12a
0521
(K12a
0521
)
A knot diagram
1
Linearized knot diagam
3 7 8 9 11 1 2 4 6 12 5 10
Solving Sequence
2,8
7 3 4 9 5 1 6 10 12 11
c
7
c
2
c
3
c
8
c
4
c
1
c
6
c
9
c
12
c
11
c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
56
− u
55
+ ··· + 2u
3
− 1i
* 1 irreducible components of dim
C
= 0, with total 56 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
56
− u
55
+ · · · + 2u
3
− 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
9
=
−u
6
− u
4
+ 1
u
6
+ 2u
4
+ u
2
a
5
=
u
9
+ 2u
7
+ u
5
− 2u
3
− u
−u
9
− 3u
7
− 3u
5
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
6
=
−u
6
− u
4
+ 1
−u
8
− 2u
6
− 2u
4
a
10
=
−u
20
− 5u
18
− 11u
16
− 10u
14
+ 2u
12
+ 13u
10
+ 9u
8
− 2u
6
− 5u
4
− u
2
+ 1
−u
22
− 6u
20
− 17u
18
− 26u
16
− 20u
14
+ 13u
10
+ 10u
8
+ 3u
6
+ 2u
4
+ u
2
a
12
=
u
37
+ 10u
35
+ ··· + 2u
3
− u
u
39
+ 11u
37
+ ··· − u
5
+ u
a
11
=
−u
54
− 15u
52
+ ··· − 7u
4
+ 1
−u
55
+ u
54
+ ··· + 2u
3
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
54
− 4u
53
+ ··· − 4u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
56
+ 33u
55
+ ··· − 10u
2
+ 1
c
2
, c
7
u
56
+ u
55
+ ··· − 2u
3
− 1
c
3
, c
4
, c
6
c
8
u
56
− u
55
+ ··· − 14u − 1
c
5
, c
11
u
56
− u
55
+ ··· − 2u − 1
c
9
u
56
+ 5u
55
+ ··· − 4088u − 1767
c
10
, c
12
u
56
− 17u
55
+ ··· + 10u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
56
− 19y
55
+ ··· − 20y + 1
c
2
, c
7
y
56
+ 33y
55
+ ··· − 10y
2
+ 1
c
3
, c
4
, c
6
c
8
y
56
− 71y
55
+ ··· + 96y + 1
c
5
, c
11
y
56
+ 17y
55
+ ··· + 10y
2
+ 1
c
9
y
56
− 27y
55
+ ··· − 19630828y + 3122289
c
10
, c
12
y
56
+ 45y
55
+ ··· + 20y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.405867 + 0.888923I
−1.83499 − 1.34944I −6.35701 + 4.05456I
u = −0.405867 − 0.888923I
−1.83499 + 1.34944I −6.35701 − 4.05456I
u = −0.021918 + 1.036680I
−4.65561 − 2.80215I −13.44960 + 3.08888I
u = −0.021918 − 1.036680I
−4.65561 + 2.80215I −13.44960 − 3.08888I
u = 0.437414 + 0.848600I
−1.27504 + 6.58005I −4.46879 − 9.66669I
u = 0.437414 − 0.848600I
−1.27504 − 6.58005I −4.46879 + 9.66669I
u = −0.917081 + 0.022864I
−13.20330 + 2.37614I −9.77936 − 0.20355I
u = −0.917081 − 0.022864I
−13.20330 − 2.37614I −9.77936 + 0.20355I
u = 0.915655 + 0.028828I
−12.4108 − 8.4052I −8.44929 + 5.11391I
u = 0.915655 − 0.028828I
−12.4108 + 8.4052I −8.44929 − 5.11391I
u = −0.307186 + 1.049610I
−1.50462 − 0.72055I −7.04900 + 0.I
u = −0.307186 − 1.049610I
−1.50462 + 0.72055I −7.04900 + 0.I
u = −0.904607
−9.25956 −9.91870
u = 0.894317 + 0.015919I
−5.91089 − 3.18842I −3.80019 + 3.52000I
u = 0.894317 − 0.015919I
−5.91089 + 3.18842I −3.80019 − 3.52000I
u = −0.435609 + 1.059800I
−0.58121 − 5.72992I −4.00000 + 8.34586I
u = −0.435609 − 1.059800I
−0.58121 + 5.72992I −4.00000 − 8.34586I
u = 0.386574 + 1.089770I
−3.86140 + 3.45773I −12.70884 − 4.98853I
u = 0.386574 − 1.089770I
−3.86140 − 3.45773I −12.70884 + 4.98853I
u = 0.398205 + 0.739047I
2.87979 + 1.78352I 2.98966 − 4.90547I
u = 0.398205 − 0.739047I
2.87979 − 1.78352I 2.98966 + 4.90547I
u = −0.196934 + 0.792679I
−0.547592 − 1.072840I −7.44811 + 6.01442I
u = −0.196934 − 0.792679I
−0.547592 + 1.072840I −7.44811 − 6.01442I
u = −0.305708 + 1.152470I
−7.27525 + 3.27832I 0
u = −0.305708 − 1.152470I
−7.27525 − 3.27832I 0
u = 0.325359 + 1.152680I
−7.79706 + 2.53101I 0
u = 0.325359 − 1.152680I
−7.79706 − 2.53101I 0
u = −0.466232 + 1.103460I
−6.06698 − 10.74040I 0
u = −0.466232 − 1.103460I
−6.06698 + 10.74040I 0
u = 0.453460 + 1.112820I
−6.83086 + 5.00850I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.453460 − 1.112820I
−6.83086 − 5.00850I 0
u = 0.421866 + 0.566053I
−0.52914 − 2.83643I −2.00972 + 2.39215I
u = 0.421866 − 0.566053I
−0.52914 + 2.83643I −2.00972 − 2.39215I
u = −0.661353 + 0.178127I
−3.45247 + 6.47746I −6.46031 − 6.34159I
u = −0.661353 − 0.178127I
−3.45247 − 6.47746I −6.46031 + 6.34159I
u = 0.662306 + 0.142602I
−4.09011 − 0.81649I −8.10360 + 0.92141I
u = 0.662306 − 0.142602I
−4.09011 + 0.81649I −8.10360 − 0.92141I
u = 0.460898 + 1.264040I
−9.81343 + 1.61572I 0
u = 0.460898 − 1.264040I
−9.81343 − 1.61572I 0
u = 0.477987 + 1.259180I
−9.68771 + 8.08384I 0
u = 0.477987 − 1.259180I
−9.68771 − 8.08384I 0
u = −0.471428 + 1.267380I
−13.12820 − 4.89278I 0
u = −0.471428 − 1.267380I
−13.12820 + 4.89278I 0
u = 0.456440 + 1.279200I
−16.4343 − 3.5507I 0
u = 0.456440 − 1.279200I
−16.4343 + 3.5507I 0
u = 0.489034 + 1.267150I
−16.1895 + 13.4198I 0
u = 0.489034 − 1.267150I
−16.1895 − 13.4198I 0
u = −0.486244 + 1.269270I
−17.0105 − 7.3815I 0
u = −0.486244 − 1.269270I
−17.0105 + 7.3815I 0
u = −0.460358 + 1.278950I
−17.2053 − 2.5025I 0
u = −0.460358 − 1.278950I
−17.2053 + 2.5025I 0
u = −0.418964 + 0.467128I
−0.73474 − 2.24620I −2.43482 + 3.10244I
u = −0.418964 − 0.467128I
−0.73474 + 2.24620I −2.43482 − 3.10244I
u = −0.530935 + 0.216533I
1.71161 + 1.86206I 0.40956 − 4.60344I
u = −0.530935 − 0.216533I
1.71161 − 1.86206I 0.40956 + 4.60344I
u = 0.517215
−1.03672 −9.70580
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
56
+ 33u
55
+ ··· − 10u
2
+ 1
c
2
, c
7
u
56
+ u
55
+ ··· − 2u
3
− 1
c
3
, c
4
, c
6
c
8
u
56
− u
55
+ ··· − 14u − 1
c
5
, c
11
u
56
− u
55
+ ··· − 2u − 1
c
9
u
56
+ 5u
55
+ ··· − 4088u − 1767
c
10
, c
12
u
56
− 17u
55
+ ··· + 10u
2
+ 1
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
56
− 19y
55
+ ··· − 20y + 1
c
2
, c
7
y
56
+ 33y
55
+ ··· − 10y
2
+ 1
c
3
, c
4
, c
6
c
8
y
56
− 71y
55
+ ··· + 96y + 1
c
5
, c
11
y
56
+ 17y
55
+ ··· + 10y
2
+ 1
c
9
y
56
− 27y
55
+ ··· − 19630828y + 3122289
c
10
, c
12
y
56
+ 45y
55
+ ··· + 20y + 1
8
