12a
0518
(K12a
0518
)
A knot diagram
1
Linearized knot diagam
3 7 8 9 11 10 2 1 12 6 5 4
Solving Sequence
2,8
7 3 4 1 9 5 12 10 6 11
c
7
c
2
c
3
c
1
c
8
c
4
c
12
c
9
c
6
c
11
c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
78
+ u
77
+ ··· + u + 1i
* 1 irreducible components of dim
C
= 0, with total 78 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
78
+ u
77
+ · · · + u + 1i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
9
=
u
8
+ u
6
+ u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
5
=
u
21
+ 4u
19
+ 9u
17
+ 12u
15
+ 12u
13
+ 10u
11
+ 9u
9
+ 6u
7
+ 3u
5
+ u
u
23
+ 5u
21
+ ··· + 2u
3
+ u
a
12
=
−u
11
− 2u
9
− 2u
7
+ u
3
u
11
+ 3u
9
+ 4u
7
+ 3u
5
+ u
3
+ u
a
10
=
u
32
+ 7u
30
+ ··· + 2u
12
+ 1
−u
32
− 8u
30
+ ··· − 12u
8
− 4u
6
a
6
=
u
66
+ 15u
64
+ ··· + u
2
+ 1
−u
66
− 16u
64
+ ··· − 4u
8
+ u
2
a
11
=
u
55
+ 12u
53
+ ··· + 5u
7
+ 2u
3
u
57
+ 13u
55
+ ··· + 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
76
− 4u
75
+ ··· − 8u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
78
+ 37u
77
+ ··· + 3u + 1
c
2
, c
7
u
78
+ u
77
+ ··· + u + 1
c
3
u
78
− u
77
+ ··· − 711u + 185
c
4
u
78
+ u
77
+ ··· − 2625u + 2061
c
5
, c
6
, c
10
c
11
u
78
+ u
77
+ ··· + 3u + 1
c
8
u
78
+ 5u
77
+ ··· + 233u + 259
c
9
u
78
+ 21u
77
+ ··· + 3489u + 187
c
12
u
78
+ 9u
77
+ ··· + 1209u + 109
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
78
+ 9y
77
+ ··· + 7y + 1
c
2
, c
7
y
78
+ 37y
77
+ ··· + 3y + 1
c
3
y
78
− 19y
77
+ ··· − 1149321y + 34225
c
4
y
78
− 27y
77
+ ··· − 82982745y + 4247721
c
5
, c
6
, c
10
c
11
y
78
+ 89y
77
+ ··· + 3y + 1
c
8
y
78
+ 17y
77
+ ··· + 3777875y + 67081
c
9
y
78
− 11y
77
+ ··· − 42057y + 34969
c
12
y
78
+ 13y
77
+ ··· + 852607y + 11881
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.176293 + 1.038900I
7.18444 − 1.67019I 0
u = −0.176293 − 1.038900I
7.18444 + 1.67019I 0
u = 0.250777 + 1.048820I
−0.743942 + 0.717138I 0
u = 0.250777 − 1.048820I
−0.743942 − 0.717138I 0
u = 0.662622 + 0.622414I
11.08790 + 7.77112I 6.08947 − 6.15924I
u = 0.662622 − 0.622414I
11.08790 − 7.77112I 6.08947 + 6.15924I
u = −0.559696 + 0.953541I
2.23246 + 0.83804I 0
u = −0.559696 − 0.953541I
2.23246 − 0.83804I 0
u = −0.644680 + 0.615060I
3.22743 − 5.55791I 3.92619 + 8.07198I
u = −0.644680 − 0.615060I
3.22743 + 5.55791I 3.92619 − 8.07198I
u = 0.578452 + 0.948103I
10.12780 − 2.94206I 0
u = 0.578452 − 0.948103I
10.12780 + 2.94206I 0
u = 0.536776 + 0.978594I
0.45641 + 2.41567I 0
u = 0.536776 − 0.978594I
0.45641 − 2.41567I 0
u = −0.494251 + 0.718740I
6.27468 − 2.02964I 2.37275 + 3.90100I
u = −0.494251 − 0.718740I
6.27468 + 2.02964I 2.37275 − 3.90100I
u = 0.679159 + 0.543155I
12.45010 − 1.87294I 7.91266 + 0.01624I
u = 0.679159 − 0.543155I
12.45010 + 1.87294I 7.91266 − 0.01624I
u = −0.248111 + 1.111540I
−4.03405 + 1.38195I 0
u = −0.248111 − 1.111540I
−4.03405 − 1.38195I 0
u = 0.617031 + 0.594101I
1.58657 + 2.14626I 0.09014 − 3.07268I
u = 0.617031 − 0.594101I
1.58657 − 2.14626I 0.09014 + 3.07268I
u = 0.231696 + 1.122790I
−2.66484 − 4.98283I 0
u = 0.231696 − 1.122790I
−2.66484 + 4.98283I 0
u = −0.220916 + 1.130910I
5.04714 + 7.32940I 0
u = −0.220916 − 1.130910I
5.04714 − 7.32940I 0
u = −0.561401 + 1.006840I
2.94053 − 5.15801I 0
u = −0.561401 − 1.006840I
2.94053 + 5.15801I 0
u = −0.646275 + 0.544185I
4.30217 + 0.43226I 6.84123 − 1.12347I
u = −0.646275 − 0.544185I
4.30217 − 0.43226I 6.84123 + 1.12347I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.769714 + 0.344540I
9.68605 + 10.02470I 4.54215 − 5.70117I
u = −0.769714 − 0.344540I
9.68605 − 10.02470I 4.54215 + 5.70117I
u = −0.330134 + 1.108880I
−4.86646 − 1.53988I 0
u = −0.330134 − 1.108880I
−4.86646 + 1.53988I 0
u = 0.294729 + 1.123740I
0.529229 − 0.096011I 0
u = 0.294729 − 1.123740I
0.529229 + 0.096011I 0
u = −0.734635 + 0.397114I
11.73850 + 0.41110I 7.04886 − 0.50949I
u = −0.734635 − 0.397114I
11.73850 − 0.41110I 7.04886 + 0.50949I
u = 0.580481 + 1.010610I
11.07250 + 6.74859I 0
u = 0.580481 − 1.010610I
11.07250 − 6.74859I 0
u = 0.758266 + 0.340646I
1.87500 − 7.66822I 2.06566 + 7.50138I
u = 0.758266 − 0.340646I
1.87500 + 7.66822I 2.06566 − 7.50138I
u = 0.353897 + 1.116280I
−3.95417 + 5.01127I 0
u = 0.353897 − 1.116280I
−3.95417 − 5.01127I 0
u = −0.369372 + 1.126780I
3.43300 − 7.21980I 0
u = −0.369372 − 1.126780I
3.43300 + 7.21980I 0
u = −0.739839 + 0.336830I
0.35594 + 4.04760I −1.60822 − 2.48146I
u = −0.739839 − 0.336830I
0.35594 − 4.04760I −1.60822 + 2.48146I
u = 0.716830 + 0.371362I
3.50311 − 1.57237I 5.70399 − 0.07635I
u = 0.716830 − 0.371362I
3.50311 + 1.57237I 5.70399 + 0.07635I
u = −0.481533 + 1.115440I
4.18249 − 0.46735I 0
u = −0.481533 − 1.115440I
4.18249 + 0.46735I 0
u = 0.501259 + 1.108890I
−2.96742 + 2.54824I 0
u = 0.501259 − 1.108890I
−2.96742 − 2.54824I 0
u = −0.520445 + 1.111910I
−3.57499 − 5.99088I 0
u = −0.520445 − 1.111910I
−3.57499 + 5.99088I 0
u = 0.562822 + 1.102750I
1.36314 + 6.46891I 0
u = 0.562822 − 1.102750I
1.36314 − 6.46891I 0
u = −0.575491 + 1.096660I
9.68171 − 5.40055I 0
u = −0.575491 − 1.096660I
9.68171 + 5.40055I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.708462 + 0.274901I
4.60181 − 3.06647I 0.82477 + 2.99322I
u = 0.708462 − 0.274901I
4.60181 + 3.06647I 0.82477 − 2.99322I
u = 0.537436 + 1.123660I
2.16314 + 7.81098I 0
u = 0.537436 − 1.123660I
2.16314 − 7.81098I 0
u = −0.562734 + 1.119050I
−1.93087 − 8.99165I 0
u = −0.562734 − 1.119050I
−1.93087 + 8.99165I 0
u = 0.569133 + 1.122950I
−0.42249 + 12.68200I 0
u = 0.569133 − 1.122950I
−0.42249 − 12.68200I 0
u = −0.573825 + 1.125130I
7.3870 − 15.0855I 0
u = −0.573825 − 1.125130I
7.3870 + 15.0855I 0
u = −0.642804 + 0.250728I
−1.16696 + 1.46809I −3.90843 − 3.90329I
u = −0.642804 − 0.250728I
−1.16696 − 1.46809I −3.90843 + 3.90329I
u = 0.243242 + 0.611279I
−0.183340 + 1.154680I −2.33672 − 6.24031I
u = 0.243242 − 0.611279I
−0.183340 − 1.154680I −2.33672 + 6.24031I
u = −0.645299 + 0.117141I
6.90376 − 3.74824I 1.81576 + 2.40321I
u = −0.645299 − 0.117141I
6.90376 + 3.74824I 1.81576 − 2.40321I
u = 0.614376 + 0.171121I
−0.44935 + 1.76162I −1.34285 − 4.29386I
u = 0.614376 − 0.171121I
−0.44935 − 1.76162I −1.34285 + 4.29386I
7
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
78
+ 37u
77
+ ··· + 3u + 1
c
2
, c
7
u
78
+ u
77
+ ··· + u + 1
c
3
u
78
− u
77
+ ··· − 711u + 185
c
4
u
78
+ u
77
+ ··· − 2625u + 2061
c
5
, c
6
, c
10
c
11
u
78
+ u
77
+ ··· + 3u + 1
c
8
u
78
+ 5u
77
+ ··· + 233u + 259
c
9
u
78
+ 21u
77
+ ··· + 3489u + 187
c
12
u
78
+ 9u
77
+ ··· + 1209u + 109
8
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
78
+ 9y
77
+ ··· + 7y + 1
c
2
, c
7
y
78
+ 37y
77
+ ··· + 3y + 1
c
3
y
78
− 19y
77
+ ··· − 1149321y + 34225
c
4
y
78
− 27y
77
+ ··· − 82982745y + 4247721
c
5
, c
6
, c
10
c
11
y
78
+ 89y
77
+ ··· + 3y + 1
c
8
y
78
+ 17y
77
+ ··· + 3777875y + 67081
c
9
y
78
− 11y
77
+ ··· − 42057y + 34969
c
12
y
78
+ 13y
77
+ ··· + 852607y + 11881
9
