12a
0743
(K12a
0743
)
A knot diagram
1
Linearized knot diagam
3 8 9 12 11 10 2 7 1 6 5 4
Solving Sequence
2,8
3 1 7 9 4 10 6 12 5 11
c
2
c
1
c
7
c
8
c
3
c
9
c
6
c
12
c
4
c
11
c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
39
+ u
38
+ ··· + 2u + 1i
* 1 irreducible components of dim
C
= 0, with total 39 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
= hu
39
+ u
38
+ · · · + 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
4
a
7
=
u
u
a
9
=
−u
3
−u
3
+ u
a
4
=
−u
8
+ u
6
− u
4
+ 1
−u
8
+ 2u
6
− 2u
4
+ 2u
2
a
10
=
−u
9
+ 2u
7
− 3u
5
+ 2u
3
− u
−u
11
+ u
9
− 2u
7
+ u
5
− u
3
+ u
a
6
=
u
21
− 4u
19
+ ··· − 2u
3
+ u
u
23
− 3u
21
+ ··· + 2u
3
+ u
a
12
=
u
20
− 3u
18
+ 7u
16
− 10u
14
+ 10u
12
− 7u
10
+ u
8
+ 2u
6
− 3u
4
+ u
2
+ 1
u
20
− 4u
18
+ 10u
16
− 18u
14
+ 23u
12
− 24u
10
+ 18u
8
− 10u
6
+ 3u
4
a
5
=
−u
32
+ 5u
30
+ ··· + 2u
2
+ 1
−u
32
+ 6u
30
+ ··· − 2u
4
+ 2u
2
a
11
=
−u
33
+ 6u
31
+ ··· + 4u
3
− u
−u
35
+ 5u
33
+ ··· + u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
38
+ 28u
36
+ 4u
35
− 120u
34
− 24u
33
+ 368u
32
+ 100u
31
− 884u
30
− 296u
29
+
1736u
28
+ 700u
27
− 2852u
26
− 1356u
25
+ 3980u
24
+ 2200u
23
− 4744u
22
− 3040u
21
+
4832u
20
+ 3580u
19
− 4176u
18
− 3616u
17
+ 3012u
16
+ 3108u
15
− 1756u
14
− 2248u
13
+
776u
12
+ 1356u
11
−220u
10
−652u
9
+ 16u
8
+ 260u
7
+ 12u
6
−88u
5
+ 32u
3
−4u
2
−20u −6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
39
+ 13u
38
+ ··· − 4u + 1
c
2
, c
7
u
39
+ u
38
+ ··· + 2u + 1
c
3
u
39
− u
38
+ ··· + 20u + 13
c
4
, c
5
, c
6
c
10
, c
11
, c
12
u
39
+ u
38
+ ··· + 2u + 1
c
9
u
39
+ 7u
38
+ ··· − 92u − 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
39
+ 27y
38
+ ··· − 4y −1
c
2
, c
7
y
39
− 13y
38
+ ··· − 4y −1
c
3
y
39
− 9y
38
+ ··· + 1752y −169
c
4
, c
5
, c
6
c
10
, c
11
, c
12
y
39
+ 55y
38
+ ··· − 4y −1
c
9
y
39
− 5y
38
+ ··· + 960y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.991889 + 0.082450I
−3.38654 − 2.37032I −8.66369 + 6.45303I
u = 0.991889 − 0.082450I
−3.38654 + 2.37032I −8.66369 − 6.45303I
u = 0.669840 + 0.792447I
−2.96109 + 4.19895I −3.45915 − 2.96726I
u = 0.669840 − 0.792447I
−2.96109 − 4.19895I −3.45915 + 2.96726I
u = −0.704805 + 0.763405I
2.32305 − 2.08285I 0.48249 + 4.55796I
u = −0.704805 − 0.763405I
2.32305 + 2.08285I 0.48249 − 4.55796I
u = −0.654698 + 0.815515I
−13.8160 − 5.2888I −3.98413 + 1.89357I
u = −0.654698 − 0.815515I
−13.8160 + 5.2888I −3.98413 − 1.89357I
u = 0.749661 + 0.738081I
3.08458 − 0.82025I 3.76726 + 3.28548I
u = 0.749661 − 0.738081I
3.08458 + 0.82025I 3.76726 − 3.28548I
u = −1.053450 + 0.104546I
−9.10965 + 4.03404I −10.86562 − 4.28987I
u = −1.053450 − 0.104546I
−9.10965 − 4.03404I −10.86562 + 4.28987I
u = −0.926938
−1.84696 −2.83260
u = 1.084680 + 0.113747I
19.2646 − 4.8999I −10.88316 + 3.33845I
u = 1.084680 − 0.113747I
19.2646 + 4.8999I −10.88316 − 3.33845I
u = 0.954725 + 0.548432I
−6.57750 − 1.92071I −8.14899 + 2.64008I
u = 0.954725 − 0.548432I
−6.57750 + 1.92071I −8.14899 − 2.64008I
u = −0.841615 + 0.719480I
−0.29950 + 2.71622I −2.52778 − 3.64683I
u = −0.841615 − 0.719480I
−0.29950 − 2.71622I −2.52778 + 3.64683I
u = −0.900993 + 0.649958I
−0.39055 + 2.53610I −4.97104 − 1.73986I
u = −0.900993 − 0.649958I
−0.39055 − 2.53610I −4.97104 + 1.73986I
u = −0.997812 + 0.527862I
−17.7602 + 1.5442I −8.40581 − 2.83679I
u = −0.997812 − 0.527862I
−17.7602 − 1.5442I −8.40581 + 2.83679I
u = 0.869652 + 0.772208I
−10.18350 − 2.90290I −2.41886 + 2.81755I
u = 0.869652 − 0.772208I
−10.18350 + 2.90290I −2.41886 − 2.81755I
u = 0.960196 + 0.698617I
2.44032 − 4.66537I 2.22438 + 2.85694I
u = 0.960196 − 0.698617I
2.44032 + 4.66537I 2.22438 − 2.85694I
u = −0.989811 + 0.703622I
1.46134 + 7.65649I −1.60381 − 9.50795I
u = −0.989811 − 0.703622I
1.46134 − 7.65649I −1.60381 + 9.50795I
u = 1.013930 + 0.706638I
−3.99832 − 9.85780I −5.28411 + 7.75743I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.013930 − 0.706638I
−3.99832 + 9.85780I −5.28411 − 7.75743I
u = −1.028300 + 0.710572I
−14.9463 + 11.0210I −5.83190 − 6.59019I
u = −1.028300 − 0.710572I
−14.9463 − 11.0210I −5.83190 + 6.59019I
u = −0.273540 + 0.656844I
−15.8067 + 2.7178I −4.22928 − 2.39792I
u = −0.273540 − 0.656844I
−15.8067 − 2.7178I −4.22928 + 2.39792I
u = 0.267258 + 0.577154I
−4.96129 − 2.11580I −3.91393 + 3.51280I
u = 0.267258 − 0.577154I
−4.96129 + 2.11580I −3.91393 − 3.51280I
u = −0.153346 + 0.396492I
0.057297 + 0.923048I 1.13344 − 7.42333I
u = −0.153346 − 0.396492I
0.057297 − 0.923048I 1.13344 + 7.42333I
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
8
u
39
+ 13u
38
+ ··· − 4u + 1
c
2
, c
7
u
39
+ u
38
+ ··· + 2u + 1
c
3
u
39
− u
38
+ ··· + 20u + 13
c
4
, c
5
, c
6
c
10
, c
11
, c
12
u
39
+ u
38
+ ··· + 2u + 1
c
9
u
39
+ 7u
38
+ ··· − 92u − 7
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
39
+ 27y
38
+ ··· − 4y −1
c
2
, c
7
y
39
− 13y
38
+ ··· − 4y −1
c
3
y
39
− 9y
38
+ ··· + 1752y −169
c
4
, c
5
, c
6
c
10
, c
11
, c
12
y
39
+ 55y
38
+ ··· − 4y −1
c
9
y
39
− 5y
38
+ ··· + 960y −49
8
