12a
0258
(K12a
0258
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 1 2 5 11 12 4 9 8
Solving Sequence
2,7
6 3
4,10
5 8 11 1 12 9
c
6
c
2
c
3
c
4
c
7
c
10
c
1
c
12
c
9
c
5
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
87
+ u
86
+ ··· + b + u, 2u
87
− 2u
86
+ ··· + a + 1, u
89
− 2u
88
+ ··· − 3u + 1i
I
u
2
= hb + 1, −u
3
− u
2
+ a − u − 1, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 94 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
87
+u
86
+· · ·+b+u, 2u
87
−2u
86
+· · ·+a+1, u
89
−2u
88
+· · ·−3u+1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
10
=
−2u
87
+ 2u
86
+ ··· + 3u − 1
u
87
− u
86
+ ··· + u
2
− u
a
5
=
−u
6
− u
4
+ 1
−u
8
− 2u
6
− 2u
4
a
8
=
u
14
+ 3u
12
+ 4u
10
+ u
8
− 2u
6
− 2u
4
+ 1
u
16
+ 4u
14
+ 8u
12
+ 8u
10
+ 4u
8
a
11
=
−u
84
+ u
83
+ ··· − u
3
+ 2u
2
−u
86
+ u
85
+ ··· + u
5
− u
2
a
1
=
u
3
u
5
+ u
3
+ u
a
12
=
u
25
+ 6u
23
+ ··· + 3u
5
− u
u
27
+ 7u
25
+ ··· + u
3
+ u
a
9
=
−u
87
+ u
86
+ ··· − u
3
+ u
u
87
− u
86
+ ··· + 4u
5
− u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
88
+ 13u
87
+ ··· − 20u + 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
89
+ 48u
88
+ ··· + u − 1
c
2
, c
6
u
89
− 2u
88
+ ··· − 3u + 1
c
3
, c
5
u
89
+ 2u
88
+ ··· + 9u + 1
c
4
, c
10
u
89
− u
88
+ ··· + 120u
2
+ 32
c
7
u
89
− 12u
88
+ ··· − 3133u + 277
c
8
, c
9
, c
11
u
89
+ 6u
88
+ ··· + 3u + 1
c
12
u
89
− 33u
88
+ ··· − 7680u + 1024
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
89
− 12y
88
+ ··· + 9y −1
c
2
, c
6
y
89
+ 48y
88
+ ··· + y −1
c
3
, c
5
y
89
− 72y
88
+ ··· + 145y −1
c
4
, c
10
y
89
+ 33y
88
+ ··· − 7680y −1024
c
7
y
89
− 12y
88
+ ··· − 1273175y −76729
c
8
, c
9
, c
11
y
89
− 76y
88
+ ··· − 21y −1
c
12
y
89
+ 37y
88
+ ··· − 14024704y −1048576
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.517663 + 0.853057I
a = 0.063168 + 1.010150I
b = 0.0075337 + 0.0380861I
3.01994 + 4.86569I 0
u = 0.517663 − 0.853057I
a = 0.063168 − 1.010150I
b = 0.0075337 − 0.0380861I
3.01994 − 4.86569I 0
u = 0.448323 + 0.897280I
a = 0.215044 − 0.933260I
b = −0.0257289 + 0.0936950I
−1.14920 + 2.08020I 0
u = 0.448323 − 0.897280I
a = 0.215044 + 0.933260I
b = −0.0257289 − 0.0936950I
−1.14920 − 2.08020I 0
u = −0.527593 + 0.875296I
a = 0.04378 + 1.59082I
b = 1.91935 − 0.02195I
0.04402 − 6.76781I 0
u = −0.527593 − 0.875296I
a = 0.04378 − 1.59082I
b = 1.91935 + 0.02195I
0.04402 + 6.76781I 0
u = 0.045374 + 1.026840I
a = −1.396320 + 0.027423I
b = 0.762961 + 0.584535I
−3.82045 + 2.55031I 0
u = 0.045374 − 1.026840I
a = −1.396320 − 0.027423I
b = 0.762961 − 0.584535I
−3.82045 − 2.55031I 0
u = −0.036660 + 0.970796I
a = 1.45328 + 0.01748I
b = −0.651637 − 0.810834I
−0.611498 − 1.156220I 0
u = −0.036660 − 0.970796I
a = 1.45328 − 0.01748I
b = −0.651637 + 0.810834I
−0.611498 + 1.156220I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.487998 + 0.832637I
a = −0.15557 − 1.88628I
b = −2.07823 + 0.10861I
2.18808 − 2.44281I 0
u = −0.487998 − 0.832637I
a = −0.15557 + 1.88628I
b = −2.07823 − 0.10861I
2.18808 + 2.44281I 0
u = −0.558903 + 0.882043I
a = −0.09933 − 1.47746I
b = −1.86773 + 0.05365I
5.18287 − 10.77750I 0
u = −0.558903 − 0.882043I
a = −0.09933 + 1.47746I
b = −1.86773 − 0.05365I
5.18287 + 10.77750I 0
u = −0.570217 + 0.765010I
a = 0.46901 + 1.53140I
b = 1.82570 − 0.18189I
9.81950 − 2.26683I 9.08032 + 0.I
u = −0.570217 − 0.765010I
a = 0.46901 − 1.53140I
b = 1.82570 + 0.18189I
9.81950 + 2.26683I 9.08032 + 0.I
u = 0.083364 + 1.074910I
a = 1.42919 − 0.06261I
b = −0.891162 − 0.475610I
0.70579 + 6.29860I 0
u = 0.083364 − 1.074910I
a = 1.42919 + 0.06261I
b = −0.891162 + 0.475610I
0.70579 − 6.29860I 0
u = 0.479594 + 1.025290I
a = −0.47420 + 1.59768I
b = −0.230291 − 0.441844I
3.30132 − 0.46498I 0
u = 0.479594 − 1.025290I
a = −0.47420 − 1.59768I
b = −0.230291 + 0.441844I
3.30132 + 0.46498I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.472021 + 0.710727I
a = −0.85417 − 1.74292I
b = −1.77547 + 0.50967I
2.55068 − 1.56908I 7.62928 + 3.25616I
u = −0.472021 − 0.710727I
a = −0.85417 + 1.74292I
b = −1.77547 − 0.50967I
2.55068 + 1.56908I 7.62928 − 3.25616I
u = −0.586040 + 0.618786I
a = −0.73818 − 1.29420I
b = −1.52005 + 0.15803I
5.92566 + 6.23892I 6.49865 − 3.39583I
u = −0.586040 − 0.618786I
a = −0.73818 + 1.29420I
b = −1.52005 − 0.15803I
5.92566 − 6.23892I 6.49865 + 3.39583I
u = 0.318408 + 1.117220I
a = −1.78251 + 0.77997I
b = 0.924881 − 0.530367I
3.57675 − 0.11414I 0
u = 0.318408 − 1.117220I
a = −1.78251 − 0.77997I
b = 0.924881 + 0.530367I
3.57675 + 0.11414I 0
u = 0.821589 + 0.144980I
a = 0.196200 − 0.728902I
b = −2.16374 + 0.76355I
1.51929 − 11.34500I 2.72775 + 7.03467I
u = 0.821589 − 0.144980I
a = 0.196200 + 0.728902I
b = −2.16374 − 0.76355I
1.51929 + 11.34500I 2.72775 − 7.03467I
u = 0.505613 + 0.657363I
a = −0.353180 − 0.878368I
b = −0.302339 + 0.026911I
3.57630 − 0.65503I 5.49721 − 0.67822I
u = 0.505613 − 0.657363I
a = −0.353180 + 0.878368I
b = −0.302339 − 0.026911I
3.57630 + 0.65503I 5.49721 + 0.67822I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.824465 + 0.058272I
a = 0.173870 + 0.454318I
b = 0.247863 + 0.493086I
−0.95526 − 1.82340I 2.00883 + 3.90836I
u = −0.824465 − 0.058272I
a = 0.173870 − 0.454318I
b = 0.247863 − 0.493086I
−0.95526 + 1.82340I 2.00883 − 3.90836I
u = 0.807857 + 0.130335I
a = −0.257591 + 0.772643I
b = 2.20223 − 0.92168I
−3.42851 − 7.01835I −1.34084 + 5.91260I
u = 0.807857 − 0.130335I
a = −0.257591 − 0.772643I
b = 2.20223 + 0.92168I
−3.42851 + 7.01835I −1.34084 − 5.91260I
u = 0.302386 + 0.751486I
a = 0.521010 + 0.407084I
b = 0.034676 − 0.191296I
−0.337993 + 1.231430I −4.05082 − 5.12187I
u = 0.302386 − 0.751486I
a = 0.521010 − 0.407084I
b = 0.034676 + 0.191296I
−0.337993 − 1.231430I −4.05082 + 5.12187I
u = −0.527637 + 0.613852I
a = 0.86201 + 1.35059I
b = 1.47210 − 0.29714I
0.77238 + 2.47093I 2.67534 − 2.90980I
u = −0.527637 − 0.613852I
a = 0.86201 − 1.35059I
b = 1.47210 + 0.29714I
0.77238 − 2.47093I 2.67534 + 2.90980I
u = −0.800030 + 0.093267I
a = −0.289719 − 0.456598I
b = −0.393360 − 0.440336I
−4.50871 + 1.54087I −3.93955 − 0.53955I
u = −0.800030 − 0.093267I
a = −0.289719 + 0.456598I
b = −0.393360 + 0.440336I
−4.50871 − 1.54087I −3.93955 + 0.53955I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.792991 + 0.129489I
a = 0.362212 + 0.510270I
b = 0.508182 + 0.450575I
−0.18794 + 5.08047I 1.48224 − 3.61614I
u = −0.792991 − 0.129489I
a = 0.362212 − 0.510270I
b = 0.508182 − 0.450575I
−0.18794 − 5.08047I 1.48224 + 3.61614I
u = 0.780635 + 0.114182I
a = 0.321976 − 0.894632I
b = −2.15166 + 1.23701I
−0.76764 − 2.40112I 1.86669 + 2.82981I
u = 0.780635 − 0.114182I
a = 0.321976 + 0.894632I
b = −2.15166 − 1.23701I
−0.76764 + 2.40112I 1.86669 − 2.82981I
u = 0.435439 + 1.137790I
a = 1.87502 − 2.52725I
b = −0.13112 + 1.68520I
−2.30698 + 1.91588I 0
u = 0.435439 − 1.137790I
a = 1.87502 + 2.52725I
b = −0.13112 − 1.68520I
−2.30698 − 1.91588I 0
u = −0.462237 + 1.138890I
a = 0.265455 − 0.514615I
b = −0.847571 − 0.105524I
−0.25321 − 3.93959I 0
u = −0.462237 − 1.138890I
a = 0.265455 + 0.514615I
b = −0.847571 + 0.105524I
−0.25321 + 3.93959I 0
u = 0.730068 + 0.216859I
a = 0.017747 + 0.955576I
b = 1.41229 − 0.75798I
7.51403 − 3.34035I 7.60444 + 3.00182I
u = 0.730068 − 0.216859I
a = 0.017747 − 0.955576I
b = 1.41229 + 0.75798I
7.51403 + 3.34035I 7.60444 − 3.00182I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.476725 + 1.153260I
a = −0.81591 + 3.58368I
b = −1.01360 − 1.84791I
−1.97776 + 6.10594I 0
u = 0.476725 − 1.153260I
a = −0.81591 − 3.58368I
b = −1.01360 + 1.84791I
−1.97776 − 6.10594I 0
u = 0.513961 + 1.149840I
a = −0.11955 − 3.03672I
b = 1.39279 + 1.07006I
4.79936 + 8.02709I 0
u = 0.513961 − 1.149840I
a = −0.11955 + 3.03672I
b = 1.39279 − 1.07006I
4.79936 − 8.02709I 0
u = 0.397557 + 1.201190I
a = 3.48700 − 0.84058I
b = −1.91274 + 1.53069I
−4.63345 + 1.62389I 0
u = 0.397557 − 1.201190I
a = 3.48700 + 0.84058I
b = −1.91274 − 1.53069I
−4.63345 − 1.62389I 0
u = −0.386693 + 1.205270I
a = −0.590438 + 0.072108I
b = 0.514506 + 0.709709I
−4.15392 + 1.08717I 0
u = −0.386693 − 1.205270I
a = −0.590438 − 0.072108I
b = 0.514506 − 0.709709I
−4.15392 − 1.08717I 0
u = −0.454517 + 1.185690I
a = −0.153045 + 0.222893I
b = 0.409003 + 0.080006I
−5.26721 − 4.29704I 0
u = −0.454517 − 1.185690I
a = −0.153045 − 0.222893I
b = 0.409003 − 0.080006I
−5.26721 + 4.29704I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.729686
a = 0.344825
b = 0.329068
−1.90133 −6.04240
u = 0.383868 + 1.214570I
a = −3.21266 + 0.41913I
b = 2.00402 − 1.16861I
−7.46477 − 2.97908I 0
u = 0.383868 − 1.214570I
a = −3.21266 − 0.41913I
b = 2.00402 + 1.16861I
−7.46477 + 2.97908I 0
u = 0.609349 + 0.393185I
a = −0.300041 − 1.003460I
b = −0.787001 + 0.295364I
5.10714 + 4.77042I 6.46745 − 4.13897I
u = 0.609349 − 0.393185I
a = −0.300041 + 1.003460I
b = −0.787001 − 0.295364I
5.10714 − 4.77042I 6.46745 + 4.13897I
u = 0.372330 + 1.222140I
a = 2.99797 − 0.26076I
b = −1.99266 + 0.96659I
−2.63241 − 7.32204I 0
u = 0.372330 − 1.222140I
a = 2.99797 + 0.26076I
b = −1.99266 − 0.96659I
−2.63241 + 7.32204I 0
u = −0.406288 + 1.213400I
a = 0.504733 − 0.021247I
b = −0.389248 − 0.659986I
−8.38898 − 2.62543I 0
u = −0.406288 − 1.213400I
a = 0.504733 + 0.021247I
b = −0.389248 + 0.659986I
−8.38898 + 2.62543I 0
u = 0.499039 + 1.190420I
a = 1.03528 + 4.02706I
b = −2.39564 − 1.39498I
−3.91311 + 7.11065I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.499039 − 1.190420I
a = 1.03528 − 4.02706I
b = −2.39564 + 1.39498I
−3.91311 − 7.11065I 0
u = −0.506285 + 1.191920I
a = 0.088919 + 0.484113I
b = 0.689757 − 0.413249I
−3.30845 − 9.86019I 0
u = −0.506285 − 1.191920I
a = 0.088919 − 0.484113I
b = 0.689757 + 0.413249I
−3.30845 + 9.86019I 0
u = −0.424698 + 1.226070I
a = −0.435838 − 0.072261I
b = 0.216144 + 0.657611I
−4.80273 − 6.19578I 0
u = −0.424698 − 1.226070I
a = −0.435838 + 0.072261I
b = 0.216144 − 0.657611I
−4.80273 + 6.19578I 0
u = −0.494122 + 1.200580I
a = −0.107151 − 0.412635I
b = −0.559713 + 0.411350I
−7.76452 − 6.26633I 0
u = −0.494122 − 1.200580I
a = −0.107151 + 0.412635I
b = −0.559713 − 0.411350I
−7.76452 + 6.26633I 0
u = 0.509568 + 1.196990I
a = −1.27750 − 3.59371I
b = 2.40882 + 1.03090I
−6.57537 + 11.85050I 0
u = 0.509568 − 1.196990I
a = −1.27750 + 3.59371I
b = 2.40882 − 1.03090I
−6.57537 − 11.85050I 0
u = 0.517965 + 1.198870I
a = 1.28587 + 3.32898I
b = −2.33265 − 0.85004I
−1.6030 + 16.2547I 0
12
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.517965 − 1.198870I
a = 1.28587 − 3.32898I
b = −2.33265 + 0.85004I
−1.6030 − 16.2547I 0
u = −0.482215 + 1.216160I
a = 0.183651 + 0.333431I
b = 0.382069 − 0.492805I
−4.39340 − 2.89778I 0
u = −0.482215 − 1.216160I
a = 0.183651 − 0.333431I
b = 0.382069 + 0.492805I
−4.39340 + 2.89778I 0
u = 0.643969 + 0.126654I
a = 0.021353 − 1.206920I
b = −0.89695 + 1.30560I
0.91693 − 1.79069I 4.82285 + 4.37475I
u = 0.643969 − 0.126654I
a = 0.021353 + 1.206920I
b = −0.89695 − 1.30560I
0.91693 + 1.79069I 4.82285 − 4.37475I
u = 0.473472 + 0.377042I
a = 0.402265 + 1.101880I
b = 0.527431 − 0.409130I
0.11891 + 1.56645I 1.87864 − 4.05770I
u = 0.473472 − 0.377042I
a = 0.402265 − 1.101880I
b = 0.527431 + 0.409130I
0.11891 − 1.56645I 1.87864 + 4.05770I
u = −0.507663 + 0.181860I
a = −1.035520 − 0.475789I
b = −0.716534 + 0.001247I
2.48913 − 0.09257I 4.02207 − 0.60784I
u = −0.507663 − 0.181860I
a = −1.035520 + 0.475789I
b = −0.716534 − 0.001247I
2.48913 + 0.09257I 4.02207 + 0.60784I
13
II. I
u
2
= hb + 1, −u
3
− u
2
+ a − u − 1, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
10
=
u
3
+ u
2
+ u + 1
−1
a
5
=
−u
3
u
3
+ u
a
8
=
−u
3
u
4
+ u
3
+ u
2
+ 1
a
11
=
u
3
+ u
2
+ u + 1
−1
a
1
=
u
3
−u
4
− u
3
− u
2
− 1
a
12
=
u
3
−u
4
− u
3
− u
2
− 1
a
9
=
u
2
+ u + 1
u
4
+ u
3
+ u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
4
− u
3
− 2u
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
2
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
3
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
4
, c
10
, c
12
u
5
c
5
, c
7
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
6
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
8
, c
9
(u + 1)
5
c
11
(u − 1)
5
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
c
2
, c
6
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
3
, c
5
, c
7
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
4
, c
10
, c
12
y
5
c
8
, c
9
, c
11
(y −1)
5
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.339110 + 0.822375I
a = 0.128779 + 1.107660I
b = −1.00000
1.31583 + 1.53058I −0.02124 − 2.62456I
u = 0.339110 − 0.822375I
a = 0.128779 − 1.107660I
b = −1.00000
1.31583 − 1.53058I −0.02124 + 2.62456I
u = −0.766826
a = 0.370286
b = −1.00000
−0.756147 2.67610
u = −0.455697 + 1.200150I
a = 1.18608 − 0.87465I
b = −1.00000
−4.22763 − 4.40083I −0.31681 + 3.97407I
u = −0.455697 − 1.200150I
a = 1.18608 + 0.87465I
b = −1.00000
−4.22763 + 4.40083I −0.31681 − 3.97407I
17
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)(u
89
+ 48u
88
+ ··· + u − 1)
c
2
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
89
− 2u
88
+ ··· − 3u + 1)
c
3
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
89
+ 2u
88
+ ··· + 9u + 1)
c
4
, c
10
u
5
(u
89
− u
88
+ ··· + 120u
2
+ 32)
c
5
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
89
+ 2u
88
+ ··· + 9u + 1)
c
6
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
89
− 2u
88
+ ··· − 3u + 1)
c
7
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
89
− 12u
88
+ ··· − 3133u + 277)
c
8
, c
9
((u + 1)
5
)(u
89
+ 6u
88
+ ··· + 3u + 1)
c
11
((u − 1)
5
)(u
89
+ 6u
88
+ ··· + 3u + 1)
c
12
u
5
(u
89
− 33u
88
+ ··· − 7680u + 1024)
18
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
89
− 12y
88
+ ··· + 9y −1)
c
2
, c
6
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
89
+ 48y
88
+ ··· + y −1)
c
3
, c
5
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
89
− 72y
88
+ ··· + 145y −1)
c
4
, c
10
y
5
(y
89
+ 33y
88
+ ··· − 7680y −1024)
c
7
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
89
− 12y
88
+ ··· − 1273175y −76729)
c
8
, c
9
, c
11
((y −1)
5
)(y
89
− 76y
88
+ ··· − 21y −1)
c
12
y
5
(y
89
+ 37y
88
+ ··· − 1.40247 × 10
7
y −1048576)
19
