12a
0775
(K12a
0775
)
A knot diagram
1
Linearized knot diagam
3 8 10 12 11 1 9 2 7 6 5 4
Solving Sequence
2,9
8 3 1 7 10 4 6 11 5 12
c
8
c
2
c
1
c
7
c
9
c
3
c
6
c
10
c
5
c
12
c
4
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
43
− u
42
+ ··· + u
2
+ 1i
* 1 irreducible components of dim
C
= 0, with total 43 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
43
− u
42
+ · · · + u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
9
=
1
0
a
8
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
7
=
u
2
+ 1
u
2
a
10
=
u
4
+ u
2
+ 1
u
4
a
4
=
−u
11
− 2u
9
− 4u
7
− 4u
5
− 3u
3
−u
11
− u
9
− 2u
7
− u
5
+ u
3
+ u
a
6
=
−u
10
− u
8
− 2u
6
− u
4
+ u
2
+ 1
−u
12
− 2u
10
− 4u
8
− 4u
6
− 3u
4
a
11
=
u
26
+ 3u
24
+ ··· + u
2
+ 1
u
28
+ 4u
26
+ ··· + 12u
8
+ u
4
a
5
=
−u
42
− 5u
40
+ ··· + u
2
+ 1
−u
42
+ u
41
+ ··· + u + 1
a
12
=
u
27
+ 4u
25
+ ··· + 12u
7
+ u
3
u
27
+ 3u
25
+ ··· + u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
42
− 20u
40
+ ··· − 8u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
9
u
43
+ 11u
42
+ ··· − 2u − 1
c
2
, c
8
u
43
− u
42
+ ··· + u
2
+ 1
c
3
, c
6
u
43
− u
42
+ ··· − 16u + 5
c
4
, c
5
, c
10
c
11
, c
12
u
43
− u
42
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
y
43
+ 43y
42
+ ··· + 22y −1
c
2
, c
8
y
43
+ 11y
42
+ ··· − 2y −1
c
3
, c
6
y
43
− 17y
42
+ ··· + 6y −25
c
4
, c
5
, c
10
c
11
, c
12
y
43
+ 55y
42
+ ··· − 2y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.277924 + 0.951909I
−3.90188 − 2.73159I −14.1982 + 4.9259I
u = −0.277924 − 0.951909I
−3.90188 + 2.73159I −14.1982 − 4.9259I
u = −0.171363 + 0.973104I
6.85402 + 1.62529I −8.25292 + 0.49765I
u = −0.171363 − 0.973104I
6.85402 − 1.62529I −8.25292 − 0.49765I
u = 0.322382 + 0.965498I
−1.09525 + 5.77920I −7.86814 − 8.38864I
u = 0.322382 − 0.965498I
−1.09525 − 5.77920I −7.86814 + 8.38864I
u = 0.218790 + 0.942416I
−1.70142 − 0.27515I −9.76443 + 0.48327I
u = 0.218790 − 0.942416I
−1.70142 + 0.27515I −9.76443 − 0.48327I
u = −0.349470 + 0.985821I
7.87866 − 7.40221I −6.00523 + 6.67601I
u = −0.349470 − 0.985821I
7.87866 + 7.40221I −6.00523 − 6.67601I
u = 0.605298 + 0.622434I
12.21750 + 2.21577I 0.00606 − 3.15863I
u = 0.605298 − 0.622434I
12.21750 − 2.21577I 0.00606 + 3.15863I
u = 0.724879 + 0.902558I
11.90850 + 2.74938I −2.70691 − 3.06690I
u = 0.724879 − 0.902558I
11.90850 − 2.74938I −2.70691 + 3.06690I
u = −0.806200 + 0.845885I
4.50177 − 2.35046I −3.49635 + 4.76108I
u = −0.806200 − 0.845885I
4.50177 + 2.35046I −3.49635 − 4.76108I
u = 0.834116 + 0.820628I
3.12542 − 0.75919I −6.95792 + 1.59263I
u = 0.834116 − 0.820628I
3.12542 + 0.75919I −6.95792 − 1.59263I
u = −0.858422 + 0.817028I
6.46803 + 3.80376I −1.40821 − 3.43849I
u = −0.858422 − 0.817028I
6.46803 − 3.80376I −1.40821 + 3.43849I
u = 0.874607 + 0.815183I
15.8059 − 5.4998I −0.14982 + 1.94040I
u = 0.874607 − 0.815183I
15.8059 + 5.4998I −0.14982 − 1.94040I
u = −0.782409 + 0.935028I
4.22457 − 3.61643I −4.15250 + 0.77790I
u = −0.782409 − 0.935028I
4.22457 + 3.61643I −4.15250 − 0.77790I
u = −0.470068 + 0.618631I
3.05318 − 1.77992I 0.16951 + 4.97625I
u = −0.470068 − 0.618631I
3.05318 + 1.77992I 0.16951 − 4.97625I
u = 0.836155 + 0.902083I
9.93224 + 3.11134I 1.49937 − 2.73102I
u = 0.836155 − 0.902083I
9.93224 − 3.11134I 1.49937 + 2.73102I
u = 0.791521 + 0.959516I
2.69617 + 6.83503I −8.00000 − 6.45647I
u = 0.791521 − 0.959516I
2.69617 − 6.83503I −8.00000 + 6.45647I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.856944 + 0.910402I
−19.5349 − 3.1770I 1.83688 + 2.53616I
u = −0.856944 − 0.910402I
−19.5349 + 3.1770I 1.83688 − 2.53616I
u = −0.802935 + 0.972136I
5.98468 − 9.98705I −2.44159 + 8.35153I
u = −0.802935 − 0.972136I
5.98468 + 9.98705I −2.44159 − 8.35153I
u = 0.810427 + 0.981246I
15.2854 + 11.7553I −1.12118 − 6.76515I
u = 0.810427 − 0.981246I
15.2854 − 11.7553I −1.12118 + 6.76515I
u = −0.637340 + 0.169138I
10.43140 + 3.87845I −0.23993 − 2.25081I
u = −0.637340 − 0.169138I
10.43140 − 3.87845I −0.23993 + 2.25081I
u = 0.178947 + 0.606236I
−0.367517 + 0.825434I −8.23829 − 8.13586I
u = 0.178947 − 0.606236I
−0.367517 − 0.825434I −8.23829 + 8.13586I
u = 0.571695 + 0.133672I
1.42336 − 2.56705I −1.44027 + 4.09761I
u = 0.571695 − 0.133672I
1.42336 + 2.56705I −1.44027 − 4.09761I
u = −0.511485
−1.21227 −8.49140
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
9
u
43
+ 11u
42
+ ··· − 2u − 1
c
2
, c
8
u
43
− u
42
+ ··· + u
2
+ 1
c
3
, c
6
u
43
− u
42
+ ··· − 16u + 5
c
4
, c
5
, c
10
c
11
, c
12
u
43
− u
42
+ ··· + 2u + 1
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
y
43
+ 43y
42
+ ··· + 22y −1
c
2
, c
8
y
43
+ 11y
42
+ ··· − 2y −1
c
3
, c
6
y
43
− 17y
42
+ ··· + 6y −25
c
4
, c
5
, c
10
c
11
, c
12
y
43
+ 55y
42
+ ··· − 2y −1
8
