11a
240
(K11a
240
)
A knot diagram
1
Linearized knot diagam
7 1 10 11 8 9 2 6 3 4 5
Solving Sequence
4,10
11 5 1 3 2
6,9
7 8
c
10
c
4
c
11
c
3
c
2
c
9
c
6
c
8
c
1
, c
5
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
31
+ u
30
+ ··· + b − u, −u
31
+ u
30
+ ··· + a + 1, u
32
− 2u
31
+ ··· − 4u + 1i
I
u
2
= hb + u, a + u + 1, u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 34 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
31
+u
30
+· · ·+b−u, −u
31
+u
30
+· · ·+a+1, u
32
−2u
31
+· · ·−4u+1i
(i) Arc colorings
a
4
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
5
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
u
u
a
2
=
−u
7
+ 4u
5
− 4u
3
+ 2u
−u
9
+ 5u
7
− 7u
5
+ 2u
3
+ u
a
6
=
u
31
− u
30
+ ··· + 3u − 1
u
31
− u
30
+ ··· − u
2
+ u
a
9
=
−u
2
+ 1
−u
2
a
7
=
2u
31
− u
30
+ ··· + 5u − 2
3u
31
− 2u
30
+ ··· + 4u − 1
a
8
=
u
30
− u
29
+ ··· − u + 1
u
31
− 20u
29
+ ··· + 3u − 1
a
8
=
u
30
− u
29
+ ··· − u + 1
u
31
− 20u
29
+ ··· + 3u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
31
− 5u
30
− 77u
29
+ 93u
28
+ 654u
27
− 766u
26
− 3221u
25
+
3695u
24
+ 10154u
23
− 11648u
22
− 21262u
21
+ 25364u
20
+ 29394u
19
− 39216u
18
−
24813u
17
+ 43171u
16
+ 8234u
15
− 32696u
14
+ 6950u
13
+ 15056u
12
− 10454u
11
−
2100u
10
+ 5696u
9
− 1812u
8
− 1168u
7
+ 978u
6
− 182u
5
− 72u
4
+ 86u
3
− 38u
2
+ 13u − 19
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
32
+ u
31
+ ··· − 12u − 4
c
2
u
32
+ 15u
31
+ ··· + 152u + 16
c
3
, c
4
, c
9
c
10
, c
11
u
32
+ 2u
31
+ ··· + 4u + 1
c
5
, c
6
, c
8
u
32
− 3u
31
+ ··· − 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
32
− 15y
31
+ ··· − 152y + 16
c
2
y
32
+ y
31
+ ··· − 2848y + 256
c
3
, c
4
, c
9
c
10
, c
11
y
32
− 42y
31
+ ··· − 4y + 1
c
5
, c
6
, c
8
y
32
− 29y
31
+ ··· − 7y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.932935 + 0.300495I
a = 0.221529 − 0.245031I
b = −0.566090 − 0.639414I
−2.01989 + 4.86523I −15.1954 − 6.8122I
u = −0.932935 − 0.300495I
a = 0.221529 + 0.245031I
b = −0.566090 + 0.639414I
−2.01989 − 4.86523I −15.1954 + 6.8122I
u = 0.946587 + 0.231196I
a = 0.96950 − 1.13998I
b = −1.46374 + 0.52554I
−4.86072 − 2.90543I −17.9793 + 3.5680I
u = 0.946587 − 0.231196I
a = 0.96950 + 1.13998I
b = −1.46374 − 0.52554I
−4.86072 + 2.90543I −17.9793 − 3.5680I
u = −0.994935 + 0.377573I
a = −0.478871 − 1.164710I
b = 1.158740 + 0.411567I
−7.38074 + 8.76774I −18.7396 − 7.0546I
u = −0.994935 − 0.377573I
a = −0.478871 + 1.164710I
b = 1.158740 − 0.411567I
−7.38074 − 8.76774I −18.7396 + 7.0546I
u = −0.910087 + 0.140122I
a = 0.945898 + 0.573430I
b = 0.426810 + 0.536403I
−3.85845 + 0.52237I −20.0729 − 1.6653I
u = −0.910087 − 0.140122I
a = 0.945898 − 0.573430I
b = 0.426810 − 0.536403I
−3.85845 − 0.52237I −20.0729 + 1.6653I
u = −1.21444
a = −0.504163
b = 1.45498
−11.2682 −22.0990
u = 0.588921 + 0.485955I
a = −1.26657 + 1.04213I
b = −0.886632 + 0.309455I
−4.95979 + 1.69559I −17.9188 + 0.0178I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.588921 − 0.485955I
a = −1.26657 − 1.04213I
b = −0.886632 − 0.309455I
−4.95979 − 1.69559I −17.9188 − 0.0178I
u = 0.718238 + 0.225952I
a = −0.459114 − 0.005471I
b = 0.496105 − 0.264918I
−0.558920 − 0.474938I −11.62959 + 1.27773I
u = 0.718238 − 0.225952I
a = −0.459114 + 0.005471I
b = 0.496105 + 0.264918I
−0.558920 + 0.474938I −11.62959 − 1.27773I
u = 0.187060 + 0.621355I
a = −0.85945 + 1.47486I
b = −1.011920 + 0.169007I
−3.74023 − 5.38912I −14.5723 + 5.7053I
u = 0.187060 − 0.621355I
a = −0.85945 − 1.47486I
b = −1.011920 − 0.169007I
−3.74023 + 5.38912I −14.5723 − 5.7053I
u = 0.109732 + 0.502858I
a = −0.133670 − 0.948522I
b = 0.000513 + 0.345169I
1.17199 − 2.12258I −8.07273 + 5.17972I
u = 0.109732 − 0.502858I
a = −0.133670 + 0.948522I
b = 0.000513 − 0.345169I
1.17199 + 2.12258I −8.07273 − 5.17972I
u = −1.53142
a = 0.299106
b = 1.47350
−11.6098 −22.5450
u = −0.130280 + 0.363295I
a = 1.12950 + 2.35629I
b = 0.957149 + 0.120157I
−1.55433 + 0.80952I −9.54426 − 1.40879I
u = −0.130280 − 0.363295I
a = 1.12950 − 2.35629I
b = 0.957149 − 0.120157I
−1.55433 − 0.80952I −9.54426 + 1.40879I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.65686 + 0.03953I
a = 1.168620 + 0.485339I
b = 1.98443 + 0.52843I
−8.99808 + 1.32195I 0
u = −1.65686 − 0.03953I
a = 1.168620 − 0.485339I
b = 1.98443 − 0.52843I
−8.99808 − 1.32195I 0
u = 0.322365
a = −0.569030
b = 0.332706
−0.607216 −16.6590
u = 1.69979 + 0.04032I
a = −0.740305 − 0.450279I
b = −1.60526 − 0.31431I
−13.16730 − 1.26120I 0
u = 1.69979 − 0.04032I
a = −0.740305 + 0.450279I
b = −1.60526 + 0.31431I
−13.16730 + 1.26120I 0
u = 1.70039 + 0.07638I
a = −1.024980 + 0.658959I
b = −1.73315 + 0.64681I
−11.32620 − 6.33717I 0
u = 1.70039 − 0.07638I
a = −1.024980 − 0.658959I
b = −1.73315 − 0.64681I
−11.32620 + 6.33717I 0
u = −1.70563 + 0.05938I
a = −3.73701 − 1.61459I
b = −6.32266 − 2.62594I
−14.2827 + 4.0537I 0
u = −1.70563 − 0.05938I
a = −3.73701 + 1.61459I
b = −6.32266 + 2.62594I
−14.2827 − 4.0537I 0
u = 1.71578 + 0.10104I
a = 2.85853 − 1.60940I
b = 4.90243 − 2.63065I
−16.9357 − 10.6981I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.71578 − 0.10104I
a = 2.85853 + 1.60940I
b = 4.90243 + 2.63065I
−16.9357 + 10.6981I 0
u = 1.75198
a = 3.58689
b = 6.06538
17.6150 0
8
II. I
u
2
= hb + u, a + u + 1, u
2
+ u − 1i
(i) Arc colorings
a
4
=
0
u
a
10
=
1
0
a
11
=
1
−u + 1
a
5
=
−u
−u + 1
a
1
=
u
u
a
3
=
u
u
a
2
=
u
u
a
6
=
−u − 1
−u
a
9
=
u
u − 1
a
7
=
−1
−1
a
8
=
−1
−1
a
8
=
−1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −15
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u
2
c
3
, c
4
u
2
− u − 1
c
5
, c
6
(u − 1)
2
c
8
(u + 1)
2
c
9
, c
10
, c
11
u
2
+ u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
y
2
c
3
, c
4
, c
9
c
10
, c
11
y
2
− 3y + 1
c
5
, c
6
, c
8
(y − 1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.61803
b = −0.618034
−2.63189 −15.0000
u = −1.61803
a = 0.618034
b = 1.61803
−10.5276 −15.0000
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
u
2
(u
32
+ u
31
+ ··· − 12u − 4)
c
2
u
2
(u
32
+ 15u
31
+ ··· + 152u + 16)
c
3
, c
4
(u
2
− u − 1)(u
32
+ 2u
31
+ ··· + 4u + 1)
c
5
, c
6
((u − 1)
2
)(u
32
− 3u
31
+ ··· − 5u − 1)
c
8
((u + 1)
2
)(u
32
− 3u
31
+ ··· − 5u − 1)
c
9
, c
10
, c
11
(u
2
+ u − 1)(u
32
+ 2u
31
+ ··· + 4u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
2
(y
32
− 15y
31
+ ··· − 152y + 16)
c
2
y
2
(y
32
+ y
31
+ ··· − 2848y + 256)
c
3
, c
4
, c
9
c
10
, c
11
(y
2
− 3y + 1)(y
32
− 42y
31
+ ··· − 4y + 1)
c
5
, c
6
, c
8
((y − 1)
2
)(y
32
− 29y
31
+ ··· − 7y + 1)
14
