12a
0842
(K12a
0842
)
A knot diagram
1
Linearized knot diagam
4 5 9 2 10 12 11 3 1 8 7 6
Solving Sequence
7,11 3,8
9 4 12 6 1 10 5 2
c
7
c
8
c
3
c
11
c
6
c
12
c
10
c
5
c
2
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
48
− 3u
47
+ ··· + b − 1, u
49
+ 32u
47
+ ··· + a + 2, u
50
− 2u
49
+ ··· + 6u − 1i
I
u
2
= hu
3
+ u
2
+ b + 2u + 1, −u
4
− 3u
2
+ a − 1, u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1i
* 2 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
= h2u
48
−3u
47
+· · ·+b−1, u
49
+32u
47
+· · ·+a+2, u
50
−2u
49
+· · ·+6u−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
−u
49
− 32u
47
+ ··· − 3u − 2
−2u
48
+ 3u
47
+ ··· − 4u + 1
a
8
=
1
−u
2
a
9
=
−u
9
− 6u
7
− 11u
5
− 6u
3
− u
−u
9
− 5u
7
− 7u
5
− 2u
3
+ u
a
4
=
−u
49
+ 2u
48
+ ··· + 2u − 3
u
47
− 2u
46
+ ··· − 3u + 1
a
12
=
u
u
a
6
=
u
2
+ 1
u
2
a
1
=
u
3
+ 2u
u
3
+ u
a
10
=
−u
u
3
+ u
a
5
=
−u
6
− 3u
4
+ 1
u
8
+ 4u
6
+ 4u
4
+ 2u
2
a
2
=
−u
49
+ u
48
+ ··· + u − 2
−u
48
+ 2u
47
+ ··· − 3u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
49
+ 2u
48
+ ··· − 3u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
50
− 6u
49
+ ··· + 4u − 1
c
3
, c
8
u
50
+ u
49
+ ··· + 64u + 32
c
5
u
50
+ 2u
49
+ ··· − 3538u − 1049
c
6
, c
7
, c
10
c
11
, c
12
u
50
+ 2u
49
+ ··· − 6u − 1
c
9
u
50
− 6u
49
+ ··· − 2094u + 279
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
50
− 50y
49
+ ··· + 16y + 1
c
3
, c
8
y
50
− 33y
49
+ ··· − 7680y + 1024
c
5
y
50
+ 18y
49
+ ··· − 9892846y + 1100401
c
6
, c
7
, c
10
c
11
, c
12
y
50
+ 66y
49
+ ··· − 22y + 1
c
9
y
50
+ 30y
49
+ ··· − 3350862y + 77841
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.241502 + 0.933682I
a = 0.521403 + 0.320826I
b = 0.450327 − 0.085914I
−2.13050 − 2.69655I 0.94523 + 4.45521I
u = −0.241502 − 0.933682I
a = 0.521403 − 0.320826I
b = 0.450327 + 0.085914I
−2.13050 + 2.69655I 0.94523 − 4.45521I
u = 0.224098 + 1.034240I
a = 2.13283 + 0.91465I
b = 1.44365 − 0.08763I
−5.96928 + 1.26789I 0
u = 0.224098 − 1.034240I
a = 2.13283 − 0.91465I
b = 1.44365 + 0.08763I
−5.96928 − 1.26789I 0
u = 0.286948 + 1.018960I
a = −2.14547 − 1.33637I
b = −1.63676 − 0.38283I
−5.25575 + 6.41850I 0
u = 0.286948 − 1.018960I
a = −2.14547 + 1.33637I
b = −1.63676 + 0.38283I
−5.25575 − 6.41850I 0
u = −0.260970 + 1.033400I
a = −0.934867 − 0.520605I
b = −0.824900 + 0.201401I
−7.72397 − 3.91870I 0
u = −0.260970 − 1.033400I
a = −0.934867 + 0.520605I
b = −0.824900 − 0.201401I
−7.72397 + 3.91870I 0
u = 0.331404 + 1.033160I
a = 1.97777 + 1.46824I
b = 1.50104 + 0.67230I
−11.6778 + 10.5158I 0
u = 0.331404 − 1.033160I
a = 1.97777 − 1.46824I
b = 1.50104 − 0.67230I
−11.6778 − 10.5158I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.356837 + 0.800466I
a = 0.088289 − 1.106300I
b = −0.427389 − 0.642251I
−5.40920 − 3.17023I −6.24629 + 4.79557I
u = −0.356837 − 0.800466I
a = 0.088289 + 1.106300I
b = −0.427389 + 0.642251I
−5.40920 + 3.17023I −6.24629 − 4.79557I
u = 0.192571 + 1.121550I
a = −1.64217 − 0.83532I
b = −0.897289 + 0.086810I
−13.33180 − 1.81050I 0
u = 0.192571 − 1.121550I
a = −1.64217 + 0.83532I
b = −0.897289 − 0.086810I
−13.33180 + 1.81050I 0
u = 0.063290 + 0.852837I
a = 1.48467 − 0.76548I
b = 0.436324 − 1.223550I
−3.68295 + 0.97203I −6.12133 + 0.71322I
u = 0.063290 − 0.852837I
a = 1.48467 + 0.76548I
b = 0.436324 + 1.223550I
−3.68295 − 0.97203I −6.12133 − 0.71322I
u = −0.168142 + 0.755805I
a = −0.665021 + 0.730022I
b = 0.037081 + 0.681718I
−0.89055 − 1.70206I 0.92164 + 5.53470I
u = −0.168142 − 0.755805I
a = −0.665021 − 0.730022I
b = 0.037081 − 0.681718I
−0.89055 + 1.70206I 0.92164 − 5.53470I
u = 0.473458 + 0.447921I
a = −0.132112 + 0.553984I
b = −1.136560 − 0.435330I
−8.33174 − 4.01114I −4.65273 − 0.13384I
u = 0.473458 − 0.447921I
a = −0.132112 − 0.553984I
b = −1.136560 + 0.435330I
−8.33174 + 4.01114I −4.65273 + 0.13384I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.567490 + 0.251389I
a = −0.16497 − 1.91694I
b = 0.866292 − 0.017602I
−7.69762 + 7.46027I −2.75082 − 6.53209I
u = 0.567490 − 0.251389I
a = −0.16497 + 1.91694I
b = 0.866292 + 0.017602I
−7.69762 − 7.46027I −2.75082 + 6.53209I
u = −0.575142
a = 1.40804
b = 0.628444
−2.98861 −1.42680
u = 0.494918 + 0.244793I
a = 0.62020 + 1.67737I
b = −0.808413 − 0.141535I
−1.34530 + 3.74735I −0.07289 − 7.40976I
u = 0.494918 − 0.244793I
a = 0.62020 − 1.67737I
b = −0.808413 + 0.141535I
−1.34530 − 3.74735I −0.07289 + 7.40976I
u = −0.457183 + 0.286466I
a = 1.10902 − 1.15755I
b = 0.256004 − 0.666486I
−3.64157 − 1.46138I −1.93416 + 4.24780I
u = −0.457183 − 0.286466I
a = 1.10902 + 1.15755I
b = 0.256004 + 0.666486I
−3.64157 + 1.46138I −1.93416 − 4.24780I
u = 0.394100 + 0.341086I
a = −0.674932 − 0.693216I
b = 0.860383 + 0.378414I
−1.76393 − 0.83267I −2.37494 − 0.63227I
u = 0.394100 − 0.341086I
a = −0.674932 + 0.693216I
b = 0.860383 − 0.378414I
−1.76393 + 0.83267I −2.37494 + 0.63227I
u = −0.425320 + 0.091898I
a = −0.743778 + 0.682945I
b = −0.228701 + 0.291111I
1.018590 − 0.418472I 8.77405 + 2.00502I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.425320 − 0.091898I
a = −0.743778 − 0.682945I
b = −0.228701 − 0.291111I
1.018590 + 0.418472I 8.77405 − 2.00502I
u = −0.06654 + 1.65140I
a = −0.502139 + 0.521102I
b = −0.41662 + 1.56312I
−13.8965 − 4.6518I 0
u = −0.06654 − 1.65140I
a = −0.502139 − 0.521102I
b = −0.41662 − 1.56312I
−13.8965 + 4.6518I 0
u = −0.02318 + 1.67172I
a = 0.751552 − 0.327513I
b = 1.43271 − 1.45403I
−9.55272 − 2.27084I 0
u = −0.02318 − 1.67172I
a = 0.751552 + 0.327513I
b = 1.43271 + 1.45403I
−9.55272 + 2.27084I 0
u = 0.01016 + 1.69219I
a = −1.348610 + 0.394636I
b = −3.15245 + 1.83141I
−12.79330 + 1.21246I 0
u = 0.01016 − 1.69219I
a = −1.348610 − 0.394636I
b = −3.15245 − 1.83141I
−12.79330 − 1.21246I 0
u = −0.06031 + 1.70334I
a = −0.294056 − 0.407025I
b = −0.897079 − 0.594388I
−11.49630 − 3.87362I 0
u = −0.06031 − 1.70334I
a = −0.294056 + 0.407025I
b = −0.897079 + 0.594388I
−11.49630 + 3.87362I 0
u = 0.07470 + 1.72501I
a = 2.31285 + 0.82742I
b = 5.82899 + 2.24727I
−15.0169 + 7.8885I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.07470 − 1.72501I
a = 2.31285 − 0.82742I
b = 5.82899 − 2.24727I
−15.0169 − 7.8885I 0
u = 0.05873 + 1.72830I
a = −2.35182 − 0.61319I
b = −5.90142 − 1.36111I
−15.8315 + 2.4299I 0
u = 0.05873 − 1.72830I
a = −2.35182 + 0.61319I
b = −5.90142 + 1.36111I
−15.8315 − 2.4299I 0
u = −0.06774 + 1.72877I
a = 0.495071 + 0.829683I
b = 1.48829 + 1.29304I
−17.5722 − 5.2629I 0
u = −0.06774 − 1.72877I
a = 0.495071 − 0.829683I
b = 1.48829 − 1.29304I
−17.5722 + 5.2629I 0
u = 0.08767 + 1.72869I
a = −2.17648 − 0.82127I
b = −5.31736 − 2.45060I
17.9962 + 12.2318I 0
u = 0.08767 − 1.72869I
a = −2.17648 + 0.82127I
b = −5.31736 + 2.45060I
17.9962 − 12.2318I 0
u = 0.04560 + 1.74772I
a = 2.09491 + 0.62616I
b = 4.94329 + 1.19565I
15.8438 − 0.8328I 0
u = 0.04560 − 1.74772I
a = 2.09491 − 0.62616I
b = 4.94329 − 1.19565I
15.8438 + 0.8328I 0
u = 0.220298
a = −3.03230
b = 0.572682
−1.27955 −10.7800
9
II.
I
u
2
= hu
3
+ u
2
+ b + 2u + 1, −u
4
− 3u
2
+ a − 1, u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
u
4
+ 3u
2
+ 1
−u
3
− u
2
− 2u − 1
a
8
=
1
−u
2
a
9
=
1
−u
2
a
4
=
u
4
+ 3u
2
+ 1
−u
3
− u
2
− 2u − 1
a
12
=
u
u
a
6
=
u
2
+ 1
u
2
a
1
=
u
3
+ 2u
u
3
+ u
a
10
=
−u
u
3
+ u
a
5
=
−u
3
− 2u
−u
3
− u
a
2
=
u
4
+ u
3
+ 3u
2
+ 2u + 1
−u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
4
+ 5u
3
+ 20u
2
+ 14u + 9
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
8
u
5
c
4
(u + 1)
5
c
5
, c
9
u
5
+ u
4
− u
2
+ u + 1
c
6
, c
7
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
10
, c
11
, c
12
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
8
y
5
c
5
, c
9
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
6
, c
7
, c
10
c
11
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.233677 + 0.885557I
a = −0.827780 − 0.637683I
b = −0.340036 − 0.807849I
−3.46474 − 2.21397I −4.37343 + 4.39306I
u = −0.233677 − 0.885557I
a = −0.827780 + 0.637683I
b = −0.340036 + 0.807849I
−3.46474 + 2.21397I −4.37343 − 4.39306I
u = −0.416284
a = 1.54991
b = −0.268586
−0.762751 6.42730
u = −0.05818 + 1.69128I
a = 0.552827 + 0.534136I
b = 1.47433 + 1.63485I
−12.60320 − 3.33174I −5.84024 + 1.26157I
u = −0.05818 − 1.69128I
a = 0.552827 − 0.534136I
b = 1.47433 − 1.63485I
−12.60320 + 3.33174I −5.84024 − 1.26157I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
5
)(u
50
− 6u
49
+ ··· + 4u − 1)
c
3
, c
8
u
5
(u
50
+ u
49
+ ··· + 64u + 32)
c
4
((u + 1)
5
)(u
50
− 6u
49
+ ··· + 4u − 1)
c
5
(u
5
+ u
4
− u
2
+ u + 1)(u
50
+ 2u
49
+ ··· − 3538u − 1049)
c
6
, c
7
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
50
+ 2u
49
+ ··· − 6u − 1)
c
9
(u
5
+ u
4
− u
2
+ u + 1)(u
50
− 6u
49
+ ··· − 2094u + 279)
c
10
, c
11
, c
12
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
50
+ 2u
49
+ ··· − 6u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
5
)(y
50
− 50y
49
+ ··· + 16y + 1)
c
3
, c
8
y
5
(y
50
− 33y
49
+ ··· − 7680y + 1024)
c
5
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)
· (y
50
+ 18y
49
+ ··· − 9892846y + 1100401)
c
6
, c
7
, c
10
c
11
, c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
50
+ 66y
49
+ ··· − 22y + 1)
c
9
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
50
+ 30y
49
+ ··· − 3350862y + 77841)
15
