12a
0169
(K12a
0169
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 1 12 11 10 4 8 7 6
Solving Sequence
4,10
9 3 8 11 7 12 6 1 2 5
c
9
c
3
c
8
c
10
c
7
c
11
c
6
c
12
c
1
c
5
c
2
, c
4
Ideals for irreducible components
2
of X
par
I
u
1
= hu
24
− u
23
+ ··· − 2u
2
+ 1i
* 1 irreducible components of dim
C
= 0, with total 24 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
24
− u
23
+ · · · − 2u
2
+ 1i
(i) Arc colorings
a
4
=
0
u
a
10
=
1
0
a
9
=
1
u
2
a
3
=
−u
−u
3
+ u
a
8
=
−u
2
+ 1
u
2
a
11
=
u
4
− u
2
+ 1
−u
4
a
7
=
−u
6
+ u
4
− 2u
2
+ 1
u
6
+ u
2
a
12
=
u
8
− u
6
+ 3u
4
− 2u
2
+ 1
−u
8
− 2u
4
a
6
=
−u
10
+ u
8
− 4u
6
+ 3u
4
− 3u
2
+ 1
u
10
+ 3u
6
+ u
2
a
1
=
u
12
− u
10
+ 5u
8
− 4u
6
+ 6u
4
− 3u
2
+ 1
−u
12
− 4u
8
− 3u
4
a
2
=
u
16
− u
14
+ 7u
12
− 6u
10
+ 15u
8
− 10u
6
+ 10u
4
− 4u
2
+ 1
u
18
− 2u
16
+ 7u
14
− 12u
12
+ 15u
10
− 20u
8
+ 10u
6
− 8u
4
+ u
2
a
5
=
−u
14
+ u
12
− 6u
10
+ 5u
8
− 10u
6
+ 6u
4
− 4u
2
+ 1
u
14
+ 5u
10
+ 6u
6
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
23
+8u
21
−4u
20
−44u
19
+4u
18
+72u
17
−32u
16
−176u
15
+28u
14
+228u
13
−88u
12
−
308u
11
+ 64u
10
+ 296u
9
− 96u
8
− 220u
7
+ 52u
6
+ 136u
5
− 28u
4
− 44u
3
+ 4u
2
+ 16u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
24
+ 15u
23
+ ··· + 4u + 1
c
2
, c
4
u
24
− u
23
+ ··· − 4u + 1
c
3
, c
9
u
24
− u
23
+ ··· − 2u
2
+ 1
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
u
24
− 3u
23
+ ··· − 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
− 11y
23
+ ··· − 20y + 1
c
2
, c
4
y
24
− 15y
23
+ ··· − 4y + 1
c
3
, c
9
y
24
− 3y
23
+ ··· − 4y + 1
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
24
+ 37y
23
+ ··· + 12y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.818053 + 0.585237I
−4.34966 + 5.76332I −0.83619 − 8.32312I
u = 0.818053 − 0.585237I
−4.34966 − 5.76332I −0.83619 + 8.32312I
u = 0.629900 + 0.676291I
−4.99244 − 1.16181I −3.45696 + 0.66594I
u = 0.629900 − 0.676291I
−4.99244 + 1.16181I −3.45696 − 0.66594I
u = −0.703557 + 0.542696I
−1.64192 − 2.01575I 2.27145 + 4.63931I
u = −0.703557 − 0.542696I
−1.64192 + 2.01575I 2.27145 − 4.63931I
u = 0.865592 + 0.818505I
−9.02499 + 3.00763I 0.51260 − 2.75465I
u = 0.865592 − 0.818505I
−9.02499 − 3.00763I 0.51260 + 2.75465I
u = −0.748084 + 0.274295I
−0.07774 − 3.02933I 5.70852 + 9.26987I
u = −0.748084 − 0.274295I
−0.07774 + 3.02933I 5.70852 − 9.26987I
u = −0.844876 + 0.858831I
−12.84350 + 1.46852I −3.16556 − 0.66920I
u = −0.844876 − 0.858831I
−12.84350 − 1.46852I −3.16556 + 0.66920I
u = −0.908318 + 0.819726I
−12.6233 − 7.6239I −2.60069 + 6.03151I
u = −0.908318 − 0.819726I
−12.6233 + 7.6239I −2.60069 − 6.03151I
u = 0.662055 + 0.056751I
0.927331 + 0.040320I 11.54599 − 0.37990I
u = 0.662055 − 0.056751I
0.927331 − 0.040320I 11.54599 + 0.37990I
u = −0.966596 + 0.956593I
18.1029 − 3.5079I 0.10153 + 2.15218I
u = −0.966596 − 0.956593I
18.1029 + 3.5079I 0.10153 − 2.15218I
u = 0.962124 + 0.965092I
14.0584 − 1.6220I −3.03720 + 0.65264I
u = 0.962124 − 0.965092I
14.0584 + 1.6220I −3.03720 − 0.65264I
u = 0.975953 + 0.955750I
14.1060 + 8.6639I −2.94392 − 4.94788I
u = 0.975953 − 0.955750I
14.1060 − 8.6639I −2.94392 + 4.94788I
u = −0.242249 + 0.453783I
−1.64107 + 0.52371I −4.09957 − 0.43757I
u = −0.242249 − 0.453783I
−1.64107 − 0.52371I −4.09957 + 0.43757I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
24
+ 15u
23
+ ··· + 4u + 1
c
2
, c
4
u
24
− u
23
+ ··· − 4u + 1
c
3
, c
9
u
24
− u
23
+ ··· − 2u
2
+ 1
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
u
24
− 3u
23
+ ··· − 4u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
24
− 11y
23
+ ··· − 20y + 1
c
2
, c
4
y
24
− 15y
23
+ ··· − 4y + 1
c
3
, c
9
y
24
− 3y
23
+ ··· − 4y + 1
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
24
+ 37y
23
+ ··· + 12y + 1
7
