 1.192050 + 0.720803I
3.23509 + 1.72545I −0.63608 − 2.52233I
u = 1.119540 + 0.118261I
a = −1.43525 + 0.93911I
b = −0.508472 + 1.022410I
−11.18060 − 5.92010I −14.7078 + 4.1258I
u = 1.119540 − 0.118261I
a = −1.43525 − 0.93911I
b = −0.508472 − 1.022410I
−11.18060 + 5.92010I −14.7078 − 4.1258I
u = −0.967598 + 0.636531I
a = −0.986041 − 0.154167I
b = −1.50085 + 0.53336I
−1.09066 + 3.19486I −9.71319 − 2.77080I
u = −0.967598 − 0.636531I
a = −0.986041 + 0.154167I
b = −1.50085 − 0.53336I
−1.09066 − 3.19486I −9.71319 + 2.77080I
u = 0.923611 + 0.710370I
a = −0.706663 + 0.972785I
b = −1.261460 − 0.179818I
2.85420 − 3.77887I −1.29814 + 3.89618I
u = 0.923611 − 0.710370I
a = −0.706663 − 0.972785I
b = −1.261460 + 0.179818I
2.85420 + 3.77887I −1.29814 − 3.89618I
u = −1.044390 + 0.520208I
a = −0.308880 − 1.070380I
b = 0.105718 − 0.418991I
−8.72011 + 1.04091I −12.84142 − 2.04561I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.044390 − 0.520208I
a = −0.308880 + 1.070380I
b = 0.105718 + 0.418991I
−8.72011 − 1.04091I −12.84142 + 2.04561I
u = −0.832533
a = −0.863173
b = −0.390504
−1.36456 −6.49720
u = 0.999651 + 0.662016I
a = 0.70686 − 2.15648I
b = 1.84700 − 0.41941I
−2.49390 − 5.84473I −8.96835 + 4.95079I
u = 0.999651 − 0.662016I
a = 0.70686 + 2.15648I
b = 1.84700 + 0.41941I
−2.49390 + 5.84473I −8.96835 − 4.95079I
u = 0.886910 + 0.817524I
a = 1.29729 + 0.86025I
b = 0.19926 + 2.00598I
0.09589 − 3.04539I −10.49856 + 3.07346I
u = 0.886910 − 0.817524I
a = 1.29729 − 0.86025I
b = 0.19926 − 2.00598I
0.09589 + 3.04539I −10.49856 − 3.07346I
u = −0.280203 + 0.733036I
a = 0.964828 − 0.842495I
b = 0.673575 − 0.185802I
−6.48734 + 3.48149I −9.01514 − 3.12997I
u = −0.280203 − 0.733036I
a = 0.964828 + 0.842495I
b = 0.673575 + 0.185802I
−6.48734 − 3.48149I −9.01514 + 3.12997I
u = −1.007860 + 0.690657I
a = 0.76872 + 1.59182I
b = 2.25197 + 0.62559I
0.21056 + 8.20034I −6.93623 − 7.67757I
u = −1.007860 − 0.690657I
a = 0.76872 − 1.59182I
b = 2.25197 − 0.62559I
0.21056 − 8.20034I −6.93623 + 7.67757I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.045240 + 0.713362I
a = −0.03530 − 2.41846I
b = −2.14450 − 1.67720I
−5.78918 + 12.39880I −9.85333 − 7.70880I
u = −1.045240 − 0.713362I
a = −0.03530 + 2.41846I
b = −2.14450 + 1.67720I
−5.78918 − 12.39880I −9.85333 + 7.70880I
u = −0.282825 + 0.410007I
a = −0.74438 + 1.30353I
b = −0.006557 + 0.477496I
−0.429568 + 1.170440I −5.16678 − 5.64189I
u = −0.282825 − 0.410007I
a = −0.74438 − 1.30353I
b = −0.006557 − 0.477496I
−0.429568 − 1.170440I −5.16678 + 5.64189I
u = 0.392648
a = 1.74649
b = −0.848760
−2.15415 −1.93570
8
II. I
u
2
= h−u
2
+ b, a − u, u
3
− u
2
+ 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
0
u
a
2
=
1
u
2
a
5
=
u
u
2
a
3
=
u + 1
2u
2
a
7
=
u
u
a
9
=
−u
2
+ 1
−u
2
+ u + 1
a
4
=
u
u
2
a
10
=
−u
2
+ 1
−u
2
+ u + 1
a
6
=
−1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
2
+ 7u − 10
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
− u
2
+ 1
c
2
, c
3
(u − 1)
3
c
4
, c
9
u
3
c
5
(u + 1)
3
c
6
, c
10
u
3
− u
2
+ 2u − 1
c
7
u
3
+ u
2
− 1
c
8
u
3
+ u
2
+ 2u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
3
− y
2
+ 2y −1
c
2
, c
3
, c
5
(y −1)
3
c
4
, c
9
y
3
c
6
, c
8
, c
10
y
3
+ 3y
2
+ 2y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.877439 + 0.744862I
b = 0.215080 + 1.307140I
1.37919 − 2.82812I −4.28809 + 2.59975I
u = 0.877439 − 0.744862I
a = 0.877439 − 0.744862I
b = 0.215080 − 1.307140I
1.37919 + 2.82812I −4.28809 − 2.59975I
u = −0.754878
a = −0.754878
b = 0.569840
−2.75839 −16.4240
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 1)(u
35
+ 2u
34
+ ··· − 2u − 1)
c
2
, c
3
((u − 1)
3
)(u
35
− 4u
34
+ ··· + 3u − 1)
c
4
, c
9
u
3
(u
35
− u
34
+ ··· − 28u − 8)
c
5
((u + 1)
3
)(u
35
− 4u
34
+ ··· + 3u − 1)
c
6
(u
3
− u
2
+ 2u − 1)(u
35
− 2u
34
+ ··· + 36u − 36)
c
7
(u
3
+ u
2
− 1)(u
35
+ 2u
34
+ ··· − 2u − 1)
c
8
(u
3
+ u
2
+ 2u + 1)(u
35
+ 12u
34
+ ··· + 10u + 1)
c
10
(u
3
− u
2
+ 2u − 1)(u
35
+ 12u
34
+ ··· + 10u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
3
− y
2
+ 2y −1)(y
35
− 12y
34
+ ··· + 10y −1)
c
2
, c
3
, c
5
((y −1)
3
)(y
35
− 34y
34
+ ··· + 19y −1)
c
4
, c
9
y
3
(y
35
+ 21y
34
+ ··· + 16y −64)
c
6
(y
3
+ 3y
2
+ 2y −1)(y
35
− 12y
34
+ ··· + 22392y −1296)
c
8
, c
10
(y
3
+ 3y
2
+ 2y −1)(y
35
+ 24y
34
+ ··· + 10y −1)
14
