12a
0124
(K12a
0124
)
A knot diagram
1
Linearized knot diagam
3 5 8 6 2 9 11 4 1 12 7 10
Solving Sequence
7,11 4,8
9 12 3 6 5 2 10 1
c
7
c
8
c
11
c
3
c
6
c
4
c
2
c
10
c
12
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h8u
82
− 5u
81
+ ··· + 2b + 6u, −8u
82
+ 24u
81
+ ··· + 2a + 13, u
83
− 3u
82
+ ··· + 3u
2
+ 1i
I
u
2
= h−u
2
a + b, −u
2
a + u
3
+ a
2
− au + 2u
2
− a + 2u, u
4
+ u
3
+ u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 91 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
= h8u
82
− 5u
81
+ · · · + 2b + 6u, −8u
82
+ 24u
81
+ · · · + 2a + 13, u
83
−
3u
82
+ · · · + 3u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
4u
82
− 12u
81
+ ··· − 15u
2
−
13
2
−4u
82
+
5
2
u
81
+ ··· −
13
2
u
2
− 3u
a
8
=
1
u
2
a
9
=
−u
7
− 2u
3
u
7
+ u
5
+ 2u
3
+ u
a
12
=
−u
u
a
3
=
4u
82
− 12u
81
+ ··· − u −
13
2
−4u
82
+
7
2
u
81
+ ··· −
13
2
u
2
− 3u
a
6
=
−u
14
− u
12
− 4u
10
− 3u
8
− 4u
6
− 2u
4
+ 1
u
14
+ 2u
12
+ 5u
10
+ 6u
8
+ 6u
6
+ 4u
4
+ u
2
a
5
=
5
2
u
82
− 8u
81
+ ··· − 12u
2
− 6
−
5
2
u
82
+ u
81
+ ··· − 6u
2
−
5
2
u
a
2
=
−
1
2
u
79
+ u
78
+ ··· − u
2
+
1
2
−
1
2
u
81
+ u
80
+ ··· − 5u
4
+
1
2
u
2
a
10
=
−u
3
u
3
+ u
a
1
=
−u
5
− u
u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
15
2
u
82
− 8u
81
+ ··· + 17u +
25
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
83
+ 27u
82
+ ··· − 22u − 1
c
2
, c
5
u
83
+ 5u
82
+ ··· − 6u − 1
c
3
, c
8
u
83
− u
82
+ ··· + 896u − 256
c
6
u
83
+ 3u
82
+ ··· − 28678u − 6329
c
7
, c
11
u
83
+ 3u
82
+ ··· − 3u
2
− 1
c
9
, c
10
, c
12
u
83
− 21u
82
+ ··· − 6u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
83
+ 63y
82
+ ··· − 782y −1
c
2
, c
5
y
83
+ 27y
82
+ ··· − 22y −1
c
3
, c
8
y
83
− 45y
82
+ ··· + 770048y −65536
c
6
y
83
− 15y
82
+ ··· − 3774261726y −40056241
c
7
, c
11
y
83
+ 21y
82
+ ··· − 6y −1
c
9
, c
10
, c
12
y
83
+ 85y
82
+ ··· + 34y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.343094 + 0.932755I
a = −2.78793 + 1.29202I
b = 1.91795 + 0.42351I
0.45934 + 5.84256I 0
u = −0.343094 − 0.932755I
a = −2.78793 − 1.29202I
b = 1.91795 − 0.42351I
0.45934 − 5.84256I 0
u = 0.298458 + 0.943559I
a = 0.810864 − 1.048410I
b = −0.268792 + 0.197880I
3.34331 − 5.43598I 0
u = 0.298458 − 0.943559I
a = 0.810864 + 1.048410I
b = −0.268792 − 0.197880I
3.34331 + 5.43598I 0
u = −0.276801 + 0.948551I
a = 2.78574 − 1.08616I
b = −1.72816 − 0.44874I
4.22365 + 2.71450I 0
u = −0.276801 − 0.948551I
a = 2.78574 + 1.08616I
b = −1.72816 + 0.44874I
4.22365 − 2.71450I 0
u = −0.151262 + 1.004710I
a = −2.47403 + 1.00449I
b = 1.40269 + 0.20603I
8.14096 − 5.43626I 0
u = −0.151262 − 1.004710I
a = −2.47403 − 1.00449I
b = 1.40269 − 0.20603I
8.14096 + 5.43626I 0
u = −0.176749 + 1.004370I
a = 2.53616 − 1.01779I
b = −1.47125 − 0.23649I
8.77851 + 0.61029I 0
u = −0.176749 − 1.004370I
a = 2.53616 + 1.01779I
b = −1.47125 + 0.23649I
8.77851 − 0.61029I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.260636 + 0.933904I
a = −0.949681 + 0.967452I
b = 0.337767 − 0.158358I
3.56863 + 0.12560I 0
u = 0.260636 − 0.933904I
a = −0.949681 − 0.967452I
b = 0.337767 + 0.158358I
3.56863 − 0.12560I 0
u = 0.701152 + 0.649791I
a = −0.157823 − 0.633968I
b = −0.626202 − 0.050636I
2.86901 + 0.22933I 0
u = 0.701152 − 0.649791I
a = −0.157823 + 0.633968I
b = −0.626202 + 0.050636I
2.86901 − 0.22933I 0
u = 0.410309 + 0.844223I
a = 0.254656 − 0.824567I
b = −0.0949998 + 0.0602684I
−1.01210 − 2.01711I 0
u = 0.410309 − 0.844223I
a = 0.254656 + 0.824567I
b = −0.0949998 − 0.0602684I
−1.01210 + 2.01711I 0
u = −0.345364 + 1.006340I
a = 2.65833 − 1.18168I
b = −1.85397 − 0.29823I
7.79448 + 5.46509I 0
u = −0.345364 − 1.006340I
a = 2.65833 + 1.18168I
b = −1.85397 + 0.29823I
7.79448 − 5.46509I 0
u = −0.364365 + 1.006860I
a = −2.63344 + 1.19214I
b = 1.87737 + 0.29191I
6.89769 + 11.50320I 0
u = −0.364365 − 1.006860I
a = −2.63344 − 1.19214I
b = 1.87737 − 0.29191I
6.89769 − 11.50320I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.692299 + 0.586886I
a = 0.105400 + 0.701418I
b = 0.612805 − 0.126857I
2.48455 − 5.49249I 4.00000 + 6.10141I
u = 0.692299 − 0.586886I
a = 0.105400 − 0.701418I
b = 0.612805 + 0.126857I
2.48455 + 5.49249I 4.00000 − 6.10141I
u = −0.227401 + 0.873532I
a = −2.94954 + 0.75439I
b = 1.52584 + 0.73742I
1.21112 − 0.90797I 9.54112 + 0.I
u = −0.227401 − 0.873532I
a = −2.94954 − 0.75439I
b = 1.52584 − 0.73742I
1.21112 + 0.90797I 9.54112 + 0.I
u = 0.671828 + 0.924378I
a = −0.622871 − 1.201050I
b = 0.424357 − 0.487883I
3.34573 + 0.48424I 0
u = 0.671828 − 0.924378I
a = −0.622871 + 1.201050I
b = 0.424357 + 0.487883I
3.34573 − 0.48424I 0
u = 0.831524 + 0.823800I
a = −0.694677 + 0.144305I
b = −1.31150 − 1.32695I
−2.75630 + 0.70086I 0
u = 0.831524 − 0.823800I
a = −0.694677 − 0.144305I
b = −1.31150 + 1.32695I
−2.75630 − 0.70086I 0
u = −0.824449 + 0.832717I
a = −0.441161 + 0.881969I
b = 0.878507 − 0.473427I
−3.19009 + 2.27190I 0
u = −0.824449 − 0.832717I
a = −0.441161 − 0.881969I
b = 0.878507 + 0.473427I
−3.19009 − 2.27190I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.702510 + 0.941716I
a = 0.79024 + 1.27635I
b = −0.635208 + 0.630772I
3.58007 − 5.50953I 0
u = 0.702510 − 0.941716I
a = 0.79024 − 1.27635I
b = −0.635208 − 0.630772I
3.58007 + 5.50953I 0
u = −0.844416 + 0.826159I
a = 0.347812 − 0.924380I
b = −0.835200 + 0.578194I
−3.88410 − 3.30015I 0
u = −0.844416 − 0.826159I
a = 0.347812 + 0.924380I
b = −0.835200 − 0.578194I
−3.88410 + 3.30015I 0
u = 0.826999 + 0.856266I
a = 1.164140 − 0.198525I
b = 1.16923 + 1.96671I
−5.26102 − 3.68755I 0
u = 0.826999 − 0.856266I
a = 1.164140 + 0.198525I
b = 1.16923 − 1.96671I
−5.26102 + 3.68755I 0
u = 0.878403 + 0.805051I
a = −0.224135 + 0.379019I
b = −1.82429 − 0.88600I
−0.18851 + 3.79252I 0
u = 0.878403 − 0.805051I
a = −0.224135 − 0.379019I
b = −1.82429 + 0.88600I
−0.18851 − 3.79252I 0
u = 0.862910 + 0.833613I
a = 0.561839 − 0.539969I
b = 1.85608 + 1.34386I
−7.19455 + 3.46271I 0
u = 0.862910 − 0.833613I
a = 0.561839 + 0.539969I
b = 1.85608 − 1.34386I
−7.19455 − 3.46271I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.887411 + 0.811360I
a = 0.175412 − 0.475350I
b = 1.95172 + 0.87958I
−1.29156 + 9.73575I 0
u = 0.887411 − 0.811360I
a = 0.175412 + 0.475350I
b = 1.95172 − 0.87958I
−1.29156 − 9.73575I 0
u = −0.826761 + 0.900804I
a = −0.472121 + 0.373985I
b = 0.586467 − 0.059273I
−5.92062 + 3.08392I 0
u = −0.826761 − 0.900804I
a = −0.472121 − 0.373985I
b = 0.586467 + 0.059273I
−5.92062 − 3.08392I 0
u = −0.863134 + 0.872566I
a = 0.142463 − 0.696056I
b = −0.501251 + 0.517205I
−8.78473 + 1.47520I 0
u = −0.863134 − 0.872566I
a = 0.142463 + 0.696056I
b = −0.501251 − 0.517205I
−8.78473 − 1.47520I 0
u = 0.801558 + 0.934083I
a = −1.66203 − 1.34142I
b = 1.40427 − 1.92771I
−5.01807 − 2.40115I 0
u = 0.801558 − 0.934083I
a = −1.66203 + 1.34142I
b = 1.40427 + 1.92771I
−5.01807 + 2.40115I 0
u = −0.790976 + 0.948890I
a = −1.000770 + 0.219273I
b = 0.906203 + 0.319706I
−2.83107 + 3.77517I 0
u = −0.790976 − 0.948890I
a = −1.000770 − 0.219273I
b = 0.906203 − 0.319706I
−2.83107 − 3.77517I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.791713 + 0.956841I
a = 1.40368 + 1.52199I
b = −1.55092 + 1.33569I
−2.34548 − 6.77030I 0
u = 0.791713 − 0.956841I
a = 1.40368 − 1.52199I
b = −1.55092 − 1.33569I
−2.34548 + 6.77030I 0
u = −0.799666 + 0.960907I
a = 1.030170 − 0.107018I
b = −0.879329 − 0.414641I
−3.46560 + 9.43293I 0
u = −0.799666 − 0.960907I
a = 1.030170 + 0.107018I
b = −0.879329 + 0.414641I
−3.46560 − 9.43293I 0
u = 0.275137 + 0.693608I
a = −0.675065 − 0.100546I
b = 0.094240 + 0.169866I
0.331802 − 1.167540I 4.13626 + 5.89733I
u = 0.275137 − 0.693608I
a = −0.675065 + 0.100546I
b = 0.094240 − 0.169866I
0.331802 + 1.167540I 4.13626 − 5.89733I
u = −0.836268 + 0.942183I
a = 0.700473 + 0.066913I
b = −0.547096 − 0.403838I
−8.56607 + 4.83405I 0
u = −0.836268 − 0.942183I
a = 0.700473 − 0.066913I
b = −0.547096 + 0.403838I
−8.56607 − 4.83405I 0
u = 0.813423 + 0.965425I
a = −1.48006 − 1.73900I
b = 2.05027 − 1.31001I
−6.78150 − 9.69525I 0
u = 0.813423 − 0.965425I
a = −1.48006 + 1.73900I
b = 2.05027 + 1.31001I
−6.78150 + 9.69525I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.881180 + 0.913291I
a = −0.276328 − 0.566432I
b = −0.052289 + 0.630239I
−5.91260 + 5.79260I 0
u = −0.881180 − 0.913291I
a = −0.276328 + 0.566432I
b = −0.052289 − 0.630239I
−5.91260 − 5.79260I 0
u = −0.874973 + 0.927104I
a = 0.412919 + 0.466829I
b = −0.118207 − 0.607991I
−5.86859 + 0.69731I 0
u = −0.874973 − 0.927104I
a = 0.412919 − 0.466829I
b = −0.118207 + 0.607991I
−5.86859 − 0.69731I 0
u = 0.807166 + 0.988107I
a = 1.30635 + 1.77940I
b = −1.96902 + 0.87719I
0.38494 − 10.04700I 0
u = 0.807166 − 0.988107I
a = 1.30635 − 1.77940I
b = −1.96902 − 0.87719I
0.38494 + 10.04700I 0
u = 0.814683 + 0.989919I
a = −1.31841 − 1.82237I
b = 2.08670 − 0.84930I
−0.7294 − 16.0429I 0
u = 0.814683 − 0.989919I
a = −1.31841 + 1.82237I
b = 2.08670 + 0.84930I
−0.7294 + 16.0429I 0
u = −0.685603 + 0.181822I
a = 0.179731 + 0.546082I
b = 1.46603 − 0.33423I
4.26360 − 7.76953I 3.54954 + 5.53049I
u = −0.685603 − 0.181822I
a = 0.179731 − 0.546082I
b = 1.46603 + 0.33423I
4.26360 + 7.76953I 3.54954 − 5.53049I
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.674643 + 0.146601I
a = −0.222141 − 0.444607I
b = −1.44805 + 0.27104I
5.06565 − 1.86935I 5.12348 + 0.51490I
u = −0.674643 − 0.146601I
a = −0.222141 + 0.444607I
b = −1.44805 − 0.27104I
5.06565 + 1.86935I 5.12348 − 0.51490I
u = −0.008776 + 0.667764I
a = −1.80337 − 0.38176I
b = 0.326945 + 0.654149I
0.85058 − 1.37420I 8.90440 + 4.62607I
u = −0.008776 − 0.667764I
a = −1.80337 + 0.38176I
b = 0.326945 − 0.654149I
0.85058 + 1.37420I 8.90440 − 4.62607I
u = 0.503583 + 0.430436I
a = 0.295380 + 0.927474I
b = 0.270810 − 0.364960I
−2.26469 − 1.48041I −3.63359 + 3.93107I
u = 0.503583 − 0.430436I
a = 0.295380 − 0.927474I
b = 0.270810 + 0.364960I
−2.26469 + 1.48041I −3.63359 − 3.93107I
u = −0.525596 + 0.233371I
a = 0.654378 + 0.773168I
b = 1.185590 − 0.435183I
−1.65887 − 2.59378I −1.71925 + 4.29721I
u = −0.525596 − 0.233371I
a = 0.654378 − 0.773168I
b = 1.185590 + 0.435183I
−1.65887 + 2.59378I −1.71925 − 4.29721I
u = −0.236010 + 0.516430I
a = 1.82352 + 1.48424I
b = 0.401128 − 1.148950I
0.06273 + 2.77066I 3.20202 + 1.07638I
u = −0.236010 − 0.516430I
a = 1.82352 − 1.48424I
b = 0.401128 + 1.148950I
0.06273 − 2.77066I 3.20202 − 1.07638I
12
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.512953
a = −0.759184
b = −1.14909
1.54099 6.17060
u = 0.482267 + 0.070594I
a = 0.085534 + 1.365730I
b = 0.057289 − 0.563567I
0.87174 + 2.58253I 1.22669 − 2.95128I
u = 0.482267 − 0.070594I
a = 0.085534 − 1.365730I
b = 0.057289 + 0.563567I
0.87174 − 2.58253I 1.22669 + 2.95128I
13
II. I
u
2
= h−u
2
a + b, −u
2
a + u
3
+ a
2
− au + 2u
2
− a + 2u, u
4
+ u
3
+ u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
a
u
2
a
a
8
=
1
u
2
a
9
=
1
u
2
a
12
=
−u
u
a
3
=
a
u
2
a
a
6
=
u
2
+ 1
−u
3
− u
2
− 1
a
5
=
u
3
a + 2a
−u
3
a + 2u
2
a − au
a
2
=
−2u
2
+ a − u − 2
u
2
a + u
3
+ u
2
+ 2
a
10
=
−u
3
u
3
+ u
a
1
=
−u
2
− 1
u
3
+ u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
3
a − 4u
2
a + u
3
− 2au + 6u
2
− a + 6u + 2
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
3
, c
8
u
8
c
6
, c
9
, c
10
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
7
(u
4
+ u
3
+ u
2
+ 1)
2
c
11
(u
4
− u
3
+ u
2
+ 1)
2
c
12
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
4
c
3
, c
8
y
8
c
6
, c
9
, c
10
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
7
, c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351808 + 0.720342I
a = 1.54112 − 0.21492I
b = −0.500000 + 0.866025I
0.21101 − 3.44499I 5.86133 + 9.77094I
u = 0.351808 + 0.720342I
a = −0.58443 + 1.44211I
b = −0.500000 − 0.866025I
0.211005 + 0.614778I 1.10064 + 1.99408I
u = 0.351808 − 0.720342I
a = 1.54112 + 0.21492I
b = −0.500000 − 0.866025I
0.21101 + 3.44499I 5.86133 − 9.77094I
u = 0.351808 − 0.720342I
a = −0.58443 − 1.44211I
b = −0.500000 + 0.866025I
0.211005 − 0.614778I 1.10064 − 1.99408I
u = −0.851808 + 0.911292I
a = 0.576953 − 0.283088I
b = −0.500000 − 0.866025I
−6.79074 + 1.13408I −1.56110 − 0.68902I
u = −0.851808 + 0.911292I
a = −0.533637 − 0.358112I
b = −0.500000 + 0.866025I
−6.79074 + 5.19385I −0.90087 − 4.17049I
u = −0.851808 − 0.911292I
a = 0.576953 + 0.283088I
b = −0.500000 + 0.866025I
−6.79074 − 1.13408I −1.56110 + 0.68902I
u = −0.851808 − 0.911292I
a = −0.533637 + 0.358112I
b = −0.500000 − 0.866025I
−6.79074 − 5.19385I −0.90087 + 4.17049I
17
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
((u
2
− u + 1)
4
)(u
83
+ 27u
82
+ ··· − 22u − 1)
c
2
((u
2
+ u + 1)
4
)(u
83
+ 5u
82
+ ··· − 6u − 1)
c
3
, c
8
u
8
(u
83
− u
82
+ ··· + 896u − 256)
c
5
((u
2
− u + 1)
4
)(u
83
+ 5u
82
+ ··· − 6u − 1)
c
6
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
83
+ 3u
82
+ ··· − 28678u − 6329)
c
7
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
83
+ 3u
82
+ ··· − 3u
2
− 1)
c
9
, c
10
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
83
− 21u
82
+ ··· − 6u + 1)
c
11
((u
4
− u
3
+ u
2
+ 1)
2
)(u
83
+ 3u
82
+ ··· − 3u
2
− 1)
c
12
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
83
− 21u
82
+ ··· − 6u + 1)
18
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
4
)(y
83
+ 63y
82
+ ··· − 782y −1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
83
+ 27y
82
+ ··· − 22y −1)
c
3
, c
8
y
8
(y
83
− 45y
82
+ ··· + 770048y −65536)
c
6
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
83
− 15y
82
+ ··· − 3774261726y −40056241)
c
7
, c
11
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
83
+ 21y
82
+ ··· − 6y −1)
c
9
, c
10
, c
12
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
83
+ 85y
82
+ ··· + 34y −1)
19
