12a
0085
(K12a
0085
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 9 11 12 4 1 6 7 10
Solving Sequence
7,12 4,8
9 3 11 6 5 2 10 1
c
7
c
8
c
3
c
11
c
6
c
5
c
2
c
10
c
12
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
77
+ 41u
75
+ ··· + 3u
2
+ b, u
76
− 41u
74
+ ··· + a + 1, u
78
− 2u
77
+ ··· + u + 1i
I
u
2
= h−u
5
+ 2u
3
− u
2
+ b + 1, u
5
− 3u
3
+ u
2
+ a + 2u − 2, u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 84 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
= h−u
77
+41u
75
+· · ·+3 u
2
+b, u
76
−41u
74
+· · ·+ a + 1, u
78
−2u
77
+· · ·+ u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−u
76
+ 41u
74
+ ··· + u − 1
u
77
− 41u
75
+ ··· − 3u
3
− 3u
2
a
8
=
1
−u
2
a
9
=
u
11
− 6u
9
+ 12u
7
− 8u
5
+ u
3
− 2u
−u
11
+ 5u
9
− 8u
7
+ 5u
5
− 3u
3
+ u
a
3
=
u
77
− u
76
+ ··· + 8u
2
− 2
−2u
77
+ 2u
76
+ ··· + 3u + 2
a
11
=
−u
u
a
6
=
−u
2
+ 1
u
2
a
5
=
u
20
− 11u
18
+ ··· − 3u
2
+ 1
−u
20
+ 10u
18
+ ··· − 5u
4
+ 2u
2
a
2
=
u
77
− u
76
+ ··· + u − 1
−u
77
+ u
76
+ ··· + 2u + 1
a
10
=
u
3
− 2u
−u
3
+ u
a
1
=
u
7
− 4u
5
+ 4u
3
−u
7
+ 3u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8u
77
− 9u
76
+ ··· − 18u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
78
+ 37u
77
+ ··· + 114u + 1
c
2
, c
4
u
78
− 7u
77
+ ··· − 14u + 1
c
3
, c
8
u
78
+ u
77
+ ··· − 800u
2
+ 64
c
5
u
78
− 2u
77
+ ··· − 14229u + 4721
c
6
, c
7
, c
10
c
11
u
78
+ 2u
77
+ ··· − u + 1
c
9
, c
12
u
78
+ 14u
77
+ ··· + 2121u + 207
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
78
+ 15y
77
+ ··· − 9330y + 1
c
2
, c
4
y
78
− 37y
77
+ ··· − 114y + 1
c
3
, c
8
y
78
− 39y
77
+ ··· − 102400y + 4096
c
5
y
78
− 14y
77
+ ··· − 388528493y + 22287841
c
6
, c
7
, c
10
c
11
y
78
− 86y
77
+ ··· − y + 1
c
9
, c
12
y
78
+ 46y
77
+ ··· + 505791y + 42849
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.599318 + 0.584746I
a = −1.27127 − 2.26281I
b = 0.406911 − 0.172691I
−0.78175 − 12.54030I 6.09746 + 10.28095I
u = −0.599318 − 0.584746I
a = −1.27127 + 2.26281I
b = 0.406911 + 0.172691I
−0.78175 + 12.54030I 6.09746 − 10.28095I
u = −0.599850 + 0.564094I
a = 1.27636 + 2.11108I
b = −0.278346 + 0.277845I
1.55983 − 6.97653I 9.24019 + 6.64171I
u = −0.599850 − 0.564094I
a = 1.27636 − 2.11108I
b = −0.278346 − 0.277845I
1.55983 + 6.97653I 9.24019 − 6.64171I
u = 0.802841 + 0.168457I
a = −2.26912 + 0.63093I
b = 0.357877 − 0.320430I
3.97087 + 7.20217I 11.97302 − 7.33202I
u = 0.802841 − 0.168457I
a = −2.26912 − 0.63093I
b = 0.357877 + 0.320430I
3.97087 − 7.20217I 11.97302 + 7.33202I
u = 0.805439 + 0.096765I
a = 2.36172 − 0.39872I
b = −0.359916 + 0.190669I
5.68267 + 1.74998I 15.1419 − 1.8440I
u = 0.805439 − 0.096765I
a = 2.36172 + 0.39872I
b = −0.359916 − 0.190669I
5.68267 − 1.74998I 15.1419 + 1.8440I
u = 0.576717 + 0.560895I
a = 1.345380 − 0.149355I
b = −0.580995 + 0.149181I
−3.14364 + 6.35997I 4.39196 − 7.72214I
u = 0.576717 − 0.560895I
a = 1.345380 + 0.149355I
b = −0.580995 − 0.149181I
−3.14364 − 6.35997I 4.39196 + 7.72214I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.520959 + 0.606760I
a = 0.872173 + 0.421611I
b = −0.423627 − 0.168563I
−4.39476 + 0.68401I 5.05755 + 1.70512I
u = 0.520959 − 0.606760I
a = 0.872173 − 0.421611I
b = −0.423627 + 0.168563I
−4.39476 − 0.68401I 5.05755 − 1.70512I
u = −0.558404 + 0.553380I
a = −1.70602 − 1.99005I
b = 0.351192 − 0.673255I
−4.00256 − 3.82936I 4.97815 + 7.00642I
u = −0.558404 − 0.553380I
a = −1.70602 + 1.99005I
b = 0.351192 + 0.673255I
−4.00256 + 3.82936I 4.97815 − 7.00642I
u = −0.616947 + 0.471605I
a = 0.92409 + 1.52711I
b = 0.200273 + 0.380587I
3.37693 − 3.48998I 11.29099 + 6.45217I
u = −0.616947 − 0.471605I
a = 0.92409 − 1.52711I
b = 0.200273 − 0.380587I
3.37693 + 3.48998I 11.29099 − 6.45217I
u = 0.463830 + 0.615489I
a = −0.400498 − 0.952066I
b = 0.193565 + 0.455914I
−4.56309 + 3.47022I 4.07696 − 8.07073I
u = 0.463830 − 0.615489I
a = −0.400498 + 0.952066I
b = 0.193565 − 0.455914I
−4.56309 − 3.47022I 4.07696 + 8.07073I
u = −0.640817 + 0.405593I
a = −0.62536 − 1.32558I
b = −0.339137 − 0.372642I
2.45756 + 1.82034I 10.30646 + 0.66101I
u = −0.640817 − 0.405593I
a = −0.62536 + 1.32558I
b = −0.339137 + 0.372642I
2.45756 − 1.82034I 10.30646 − 0.66101I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.541710 + 0.519985I
a = −1.108600 + 0.518660I
b = 0.443108 − 0.259777I
−1.72128 + 2.09820I 6.19910 − 2.97697I
u = 0.541710 − 0.519985I
a = −1.108600 − 0.518660I
b = 0.443108 + 0.259777I
−1.72128 − 2.09820I 6.19910 + 2.97697I
u = −0.364533 + 0.621473I
a = −0.308159 − 0.767887I
b = 1.44780 + 0.32955I
−1.47143 + 8.43896I 4.23505 − 4.36846I
u = −0.364533 − 0.621473I
a = −0.308159 + 0.767887I
b = 1.44780 − 0.32955I
−1.47143 − 8.43896I 4.23505 + 4.36846I
u = −0.710687 + 0.078003I
a = −0.060076 − 0.763572I
b = −0.130706 − 0.534050I
0.94742 − 1.90115I 11.49371 + 4.32216I
u = −0.710687 − 0.078003I
a = −0.060076 + 0.763572I
b = −0.130706 + 0.534050I
0.94742 + 1.90115I 11.49371 − 4.32216I
u = −0.408883 + 0.562647I
a = 0.668680 − 0.785887I
b = 1.29216 + 0.60980I
−4.44211 − 0.01380I 2.92184 − 0.24930I
u = −0.408883 − 0.562647I
a = 0.668680 + 0.785887I
b = 1.29216 − 0.60980I
−4.44211 + 0.01380I 2.92184 + 0.24930I
u = 0.383552 + 0.577713I
a = 0.12013 − 1.41029I
b = −0.127003 + 0.649722I
−3.70836 − 2.45187I 2.28902 + 1.10557I
u = 0.383552 − 0.577713I
a = 0.12013 + 1.41029I
b = −0.127003 − 0.649722I
−3.70836 + 2.45187I 2.28902 − 1.10557I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.350709 + 0.593744I
a = 0.107351 + 0.504864I
b = −1.343700 − 0.316084I
0.83358 + 3.01735I 7.25837 − 0.49247I
u = −0.350709 − 0.593744I
a = 0.107351 − 0.504864I
b = −1.343700 + 0.316084I
0.83358 − 3.01735I 7.25837 + 0.49247I
u = 0.442910 + 0.518440I
a = −0.521192 + 1.076400I
b = 0.241180 − 0.446445I
−2.02036 + 1.50879I 4.97629 − 5.09309I
u = 0.442910 − 0.518440I
a = −0.521192 − 1.076400I
b = 0.241180 + 0.446445I
−2.02036 − 1.50879I 4.97629 + 5.09309I
u = 0.645827
a = −3.49684
b = 0.752757
−0.308788 16.2550
u = 1.42939 + 0.12884I
a = −0.715223 − 0.199859I
b = 0.242567 + 1.308940I
4.19828 − 5.82838I 0
u = 1.42939 − 0.12884I
a = −0.715223 + 0.199859I
b = 0.242567 − 1.308940I
4.19828 + 5.82838I 0
u = 1.44778 + 0.09666I
a = 0.890808 + 0.121968I
b = −0.777252 − 0.958438I
6.48020 − 0.70488I 0
u = 1.44778 − 0.09666I
a = 0.890808 − 0.121968I
b = −0.777252 + 0.958438I
6.48020 + 0.70488I 0
u = −0.219534 + 0.501040I
a = 0.048525 − 0.591352I
b = −0.992560 + 0.039651I
2.27210 + 0.14195I 7.85107 + 0.40000I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.219534 − 0.501040I
a = 0.048525 + 0.591352I
b = −0.992560 − 0.039651I
2.27210 − 0.14195I 7.85107 − 0.40000I
u = −0.128219 + 0.511830I
a = −0.235116 + 0.964407I
b = 0.956059 − 0.264958I
0.94305 − 4.90661I 4.88398 + 5.62400I
u = −0.128219 − 0.511830I
a = −0.235116 − 0.964407I
b = 0.956059 + 0.264958I
0.94305 + 4.90661I 4.88398 − 5.62400I
u = −1.47226 + 0.12075I
a = −0.646554 − 0.603418I
b = 1.07448 + 1.96891I
2.26225 + 0.10898I 0
u = −1.47226 − 0.12075I
a = −0.646554 + 0.603418I
b = 1.07448 − 1.96891I
2.26225 − 0.10898I 0
u = 1.48782 + 0.13055I
a = −1.266260 − 0.496751I
b = 1.79248 + 2.09730I
1.73920 + 2.36195I 0
u = 1.48782 − 0.13055I
a = −1.266260 + 0.496751I
b = 1.79248 − 2.09730I
1.73920 − 2.36195I 0
u = −1.49302 + 0.17302I
a = −0.365143 − 0.576564I
b = 0.27170 + 1.51342I
1.82198 − 6.26698I 0
u = −1.49302 − 0.17302I
a = −0.365143 + 0.576564I
b = 0.27170 − 1.51342I
1.82198 + 6.26698I 0
u = 1.51505
a = 0.596370
b = −0.787410
7.16001 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.51198 + 0.13240I
a = 0.718360 + 0.295013I
b = −1.39217 − 1.12726I
4.46397 − 3.74584I 0
u = −1.51198 − 0.13240I
a = 0.718360 − 0.295013I
b = −1.39217 + 1.12726I
4.46397 + 3.74584I 0
u = −1.52531 + 0.18019I
a = −0.095957 + 0.602649I
b = 0.632159 − 1.131610I
2.35354 − 3.51433I 0
u = −1.52531 − 0.18019I
a = −0.095957 − 0.602649I
b = 0.632159 + 1.131610I
2.35354 + 3.51433I 0
u = −1.54784 + 0.15010I
a = 0.800421 − 0.348125I
b = −1.84286 + 0.24135I
5.27378 − 4.50564I 0
u = −1.54784 − 0.15010I
a = 0.800421 + 0.348125I
b = −1.84286 − 0.24135I
5.27378 + 4.50564I 0
u = 1.54914 + 0.16225I
a = 2.40899 − 1.35253I
b = −5.46000 + 2.34055I
3.02625 + 6.41510I 0
u = 1.54914 − 0.16225I
a = 2.40899 + 1.35253I
b = −5.46000 − 2.34055I
3.02625 − 6.41510I 0
u = −1.55516 + 0.16682I
a = −0.698192 + 0.656632I
b = 1.77866 − 0.87693I
3.96996 − 9.00544I 0
u = −1.55516 − 0.16682I
a = −0.698192 − 0.656632I
b = 1.77866 + 0.87693I
3.96996 + 9.00544I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.56285 + 0.17763I
a = 2.12209 − 1.39087I
b = −4.66821 + 2.86534I
6.4302 + 15.3326I 0
u = 1.56285 − 0.17763I
a = 2.12209 + 1.39087I
b = −4.66821 − 2.86534I
6.4302 − 15.3326I 0
u = 1.56376 + 0.16947I
a = −2.16477 + 1.35365I
b = 4.71147 − 2.63450I
8.79025 + 9.66014I 0
u = 1.56376 − 0.16947I
a = −2.16477 − 1.35365I
b = 4.71147 + 2.63450I
8.79025 − 9.66014I 0
u = −1.57499
a = 3.21218
b = −7.08625
7.27373 0
u = 1.57066 + 0.11845I
a = 1.72662 − 1.03255I
b = −3.35704 + 1.54534I
9.89816 + 0.10218I 0
u = 1.57066 − 0.11845I
a = 1.72662 + 1.03255I
b = −3.35704 − 1.54534I
9.89816 − 0.10218I 0
u = 1.56925 + 0.13763I
a = −1.97809 + 1.10921I
b = 4.00616 − 1.77495I
10.73110 + 5.71945I 0
u = 1.56925 − 0.13763I
a = −1.97809 − 1.10921I
b = 4.00616 + 1.77495I
10.73110 − 5.71945I 0
u = −0.417322
a = −0.507402
b = −0.228914
0.609628 16.4390
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.58333 + 0.01332I
a = 0.212622 − 1.085020I
b = −0.36367 + 1.69619I
8.73171 + 2.18417I 0
u = 1.58333 − 0.01332I
a = 0.212622 + 1.085020I
b = −0.36367 − 1.69619I
8.73171 − 2.18417I 0
u = −1.60086 + 0.01874I
a = −2.81392 − 0.07195I
b = 5.91931 + 0.33569I
13.82740 − 2.12508I 0
u = −1.60086 − 0.01874I
a = −2.81392 + 0.07195I
b = 5.91931 − 0.33569I
13.82740 + 2.12508I 0
u = −1.60129 + 0.03305I
a = 2.69061 + 0.07587I
b = −5.63287 − 0.46743I
12.1159 − 7.8628I 0
u = −1.60129 − 0.03305I
a = 2.69061 − 0.07587I
b = −5.63287 + 0.46743I
12.1159 + 7.8628I 0
u = 0.119394 + 0.283485I
a = 0.05243 + 2.15878I
b = 0.425883 − 0.450283I
−1.64524 + 0.68396I −1.89365 − 2.07297I
u = 0.119394 − 0.283485I
a = 0.05243 − 2.15878I
b = 0.425883 + 0.450283I
−1.64524 − 0.68396I −1.89365 + 2.07297I
12
II. I
u
2
=
h−u
5
+2u
3
−u
2
+b+1, u
5
−3u
3
+u
2
+a+2u−2, u
6
+u
5
−3u
4
−2u
3
+2u
2
−u−1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−u
5
+ 3u
3
− u
2
− 2u + 2
u
5
− 2u
3
+ u
2
− 1
a
8
=
1
−u
2
a
9
=
1
−u
2
a
3
=
−u
5
+ 3u
3
− u
2
− 2u + 2
u
5
− 2u
3
+ u
2
− 1
a
11
=
−u
u
a
6
=
−u
2
+ 1
u
2
a
5
=
u
4
− 3u
2
+ 1
u
5
− u
4
− 2u
3
+ 3u
2
− u − 1
a
2
=
−u
5
− u
4
+ 3u
3
+ 2u
2
− 2u + 1
u
4
− 2u
2
+ u
a
10
=
u
3
− 2u
−u
3
+ u
a
1
=
−u
4
+ 3u
2
− 1
−u
5
+ u
4
+ 2u
3
− 3u
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
5
+ u
4
− 6u
3
− u
2
− 2u + 2
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
8
u
6
c
4
(u + 1)
6
c
5
, c
9
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
6
, c
7
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
10
, c
11
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
12
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
8
y
6
c
5
, c
9
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
c
6
, c
7
, c
10
c
11
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.493180 + 0.575288I
a = 0.089969 − 0.799962I
b = −0.446039 − 0.121233I
−4.60518 + 1.97241I 2.71215 − 3.88360I
u = 0.493180 − 0.575288I
a = 0.089969 + 0.799962I
b = −0.446039 + 0.121233I
−4.60518 − 1.97241I 2.71215 + 3.88360I
u = −0.483672
a = 2.42043
b = −0.566232
−0.906083 3.38760
u = −1.52087 + 0.16310I
a = −0.227586 − 0.710576I
b = 0.87287 + 1.51178I
2.05064 − 4.59213I 6.49628 + 3.92496I
u = −1.52087 − 0.16310I
a = −0.227586 + 0.710576I
b = 0.87287 − 1.51178I
2.05064 + 4.59213I 6.49628 − 3.92496I
u = 1.53904
a = −1.14519
b = 2.71257
6.01515 6.19550
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
78
+ 37u
77
+ ··· + 114u + 1)
c
2
((u − 1)
6
)(u
78
− 7u
77
+ ··· − 14u + 1)
c
3
, c
8
u
6
(u
78
+ u
77
+ ··· − 800u
2
+ 64)
c
4
((u + 1)
6
)(u
78
− 7u
77
+ ··· − 14u + 1)
c
5
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
78
− 2u
77
+ ··· − 14229u + 4721)
c
6
, c
7
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
78
+ 2u
77
+ ··· − u + 1)
c
9
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
78
+ 14u
77
+ ··· + 2121u + 207)
c
10
, c
11
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
78
+ 2u
77
+ ··· − u + 1)
c
12
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
78
+ 14u
77
+ ··· + 2121u + 207)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
78
+ 15y
77
+ ··· − 9330y + 1)
c
2
, c
4
((y −1)
6
)(y
78
− 37y
77
+ ··· − 114y + 1)
c
3
, c
8
y
6
(y
78
− 39y
77
+ ··· − 102400y + 4096)
c
5
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
78
− 14y
77
+ ··· − 388528493y + 22287841)
c
6
, c
7
, c
10
c
11
(y
6
− 7y
5
+ ··· − 5y + 1)(y
78
− 86y
77
+ ··· − y + 1)
c
9
, c
12
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
78
+ 46y
77
+ ··· + 505791y + 42849)
18
