12a
0161
(K12a
0161
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 11 12 1 10 4 8 7 6
Solving Sequence
7,11
12 6 1 8
2,5
3 4 10 9
c
11
c
6
c
12
c
7
c
5
c
2
c
4
c
10
c
9
c
1
, c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
45
− 19u
43
+ ··· + b − u, u
77
+ u
76
+ ··· + a − 1, u
78
+ 2u
77
+ ··· − 4u
2
− 1i
I
u
2
= hu
2
+ b − u + 1, −2u
2
+ a − 2, u
3
− u
2
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 81 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
45
−19u
43
+· · ·+b−u, u
77
+u
76
+· · ·+a−1, u
78
+2u
77
+· · ·−4u
2
−1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
6
=
−u
u
3
+ u
a
1
=
u
2
+ 1
−u
4
− 2u
2
a
8
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
2
=
−u
77
− u
76
+ ··· + 4u + 1
u
45
+ 19u
43
+ ··· − 8u
2
+ u
a
5
=
−u
3
− 2u
u
3
+ u
a
3
=
−u
51
− 22u
49
+ ··· − 16u
2
+ 6u
−u
77
− 2u
76
+ ··· − 2u
2
+ 1
a
4
=
2u
77
+ 2u
76
+ ··· − 4u − 1
−u
77
− 2u
76
+ ··· − u + 1
a
10
=
−u
12
− 5u
10
− 9u
8
− 6u
6
+ u
2
+ 1
u
14
+ 6u
12
+ 13u
10
+ 10u
8
− 2u
6
− 4u
4
+ u
2
a
9
=
u
19
+ 8u
17
+ 26u
15
+ 42u
13
+ 31u
11
+ 2u
9
− 10u
7
− 4u
5
+ u
3
+ 2u
−u
21
− 9u
19
+ ··· − u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
77
− 2u
76
+ ··· + 19u + 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
78
+ 44u
77
+ ··· + 17u + 1
c
2
, c
4
u
78
− 4u
77
+ ··· + 9u − 1
c
3
, c
9
u
78
− u
77
+ ··· + 4u + 8
c
5
, c
7
u
78
− 2u
77
+ ··· + 4u − 1
c
6
, c
11
, c
12
u
78
+ 2u
77
+ ··· − 4u
2
− 1
c
8
, c
10
u
78
− 21u
77
+ ··· − 1232u + 64
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
78
− 16y
77
+ ··· − 329y + 1
c
2
, c
4
y
78
− 44y
77
+ ··· − 17y + 1
c
3
, c
9
y
78
− 21y
77
+ ··· − 1232y + 64
c
5
, c
7
y
78
− 38y
77
+ ··· + 8y + 1
c
6
, c
11
, c
12
y
78
+ 66y
77
+ ··· + 8y + 1
c
8
, c
10
y
78
+ 67y
77
+ ··· − 68864y + 4096
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.221840 + 0.947945I
a = −0.90854 − 2.19426I
b = −0.63575 + 1.79124I
−6.03562 − 1.43380I 0
u = 0.221840 − 0.947945I
a = −0.90854 + 2.19426I
b = −0.63575 − 1.79124I
−6.03562 + 1.43380I 0
u = −0.322913 + 0.983251I
a = −0.59097 + 2.36662I
b = −0.89380 − 1.94580I
−5.23022 + 7.49440I 0
u = −0.322913 − 0.983251I
a = −0.59097 − 2.36662I
b = −0.89380 + 1.94580I
−5.23022 − 7.49440I 0
u = −0.266822 + 1.014570I
a = −0.265621 − 0.087182I
b = 0.763129 − 0.052329I
−2.10877 + 2.64422I 0
u = −0.266822 − 1.014570I
a = −0.265621 + 0.087182I
b = 0.763129 + 0.052329I
−2.10877 − 2.64422I 0
u = −0.201285 + 0.867169I
a = −0.61079 − 1.89983I
b = 0.17550 + 1.40684I
−6.10936 − 1.32608I 0.73693 + 1.24858I
u = −0.201285 − 0.867169I
a = −0.61079 + 1.89983I
b = 0.17550 − 1.40684I
−6.10936 + 1.32608I 0.73693 − 1.24858I
u = 0.316364 + 0.786903I
a = −0.12641 + 1.88878I
b = 0.12525 − 1.63413I
−5.61969 + 7.17353I 1.91318 − 7.13043I
u = 0.316364 − 0.786903I
a = −0.12641 − 1.88878I
b = 0.12525 + 1.63413I
−5.61969 − 7.17353I 1.91318 + 7.13043I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.785147 + 0.185065I
a = −3.20268 + 3.17739I
b = 1.37992 − 2.53027I
−2.75716 − 11.61650I 5.77587 + 8.60798I
u = −0.785147 − 0.185065I
a = −3.20268 − 3.17739I
b = 1.37992 + 2.53027I
−2.75716 + 11.61650I 5.77587 − 8.60798I
u = −0.770791 + 0.171159I
a = 0.857137 + 0.494336I
b = −0.743484 + 0.108436I
0.46016 − 6.59193I 9.04398 + 5.71613I
u = −0.770791 − 0.171159I
a = 0.857137 − 0.494336I
b = −0.743484 − 0.108436I
0.46016 + 6.59193I 9.04398 − 5.71613I
u = 0.216613 + 0.756610I
a = −0.036625 + 0.331435I
b = 0.702068 − 0.142190I
−2.39928 + 2.61834I 4.76572 − 3.85328I
u = 0.216613 − 0.756610I
a = −0.036625 − 0.331435I
b = 0.702068 + 0.142190I
−2.39928 − 2.61834I 4.76572 + 3.85328I
u = −0.782132 + 0.079591I
a = −1.50273 + 2.21168I
b = 0.47522 − 1.74253I
4.89842 − 5.92763I 11.50709 + 7.07473I
u = −0.782132 − 0.079591I
a = −1.50273 − 2.21168I
b = 0.47522 + 1.74253I
4.89842 + 5.92763I 11.50709 − 7.07473I
u = −0.320177 + 1.177210I
a = −0.373766 + 1.202460I
b = −0.73975 − 1.50874I
1.55633 + 1.91915I 0
u = −0.320177 − 1.177210I
a = −0.373766 − 1.202460I
b = −0.73975 + 1.50874I
1.55633 − 1.91915I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.756990 + 0.181313I
a = −2.99011 − 3.55109I
b = 1.19344 + 2.71479I
−3.62530 + 5.27142I 4.61455 − 4.56249I
u = 0.756990 − 0.181313I
a = −2.99011 + 3.55109I
b = 1.19344 − 2.71479I
−3.62530 − 5.27142I 4.61455 + 4.56249I
u = −0.773908 + 0.039316I
a = 0.028490 − 0.517470I
b = −0.307660 + 0.629701I
6.03009 − 1.10865I 14.3285 + 0.8012I
u = −0.773908 − 0.039316I
a = 0.028490 + 0.517470I
b = −0.307660 − 0.629701I
6.03009 + 1.10865I 14.3285 − 0.8012I
u = −0.051004 + 1.229700I
a = −0.65147 − 1.75217I
b = −0.546978 − 0.264959I
−5.84021 − 1.10052I 0
u = −0.051004 − 1.229700I
a = −0.65147 + 1.75217I
b = −0.546978 + 0.264959I
−5.84021 + 1.10052I 0
u = −0.743598 + 0.185676I
a = 3.33465 − 1.56772I
b = −1.28958 + 1.13847I
−3.82604 − 2.40404I 4.47486 + 3.48448I
u = −0.743598 − 0.185676I
a = 3.33465 + 1.56772I
b = −1.28958 − 1.13847I
−3.82604 + 2.40404I 4.47486 − 3.48448I
u = 0.724926 + 0.227700I
a = 2.97783 + 1.77944I
b = −1.03439 − 1.28165I
−3.74163 − 3.33148I 4.54329 + 2.27222I
u = 0.724926 − 0.227700I
a = 2.97783 − 1.77944I
b = −1.03439 + 1.28165I
−3.74163 + 3.33148I 4.54329 − 2.27222I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.163517 + 1.233490I
a = −0.418961 + 0.316278I
b = 0.741466 + 0.295231I
−2.91541 + 2.01642I 0
u = 0.163517 − 1.233490I
a = −0.418961 − 0.316278I
b = 0.741466 − 0.295231I
−2.91541 − 2.01642I 0
u = 0.717043 + 0.182657I
a = 1.087960 − 0.258300I
b = −0.815686 − 0.256787I
−0.315586 + 0.899815I 8.05190 − 1.09209I
u = 0.717043 − 0.182657I
a = 1.087960 + 0.258300I
b = −0.815686 + 0.256787I
−0.315586 − 0.899815I 8.05190 + 1.09209I
u = −0.323388 + 1.222800I
a = 0.540462 − 0.141703I
b = 0.601785 + 0.420796I
2.39590 − 2.86000I 0
u = −0.323388 − 1.222800I
a = 0.540462 + 0.141703I
b = 0.601785 − 0.420796I
2.39590 + 2.86000I 0
u = 0.272850 + 1.244810I
a = −1.084010 − 0.348222I
b = 0.12739 + 1.81207I
−2.00980 + 1.83090I 0
u = 0.272850 − 1.244810I
a = −1.084010 + 0.348222I
b = 0.12739 − 1.81207I
−2.00980 − 1.83090I 0
u = 0.719177 + 0.040036I
a = 0.37747 − 2.59587I
b = −0.66798 + 1.81340I
1.67961 + 1.74944I 8.87795 − 3.96310I
u = 0.719177 − 0.040036I
a = 0.37747 + 2.59587I
b = −0.66798 − 1.81340I
1.67961 − 1.74944I 8.87795 + 3.96310I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.274784 + 1.281070I
a = −1.74319 − 1.43094I
b = 1.33894 − 0.46268I
−3.63721 − 3.47711I 0
u = −0.274784 − 1.281070I
a = −1.74319 + 1.43094I
b = 1.33894 + 0.46268I
−3.63721 + 3.47711I 0
u = −0.688313
a = 3.35104
b = −1.29655
0.354274 12.4140
u = 0.298521 + 1.292870I
a = 1.55001 + 1.06840I
b = 1.15050 − 1.83635I
−2.48033 + 5.43449I 0
u = 0.298521 − 1.292870I
a = 1.55001 − 1.06840I
b = 1.15050 + 1.83635I
−2.48033 − 5.43449I 0
u = −0.330502 + 1.287190I
a = −0.148188 − 0.049151I
b = 0.044709 − 0.772496I
1.89914 − 5.09234I 0
u = −0.330502 − 1.287190I
a = −0.148188 + 0.049151I
b = 0.044709 + 0.772496I
1.89914 + 5.09234I 0
u = −0.337500 + 1.313270I
a = 2.00680 − 0.11521I
b = −0.20506 + 1.85852I
0.53996 − 9.96745I 0
u = −0.337500 − 1.313270I
a = 2.00680 + 0.11521I
b = −0.20506 − 1.85852I
0.53996 + 9.96745I 0
u = 0.118296 + 1.357210I
a = −0.617226 + 1.056650I
b = 0.33291 + 1.37529I
−5.11060 + 4.79864I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.118296 − 1.357210I
a = −0.617226 − 1.056650I
b = 0.33291 − 1.37529I
−5.11060 − 4.79864I 0
u = 0.212992 + 1.351290I
a = −1.31861 + 0.66037I
b = 0.840813 + 1.053440I
−4.03525 + 2.28010I 0
u = 0.212992 − 1.351290I
a = −1.31861 − 0.66037I
b = 0.840813 − 1.053440I
−4.03525 − 2.28010I 0
u = 0.301735 + 1.364260I
a = −0.299894 + 0.766680I
b = 0.805459 + 0.404103I
−5.19764 + 4.61194I 0
u = 0.301735 − 1.364260I
a = −0.299894 − 0.766680I
b = 0.805459 − 0.404103I
−5.19764 − 4.61194I 0
u = −0.311251 + 1.368930I
a = −2.40297 − 0.78639I
b = 1.81013 − 1.33373I
−8.73778 − 6.23313I 0
u = −0.311251 − 1.368930I
a = −2.40297 + 0.78639I
b = 1.81013 + 1.33373I
−8.73778 + 6.23313I 0
u = −0.324957 + 1.366060I
a = −0.121142 − 0.795607I
b = 0.685934 − 0.203092I
−4.39480 − 10.55880I 0
u = −0.324957 − 1.366060I
a = −0.121142 + 0.795607I
b = 0.685934 + 0.203092I
−4.39480 + 10.55880I 0
u = 0.317497 + 1.368890I
a = 3.12911 − 0.00654I
b = −1.24074 − 3.25839I
−8.52354 + 9.16648I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.317497 − 1.368890I
a = 3.12911 + 0.00654I
b = −1.24074 + 3.25839I
−8.52354 − 9.16648I 0
u = −0.330057 + 1.374490I
a = 3.07659 + 0.21604I
b = −1.51815 + 2.97295I
−7.6876 − 15.6495I 0
u = −0.330057 − 1.374490I
a = 3.07659 − 0.21604I
b = −1.51815 − 2.97295I
−7.6876 + 15.6495I 0
u = 0.295867 + 1.382440I
a = −2.29433 + 0.57982I
b = 1.61811 + 1.49611I
−8.83720 + 0.37277I 0
u = 0.295867 − 1.382440I
a = −2.29433 − 0.57982I
b = 1.61811 − 1.49611I
−8.83720 − 0.37277I 0
u = 0.01557 + 1.41703I
a = −0.590163 − 0.023145I
b = −0.502778 + 0.138020I
−8.98429 + 3.00952I 0
u = 0.01557 − 1.41703I
a = −0.590163 + 0.023145I
b = −0.502778 − 0.138020I
−8.98429 − 3.00952I 0
u = −0.00417 + 1.42134I
a = 0.267222 − 0.793896I
b = −0.64326 − 2.84980I
−12.84860 − 1.47997I 0
u = −0.00417 − 1.42134I
a = 0.267222 + 0.793896I
b = −0.64326 + 2.84980I
−12.84860 + 1.47997I 0
u = 0.02254 + 1.43315I
a = 0.103861 + 0.659204I
b = −0.28487 + 2.87993I
−12.4992 + 7.7676I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.02254 − 1.43315I
a = 0.103861 − 0.659204I
b = −0.28487 − 2.87993I
−12.4992 − 7.7676I 0
u = 0.359180 + 0.436801I
a = 0.662229 + 0.372982I
b = −0.248173 − 1.100090I
0.37969 + 3.21597I 7.04752 − 8.62566I
u = 0.359180 − 0.436801I
a = 0.662229 − 0.372982I
b = −0.248173 + 1.100090I
0.37969 − 3.21597I 7.04752 + 8.62566I
u = 0.472822 + 0.269583I
a = 1.58775 + 0.80986I
b = −0.165683 − 0.644607I
0.913188 − 0.312563I 10.41614 − 0.13435I
u = 0.472822 − 0.269583I
a = 1.58775 − 0.80986I
b = −0.165683 + 0.644607I
0.913188 + 0.312563I 10.41614 + 0.13435I
u = 0.419349
a = 1.23689
b = −0.399405
0.742145 13.5430
u = −0.135471 + 0.223794I
a = 0.41686 + 2.11056I
b = 0.419076 + 0.683835I
−1.63000 − 0.53870I −3.53001 + 1.26153I
u = −0.135471 − 0.223794I
a = 0.41686 − 2.11056I
b = 0.419076 − 0.683835I
−1.63000 + 0.53870I −3.53001 − 1.26153I
12
II. I
u
2
= hu
2
+ b − u + 1, −2u
2
+ a − 2, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
6
=
−u
u
2
− u + 1
a
1
=
u
2
+ 1
−u
2
+ u − 1
a
8
=
1
0
a
2
=
2u
2
+ 2
−u
2
+ u − 1
a
5
=
−u
2
− 1
u
2
− u + 1
a
3
=
u
2
+ 1
0
a
4
=
u
2
+ 1
0
a
10
=
1
0
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
2
− 4u + 4
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
8
, c
9
c
10
u
3
c
4
(u + 1)
3
c
5
, c
7
u
3
− u
2
+ 1
c
6
u
3
+ u
2
+ 2u + 1
c
11
, c
12
u
3
− u
2
+ 2u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
8
, c
9
c
10
y
3
c
5
, c
7
y
3
− y
2
+ 2y −1
c
6
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −1.32472 + 1.12456I
b = 0.877439 + 0.744862I
−4.66906 + 2.82812I −1.84740 − 3.54173I
u = 0.215080 − 1.307140I
a = −1.32472 − 1.12456I
b = 0.877439 − 0.744862I
−4.66906 − 2.82812I −1.84740 + 3.54173I
u = 0.569840
a = 2.64944
b = −0.754878
−0.531480 2.69480
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
78
+ 44u
77
+ ··· + 17u + 1)
c
2
((u − 1)
3
)(u
78
− 4u
77
+ ··· + 9u − 1)
c
3
, c
9
u
3
(u
78
− u
77
+ ··· + 4u + 8)
c
4
((u + 1)
3
)(u
78
− 4u
77
+ ··· + 9u − 1)
c
5
, c
7
(u
3
− u
2
+ 1)(u
78
− 2u
77
+ ··· + 4u − 1)
c
6
(u
3
+ u
2
+ 2u + 1)(u
78
+ 2u
77
+ ··· − 4u
2
− 1)
c
8
, c
10
u
3
(u
78
− 21u
77
+ ··· − 1232u + 64)
c
11
, c
12
(u
3
− u
2
+ 2u − 1)(u
78
+ 2u
77
+ ··· − 4u
2
− 1)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
3
)(y
78
− 16y
77
+ ··· − 329y + 1)
c
2
, c
4
((y −1)
3
)(y
78
− 44y
77
+ ··· − 17y + 1)
c
3
, c
9
y
3
(y
78
− 21y
77
+ ··· − 1232y + 64)
c
5
, c
7
(y
3
− y
2
+ 2y −1)(y
78
− 38y
77
+ ··· + 8y + 1)
c
6
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)(y
78
+ 66y
77
+ ··· + 8y + 1)
c
8
, c
10
y
3
(y
78
+ 67y
77
+ ··· − 68864y + 4096)
18
