12a
0528
(K12a
0528
)
A knot diagram
1
Linearized knot diagam
3 7 8 10 9 2 1 12 11 5 4 6
Solving Sequence
4,10
5 11 12 9 6 1 8 3 7 2
c
4
c
10
c
11
c
9
c
5
c
12
c
8
c
3
c
7
c
2
c
1
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
90
+ 2u
89
+ ··· + 3u + 1i
I
u
2
= hu − 1i
* 2 irreducible components of dim
C
= 0, with total 91 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
90
+ 2u
89
+ · · · + 3u + 1i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
−u
2
a
11
=
u
−u
3
+ u
a
12
=
u
3
−u
3
+ u
a
9
=
−u
3
u
5
− u
3
+ u
a
6
=
u
6
− u
4
+ 1
−u
8
+ 2u
6
− 2u
4
a
1
=
u
17
− 4u
15
+ 7u
13
− 4u
11
− 3u
9
+ 6u
7
− 2u
5
+ u
−u
19
+ 5u
17
− 12u
15
+ 15u
13
− 9u
11
− u
9
+ 4u
7
− 2u
5
− u
3
+ u
a
8
=
−u
11
+ 2u
9
− 2u
7
− u
3
u
11
− 3u
9
+ 4u
7
− u
5
− u
3
+ u
a
3
=
−u
22
+ 5u
20
− 12u
18
+ 15u
16
− 10u
14
+ 2u
12
− u
8
+ u
6
− u
4
+ 1
u
22
− 6u
20
+ 17u
18
− 26u
16
+ 20u
14
− 13u
10
+ 10u
8
− u
6
− 2u
4
+ u
2
a
7
=
u
47
− 12u
45
+ ··· + 4u
7
− 2u
3
−u
49
+ 13u
47
+ ··· − 2u
3
+ u
a
2
=
u
63
− 16u
61
+ ··· − 6u
7
+ 2u
3
−u
63
+ 17u
61
+ ··· − 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
89
− 96u
87
+ ··· + 8u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
90
+ 40u
89
+ ··· + u + 1
c
2
, c
6
u
90
− 20u
88
+ ··· − u + 1
c
3
, c
12
u
90
+ 2u
89
+ ··· + 35u + 25
c
4
, c
10
u
90
+ 2u
89
+ ··· + 3u + 1
c
5
, c
11
u
90
+ 3u
89
+ ··· − 37u + 13
c
7
u
90
− 3u
89
+ ··· − 69u + 13
c
8
u
90
+ 14u
89
+ ··· + 26531u + 1493
c
9
u
90
− 48u
89
+ ··· − u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
90
+ 20y
89
+ ··· − 5y + 1
c
2
, c
6
y
90
− 40y
89
+ ··· − y + 1
c
3
, c
12
y
90
− 72y
89
+ ··· + 675y + 625
c
4
, c
10
y
90
− 48y
89
+ ··· − y + 1
c
5
, c
11
y
90
+ 75y
89
+ ··· − 21571y + 169
c
7
y
90
+ 3y
89
+ ··· + 8941y + 169
c
8
y
90
− 24y
89
+ ··· − 76057601y + 2229049
c
9
y
90
− 12y
89
+ ··· + 3y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.873411 + 0.513697I
−1.81913 − 4.09974I 0
u = −0.873411 − 0.513697I
−1.81913 + 4.09974I 0
u = 0.902181 + 0.525801I
2.99810 + 6.09081I 0
u = 0.902181 − 0.525801I
2.99810 − 6.09081I 0
u = −0.897963 + 0.536251I
1.02275 − 11.20450I 0
u = −0.897963 − 0.536251I
1.02275 + 11.20450I 0
u = 0.922566 + 0.496402I
3.48059 + 3.55498I 0
u = 0.922566 − 0.496402I
3.48059 − 3.55498I 0
u = −0.903572 + 0.276220I
0.20740 − 3.77187I 5.01166 + 7.48596I
u = −0.903572 − 0.276220I
0.20740 + 3.77187I 5.01166 − 7.48596I
u = 0.788143 + 0.520655I
−4.28906 + 5.70288I −4.46481 − 8.34300I
u = 0.788143 − 0.520655I
−4.28906 − 5.70288I −4.46481 + 8.34300I
u = −0.941169 + 0.480866I
1.91087 + 1.41955I 0
u = −0.941169 − 0.480866I
1.91087 − 1.41955I 0
u = −1.059210 + 0.019229I
6.75320 − 1.38707I 0
u = −1.059210 − 0.019229I
6.75320 + 1.38707I 0
u = 1.063040 + 0.035242I
4.97510 + 6.54466I 0
u = 1.063040 − 0.035242I
4.97510 − 6.54466I 0
u = −0.770968 + 0.482215I
−1.57766 − 2.01209I −0.85760 + 4.47164I
u = −0.770968 − 0.482215I
−1.57766 + 2.01209I −0.85760 − 4.47164I
u = 0.738456 + 0.517645I
−4.43144 − 1.45967I −5.28174 + 0.47374I
u = 0.738456 − 0.517645I
−4.43144 + 1.45967I −5.28174 − 0.47374I
u = 0.831064 + 0.096380I
1.270970 + 0.122716I 8.82726 − 0.42041I
u = 0.831064 − 0.096380I
1.270970 − 0.122716I 8.82726 + 0.42041I
u = −0.126989 + 0.822033I
4.77347 + 11.44800I 3.37132 − 7.61382I
u = −0.126989 − 0.822033I
4.77347 − 11.44800I 3.37132 + 7.61382I
u = 0.120864 + 0.820667I
6.74638 − 6.20589I 6.36909 + 3.29581I
u = 0.120864 − 0.820667I
6.74638 + 6.20589I 6.36909 − 3.29581I
u = 0.103336 + 0.819213I
7.26934 − 3.28559I 7.18981 + 2.93019I
u = 0.103336 − 0.819213I
7.26934 + 3.28559I 7.18981 − 2.93019I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.094198 + 0.819093I
5.74525 − 1.91608I 4.94203 + 2.18879I
u = −0.094198 − 0.819093I
5.74525 + 1.91608I 4.94203 − 2.18879I
u = −0.120401 + 0.803953I
1.59070 + 4.18357I 0.16117 − 3.00795I
u = −0.120401 − 0.803953I
1.59070 − 4.18357I 0.16117 + 3.00795I
u = −0.578569 + 0.559212I
0.13368 + 6.80952I −0.27186 − 4.68036I
u = −0.578569 − 0.559212I
0.13368 − 6.80952I −0.27186 + 4.68036I
u = −0.620498 + 0.503672I
−2.52931 − 0.09500I −4.25506 + 0.72855I
u = −0.620498 − 0.503672I
−2.52931 + 0.09500I −4.25506 − 0.72855I
u = −1.130970 + 0.433801I
0.70255 − 4.46851I 0
u = −1.130970 − 0.433801I
0.70255 + 4.46851I 0
u = −1.152640 + 0.389106I
1.84992 + 2.50588I 0
u = −1.152640 − 0.389106I
1.84992 − 2.50588I 0
u = 0.561042 + 0.544183I
2.05754 − 1.76920I 2.90604 + 0.28243I
u = 0.561042 − 0.544183I
2.05754 + 1.76920I 2.90604 − 0.28243I
u = 1.162250 + 0.412550I
4.17297 + 1.71901I 0
u = 1.162250 − 0.412550I
4.17297 − 1.71901I 0
u = −0.024292 + 0.753892I
2.47216 + 2.13571I 5.91843 − 3.73421I
u = −0.024292 − 0.753892I
2.47216 − 2.13571I 5.91843 + 3.73421I
u = 1.152150 + 0.484969I
0.27112 + 3.48997I 0
u = 1.152150 − 0.484969I
0.27112 − 3.48997I 0
u = 0.153502 + 0.729188I
−1.85254 − 6.16807I −1.72016 + 7.05399I
u = 0.153502 − 0.729188I
−1.85254 + 6.16807I −1.72016 − 7.05399I
u = −1.170070 + 0.486533I
3.64084 − 6.60935I 0
u = −1.170070 − 0.486533I
3.64084 + 6.60935I 0
u = 1.167280 + 0.498284I
1.08161 + 10.76960I 0
u = 1.167280 − 0.498284I
1.08161 − 10.76960I 0
u = 1.191320 + 0.443194I
5.96752 + 2.13257I 0
u = 1.191320 − 0.443194I
5.96752 − 2.13257I 0
u = 1.213250 + 0.391111I
5.57380 − 0.11285I 0
u = 1.213250 − 0.391111I
5.57380 + 0.11285I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.116850 + 0.712914I
0.62172 + 2.11128I 2.36589 − 3.42722I
u = −0.116850 − 0.712914I
0.62172 − 2.11128I 2.36589 + 3.42722I
u = −1.191720 + 0.461019I
5.84080 − 6.54472I 0
u = −1.191720 − 0.461019I
5.84080 + 6.54472I 0
u = 0.485978 + 0.528952I
2.30152 + 0.59676I 3.38135 − 0.44638I
u = 0.485978 − 0.528952I
2.30152 − 0.59676I 3.38135 + 0.44638I
u = 1.224080 + 0.384721I
8.85923 − 7.34311I 0
u = 1.224080 − 0.384721I
8.85923 + 7.34311I 0
u = −1.223690 + 0.388798I
10.80400 + 2.07981I 0
u = −1.223690 − 0.388798I
10.80400 − 2.07981I 0
u = −1.223810 + 0.399626I
11.25850 − 0.90671I 0
u = −1.223810 − 0.399626I
11.25850 + 0.90671I 0
u = 1.224080 + 0.404848I
9.70226 + 6.14308I 0
u = 1.224080 − 0.404848I
9.70226 − 6.14308I 0
u = −0.450612 + 0.548105I
0.55266 − 5.54561I 0.22267 + 5.35237I
u = −0.450612 − 0.548105I
0.55266 + 5.54561I 0.22267 − 5.35237I
u = −1.197770 + 0.505207I
4.76470 − 8.98285I 0
u = −1.197770 − 0.505207I
4.76470 + 8.98285I 0
u = −1.208040 + 0.497568I
9.04174 − 2.87641I 0
u = −1.208040 − 0.497568I
9.04174 + 2.87641I 0
u = 1.206570 + 0.501359I
10.53440 + 8.09975I 0
u = 1.206570 − 0.501359I
10.53440 − 8.09975I 0
u = 1.203810 + 0.508612I
9.9530 + 11.0635I 0
u = 1.203810 − 0.508612I
9.9530 − 11.0635I 0
u = −1.203080 + 0.511255I
7.9613 − 16.3232I 0
u = −1.203080 − 0.511255I
7.9613 + 16.3232I 0
u = 0.164684 + 0.661302I
−2.55538 + 0.91259I −3.81773 − 0.78733I
u = 0.164684 − 0.661302I
−2.55538 − 0.91259I −3.81773 + 0.78733I
u = −0.299147 + 0.470065I
−1.76511 + 0.79975I −3.82418 − 0.77073I
u = −0.299147 − 0.470065I
−1.76511 − 0.79975I −3.82418 + 0.77073I
7
II. I
u
2
= hu − 1i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
1
a
5
=
1
−1
a
11
=
1
0
a
12
=
1
0
a
9
=
−1
1
a
6
=
1
−1
a
1
=
2
−1
a
8
=
−2
1
a
3
=
−1
1
a
7
=
−2
1
a
2
=
1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u + 1
c
2
, c
3
, c
4
c
6
, c
8
, c
9
c
10
, c
12
u − 1
c
5
, c
7
, c
11
u
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
c
9
, c
10
, c
12
y −1
c
5
, c
7
, c
11
y
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
1.64493 6.00000
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
90
+ 40u
89
+ ··· + u + 1)
c
2
, c
6
(u − 1)(u
90
− 20u
88
+ ··· − u + 1)
c
3
, c
12
(u − 1)(u
90
+ 2u
89
+ ··· + 35u + 25)
c
4
, c
10
(u − 1)(u
90
+ 2u
89
+ ··· + 3u + 1)
c
5
, c
11
u(u
90
+ 3u
89
+ ··· − 37u + 13)
c
7
u(u
90
− 3u
89
+ ··· − 69u + 13)
c
8
(u − 1)(u
90
+ 14u
89
+ ··· + 26531u + 1493)
c
9
(u − 1)(u
90
− 48u
89
+ ··· − u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
90
+ 20y
89
+ ··· − 5y + 1)
c
2
, c
6
(y −1)(y
90
− 40y
89
+ ··· − y + 1)
c
3
, c
12
(y −1)(y
90
− 72y
89
+ ··· + 675y + 625)
c
4
, c
10
(y −1)(y
90
− 48y
89
+ ··· − y + 1)
c
5
, c
11
y(y
90
+ 75y
89
+ ··· − 21571y + 169)
c
7
y(y
90
+ 3y
89
+ ··· + 8941y + 169)
c
8
(y −1)(y
90
− 24y
89
+ ··· − 7.60576 × 10
7
y + 2229049)
c
9
(y −1)(y
90
− 12y
89
+ ··· + 3y + 1)
13
