12a
0760
(K12a
0760
)
A knot diagram
1
Linearized knot diagam
3 8 10 11 12 9 2 7 1 4 6 5
Solving Sequence
6,11
12 5 1 4 10 3 2 9 7 8
c
11
c
5
c
12
c
4
c
10
c
3
c
1
c
9
c
6
c
8
c
2
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
55
− u
54
+ ··· + 2u
2
− 1i
* 1 irreducible components of dim
C
= 0, with total 55 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
55
− u
54
+ · · · + 2u
2
− 1i
(i) Arc colorings
a
6
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
5
=
−u
u
3
+ u
a
1
=
u
2
+ 1
−u
4
− 2u
2
a
4
=
−u
3
− 2u
u
3
+ u
a
10
=
−u
6
− 3u
4
− 2u
2
+ 1
u
6
+ 2u
4
+ u
2
a
3
=
u
9
+ 4u
7
+ 5u
5
− 3u
−u
9
− 3u
7
− 3u
5
+ u
a
2
=
u
22
+ 9u
20
+ ··· − 2u
2
+ 1
−u
22
− 8u
20
+ ··· − 6u
4
− u
2
a
9
=
u
12
+ 5u
10
+ 9u
8
+ 4u
6
− 6u
4
− 5u
2
+ 1
−u
14
− 6u
12
− 13u
10
− 10u
8
+ 4u
6
+ 8u
4
+ u
2
a
7
=
−u
25
− 10u
23
+ ··· + 10u
3
− u
u
27
+ 11u
25
+ ··· − u
3
+ u
a
8
=
u
38
+ 15u
36
+ ··· − 4u
2
+ 1
−u
40
− 16u
38
+ ··· + 8u
6
+ 14u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
54
− 4u
53
+ ··· + 16u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
8
u
55
+ 13u
54
+ ··· + 4u + 1
c
2
, c
7
u
55
+ u
54
+ ··· + 2u
2
− 1
c
3
, c
4
, c
10
u
55
+ u
54
+ ··· − 11u − 2
c
5
, c
11
, c
12
u
55
− u
54
+ ··· + 2u
2
− 1
c
9
u
55
− 7u
54
+ ··· − 20988u + 4921
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
y
55
+ 59y
54
+ ··· − 36y −1
c
2
, c
7
y
55
− 13y
54
+ ··· + 4y −1
c
3
, c
4
, c
10
y
55
− 57y
54
+ ··· − 91y −4
c
5
, c
11
, c
12
y
55
+ 43y
54
+ ··· + 4y −1
c
9
y
55
− 29y
54
+ ··· + 402466656y −24216241
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.886995 + 0.045862I
14.3413 − 2.3988I 10.05032 + 0.51244I
u = −0.886995 − 0.045862I
14.3413 + 2.3988I 10.05032 − 0.51244I
u = 0.885350 + 0.051007I
13.9632 + 8.8219I 9.36443 − 5.35751I
u = 0.885350 − 0.051007I
13.9632 − 8.8219I 9.36443 + 5.35751I
u = −0.864228 + 0.018685I
7.62020 − 1.13554I 10.36949 + 0.23646I
u = −0.864228 − 0.018685I
7.62020 + 1.13554I 10.36949 − 0.23646I
u = 0.079677 + 1.134880I
−1.86120 + 1.73369I 0. − 4.71715I
u = 0.079677 − 1.134880I
−1.86120 − 1.73369I 0. + 4.71715I
u = 0.856539 + 0.039628I
5.73343 + 5.16675I 5.71941 − 6.13440I
u = 0.856539 − 0.039628I
5.73343 − 5.16675I 5.71941 + 6.13440I
u = 0.124104 + 0.821653I
5.01486 + 3.10577I 5.36912 − 3.19461I
u = 0.124104 − 0.821653I
5.01486 − 3.10577I 5.36912 + 3.19461I
u = 0.830617
3.31017 1.79020
u = −0.134449 + 0.719876I
4.98402 + 3.12049I 4.83515 − 1.88613I
u = −0.134449 − 0.719876I
4.98402 − 3.12049I 4.83515 + 1.88613I
u = 0.140602 + 1.264520I
−3.20902 + 2.28111I 0
u = 0.140602 − 1.264520I
−3.20902 − 2.28111I 0
u = 0.398432 + 1.236750I
2.03590 − 0.66546I 0
u = 0.398432 − 1.236750I
2.03590 + 0.66546I 0
u = 0.431188 + 1.231470I
10.32020 − 4.11571I 0
u = 0.431188 − 1.231470I
10.32020 + 4.11571I 0
u = −0.091117 + 1.302380I
−6.13952 − 0.32984I 0
u = −0.091117 − 1.302380I
−6.13952 + 0.32984I 0
u = −0.431390 + 1.236960I
10.66320 − 2.31354I 0
u = −0.431390 − 1.236960I
10.66320 + 2.31354I 0
u = −0.151057 + 1.311240I
−5.41307 − 5.33764I 0
u = −0.151057 − 1.311240I
−5.41307 + 5.33764I 0
u = −0.404078 + 1.258270I
3.77988 − 3.40970I 0
u = −0.404078 − 1.258270I
3.77988 + 3.40970I 0
u = −0.010529 + 1.328650I
−0.68155 + 2.96620I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.010529 − 1.328650I
−0.68155 − 2.96620I 0
u = 0.375169 + 1.274910I
−0.65198 + 4.33276I 0
u = 0.375169 − 1.274910I
−0.65198 − 4.33276I 0
u = 0.207095 + 1.316430I
1.91268 + 2.83669I 0
u = 0.207095 − 1.316430I
1.91268 − 2.83669I 0
u = −0.199631 + 1.326110I
1.57882 − 9.00165I 0
u = −0.199631 − 1.326110I
1.57882 + 9.00165I 0
u = −0.399305 + 1.288770I
3.55054 − 5.66585I 0
u = −0.399305 − 1.288770I
3.55054 + 5.66585I 0
u = 0.391698 + 1.302460I
1.54641 + 9.64694I 0
u = 0.391698 − 1.302460I
1.54641 − 9.64694I 0
u = −0.577636 + 0.255967I
6.50085 − 6.27951I 7.73444 + 7.40899I
u = −0.577636 − 0.255967I
6.50085 + 6.27951I 7.73444 − 7.40899I
u = 0.585657 + 0.234755I
6.72459 + 0.04895I 8.43887 − 2.19191I
u = 0.585657 − 0.234755I
6.72459 − 0.04895I 8.43887 + 2.19191I
u = −0.410657 + 1.311660I
10.10380 − 7.04613I 0
u = −0.410657 − 1.311660I
10.10380 + 7.04613I 0
u = 0.408466 + 1.314810I
9.6974 + 13.4566I 0
u = 0.408466 − 1.314810I
9.6974 − 13.4566I 0
u = −0.449466 + 0.260084I
−0.59638 − 3.23489I 2.34241 + 9.68420I
u = −0.449466 − 0.260084I
−0.59638 + 3.23489I 2.34241 − 9.68420I
u = 0.435627 + 0.099986I
0.914015 + 0.263108I 10.61137 − 1.72545I
u = 0.435627 − 0.099986I
0.914015 − 0.263108I 10.61137 + 1.72545I
u = −0.224373 + 0.350383I
−1.27930 + 0.82968I −2.41349 − 0.37291I
u = −0.224373 − 0.350383I
−1.27930 − 0.82968I −2.41349 + 0.37291I
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
8
u
55
+ 13u
54
+ ··· + 4u + 1
c
2
, c
7
u
55
+ u
54
+ ··· + 2u
2
− 1
c
3
, c
4
, c
10
u
55
+ u
54
+ ··· − 11u − 2
c
5
, c
11
, c
12
u
55
− u
54
+ ··· + 2u
2
− 1
c
9
u
55
− 7u
54
+ ··· − 20988u + 4921
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
y
55
+ 59y
54
+ ··· − 36y −1
c
2
, c
7
y
55
− 13y
54
+ ··· + 4y −1
c
3
, c
4
, c
10
y
55
− 57y
54
+ ··· − 91y −4
c
5
, c
11
, c
12
y
55
+ 43y
54
+ ··· + 4y −1
c
9
y
55
− 29y
54
+ ··· + 402466656y −24216241
8
