12a
0042
(K12a
0042
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 8 11 4 1 12 6 10 9
Solving Sequence
6,10
11
4,7
8 12 3 5 2 9 1
c
10
c
6
c
7
c
11
c
3
c
5
c
2
c
9
c
12
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
69
+ u
68
+ ··· + b − 1, u
69
+ u
68
+ ··· + a − 1, u
70
+ 2u
69
+ ··· + 7u
3
− 1i
I
u
2
= h−u
4
+ u
3
+ b − 1, −u
4
+ u
3
+ a − u − 1, u
5
− u
4
+ u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 75 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
= hu
69
+ u
68
+ · · · + b − 1, u
69
+ u
68
+ · · · + a − 1, u
70
+ 2u
69
+ · · · + 7u
3
− 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
4
=
−u
69
− u
68
+ ··· + u + 1
−u
69
− u
68
+ ··· − u + 1
a
7
=
u
−u
3
+ u
a
8
=
u
8
− u
6
+ 3u
4
− 2u
2
+ 1
u
8
+ 2u
4
a
12
=
−u
2
+ 1
−u
2
a
3
=
−u
69
+ 6u
67
+ ··· − 2u
2
+ 2u
u
69
+ 2u
68
+ ··· − u
2
− 1
a
5
=
−u
17
+ 2u
15
− 7u
13
+ 10u
11
− 15u
9
+ 14u
7
− 10u
5
+ 4u
3
− u
−u
17
+ u
15
− 5u
13
+ 4u
11
− 7u
9
+ 4u
7
− 2u
5
+ u
a
2
=
−u
69
− u
68
+ ··· + u + 1
−u
67
+ 7u
65
+ ··· − u
3
− 2u
2
a
9
=
u
4
− u
2
+ 1
u
4
a
1
=
−u
6
+ u
4
− 2u
2
+ 1
−u
6
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
69
+ 3u
68
+ ··· + 5u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
70
+ 34u
69
+ ··· + 90u + 1
c
2
, c
4
u
70
− 6u
69
+ ··· + 10u − 1
c
3
, c
7
u
70
− u
69
+ ··· − 160u + 32
c
5
u
70
+ 2u
69
+ ··· + 1156u − 809
c
6
, c
10
u
70
+ 2u
69
+ ··· + 7u
3
− 1
c
8
, c
9
, c
11
c
12
u
70
− 14u
69
+ ··· − 6u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
70
+ 10y
69
+ ··· − 6198y + 1
c
2
, c
4
y
70
− 34y
69
+ ··· − 90y + 1
c
3
, c
7
y
70
− 33y
69
+ ··· − 19968y + 1024
c
5
y
70
+ 2y
69
+ ··· − 21431896y + 654481
c
6
, c
10
y
70
− 14y
69
+ ··· − 6y
2
+ 1
c
8
, c
9
, c
11
c
12
y
70
+ 86y
69
+ ··· − 12y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.848963 + 0.526476I
a = −2.22564 + 0.01776I
b = −2.09137 + 2.15795I
−2.96182 + 3.57403I 4.58642 − 7.96880I
u = 0.848963 − 0.526476I
a = −2.22564 − 0.01776I
b = −2.09137 − 2.15795I
−2.96182 − 3.57403I 4.58642 + 7.96880I
u = −0.703772 + 0.704213I
a = 0.381569 − 0.170222I
b = −0.375365 − 0.155977I
−3.21240 − 4.24808I 2.10319 + 6.98947I
u = −0.703772 − 0.704213I
a = 0.381569 + 0.170222I
b = −0.375365 + 0.155977I
−3.21240 + 4.24808I 2.10319 − 6.98947I
u = 0.891060 + 0.401648I
a = 1.315420 − 0.011957I
b = 1.54143 − 1.57501I
3.91808 + 3.09369I 11.08096 − 6.18938I
u = 0.891060 − 0.401648I
a = 1.315420 + 0.011957I
b = 1.54143 + 1.57501I
3.91808 − 3.09369I 11.08096 + 6.18938I
u = −0.883258 + 0.525678I
a = −0.476131 + 0.081454I
b = −0.446033 + 0.623341I
−2.08503 − 5.95261I 3.66675 + 8.33839I
u = −0.883258 − 0.525678I
a = −0.476131 − 0.081454I
b = −0.446033 − 0.623341I
−2.08503 + 5.95261I 3.66675 − 8.33839I
u = 0.891599 + 0.333493I
a = −1.009990 + 0.050347I
b = −1.28724 + 1.45541I
2.79991 − 2.14652I 9.71265 − 0.23487I
u = 0.891599 − 0.333493I
a = −1.009990 − 0.050347I
b = −1.28724 − 1.45541I
2.79991 + 2.14652I 9.71265 + 0.23487I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.918259 + 0.512797I
a = 1.79709 − 0.31181I
b = 1.65385 − 2.00100I
2.56157 + 6.39701I 8.79876 − 6.86852I
u = 0.918259 − 0.512797I
a = 1.79709 + 0.31181I
b = 1.65385 + 2.00100I
2.56157 − 6.39701I 8.79876 + 6.86852I
u = −0.829178 + 0.656951I
a = −0.207393 + 0.091221I
b = 0.224040 + 0.560623I
−2.80650 − 0.77144I 0
u = −0.829178 − 0.656951I
a = −0.207393 − 0.091221I
b = 0.224040 − 0.560623I
−2.80650 + 0.77144I 0
u = −0.809134 + 0.473685I
a = 0.493245 + 0.198012I
b = 0.542821 − 0.286131I
−0.89626 − 1.96724I 5.61742 + 3.44733I
u = −0.809134 − 0.473685I
a = 0.493245 − 0.198012I
b = 0.542821 + 0.286131I
−0.89626 + 1.96724I 5.61742 − 3.44733I
u = −0.925343 + 0.107219I
a = 1.68417 + 0.13957I
b = 1.58116 − 0.07754I
4.00519 − 7.11466I 11.47294 + 7.09856I
u = −0.925343 − 0.107219I
a = 1.68417 − 0.13957I
b = 1.58116 + 0.07754I
4.00519 + 7.11466I 11.47294 − 7.09856I
u = 0.933419 + 0.540532I
a = −1.82278 + 0.45018I
b = −1.56950 + 2.06664I
0.36752 + 11.87470I 0. − 10.68011I
u = 0.933419 − 0.540532I
a = −1.82278 − 0.45018I
b = −1.56950 − 2.06664I
0.36752 − 11.87470I 0. + 10.68011I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.913045 + 0.060749I
a = −1.77668 − 0.10528I
b = −1.62183 + 0.02869I
5.72534 − 1.70594I 14.7293 + 1.6894I
u = −0.913045 − 0.060749I
a = −1.77668 + 0.10528I
b = −1.62183 − 0.02869I
5.72534 + 1.70594I 14.7293 − 1.6894I
u = 0.510558 + 0.701915I
a = −0.49728 + 1.97299I
b = 0.99266 + 1.29279I
−1.00091 − 7.27047I 1.97102 + 4.94885I
u = 0.510558 − 0.701915I
a = −0.49728 − 1.97299I
b = 0.99266 − 1.29279I
−1.00091 + 7.27047I 1.97102 − 4.94885I
u = −0.682836 + 0.516888I
a = 0.113681 + 0.487864I
b = 0.399773 + 0.054104I
−1.31076 − 1.82986I 3.73022 + 5.65801I
u = −0.682836 − 0.516888I
a = 0.113681 − 0.487864I
b = 0.399773 − 0.054104I
−1.31076 + 1.82986I 3.73022 − 5.65801I
u = 0.605074 + 0.592671I
a = −0.09109 + 2.15794I
b = 1.70767 + 1.21369I
−3.74961 + 0.67285I 0.349013 + 0.900580I
u = 0.605074 − 0.592671I
a = −0.09109 − 2.15794I
b = 1.70767 − 1.21369I
−3.74961 − 0.67285I 0.349013 − 0.900580I
u = 0.844028 + 0.060120I
a = −0.134807 + 0.151297I
b = −0.241531 + 1.133100I
1.03953 + 1.86544I 10.82292 − 4.22163I
u = 0.844028 − 0.060120I
a = −0.134807 − 0.151297I
b = −0.241531 − 1.133100I
1.03953 − 1.86544I 10.82292 + 4.22163I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.550080 + 0.618252I
a = 0.389492 − 0.684495I
b = −0.292647 − 0.181492I
−3.14503 + 1.63469I −0.27409 − 1.40763I
u = −0.550080 − 0.618252I
a = 0.389492 + 0.684495I
b = −0.292647 + 0.181492I
−3.14503 − 1.63469I −0.27409 + 1.40763I
u = 0.482045 + 0.648946I
a = 0.44168 − 1.88279I
b = −1.04226 − 1.10595I
1.18543 − 2.05480I 5.19934 + 0.85632I
u = 0.482045 − 0.648946I
a = 0.44168 + 1.88279I
b = −1.04226 + 1.10595I
1.18543 + 2.05480I 5.19934 − 0.85632I
u = −0.786558
a = 2.34128
b = 1.76553
−0.192753 15.7720
u = −0.896752 + 0.854059I
a = 0.459172 − 0.811143I
b = 2.34273 + 0.72734I
−3.69892 − 0.39441I 0
u = −0.896752 − 0.854059I
a = 0.459172 + 0.811143I
b = 2.34273 − 0.72734I
−3.69892 + 0.39441I 0
u = −0.929010 + 0.846151I
a = −0.927478 + 0.420395I
b = −2.24639 − 1.74550I
−3.60036 − 5.92816I 0
u = −0.929010 − 0.846151I
a = −0.927478 − 0.420395I
b = −2.24639 + 1.74550I
−3.60036 + 5.92816I 0
u = −0.888018 + 0.909430I
a = −0.29657 − 1.69935I
b = 2.75886 − 1.26977I
−6.66789 + 2.80518I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.888018 − 0.909430I
a = −0.29657 + 1.69935I
b = 2.75886 + 1.26977I
−6.66789 − 2.80518I 0
u = 0.911261 + 0.892581I
a = −0.290253 + 0.624136I
b = 0.068856 + 0.705184I
−9.41866 + 2.57034I 0
u = 0.911261 − 0.892581I
a = −0.290253 − 0.624136I
b = 0.068856 − 0.705184I
−9.41866 − 2.57034I 0
u = 0.898692 + 0.907339I
a = 0.140709 − 0.798542I
b = −0.287738 − 0.771476I
−11.28450 − 1.94871I 0
u = 0.898692 − 0.907339I
a = 0.140709 + 0.798542I
b = −0.287738 + 0.771476I
−11.28450 + 1.94871I 0
u = −0.905732 + 0.903520I
a = −0.07637 + 2.14749I
b = −3.85473 + 1.23788I
−12.02480 − 0.74463I 0
u = −0.905732 − 0.903520I
a = −0.07637 − 2.14749I
b = −3.85473 − 1.23788I
−12.02480 + 0.74463I 0
u = −0.889133 + 0.920620I
a = 0.53328 + 1.76854I
b = −2.56436 + 1.69325I
−9.24667 + 8.31125I 0
u = −0.889133 − 0.920620I
a = 0.53328 − 1.76854I
b = −2.56436 − 1.69325I
−9.24667 − 8.31125I 0
u = 0.940923 + 0.878414I
a = −0.602531 + 0.307679I
b = −0.386679 + 0.597282I
−9.32294 + 3.97310I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.940923 − 0.878414I
a = −0.602531 − 0.307679I
b = −0.386679 − 0.597282I
−9.32294 − 3.97310I 0
u = −0.951721 + 0.881216I
a = 2.16536 − 0.07906I
b = 2.78923 + 3.81036I
−11.87630 − 5.84179I 0
u = −0.951721 − 0.881216I
a = 2.16536 + 0.07906I
b = 2.78923 − 3.81036I
−11.87630 + 5.84179I 0
u = 0.958325 + 0.878505I
a = 0.755872 − 0.166096I
b = 0.610563 − 0.547751I
−11.09220 + 8.53782I 0
u = 0.958325 − 0.878505I
a = 0.755872 + 0.166096I
b = 0.610563 + 0.547751I
−11.09220 − 8.53782I 0
u = −0.965439 + 0.872470I
a = −1.71394 − 0.28873I
b = −1.90629 − 3.41650I
−6.41881 − 9.38008I 0
u = −0.965439 − 0.872470I
a = −1.71394 + 0.28873I
b = −1.90629 + 3.41650I
−6.41881 + 9.38008I 0
u = 0.928310 + 0.917655I
a = −0.170687 − 0.499826I
b = −0.403670 − 0.360916I
−12.77690 + 4.46172I 0
u = 0.928310 − 0.917655I
a = −0.170687 + 0.499826I
b = −0.403670 + 0.360916I
−12.77690 − 4.46172I 0
u = −0.972378 + 0.878766I
a = 1.76827 + 0.52427I
b = 1.60446 + 3.65424I
−8.9768 − 14.9420I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.972378 − 0.878766I
a = 1.76827 − 0.52427I
b = 1.60446 − 3.65424I
−8.9768 + 14.9420I 0
u = 0.949802 + 0.906372I
a = 0.487273 + 0.146616I
b = 0.517570 − 0.112853I
−12.70680 + 2.25291I 0
u = 0.949802 − 0.906372I
a = 0.487273 − 0.146616I
b = 0.517570 + 0.112853I
−12.70680 − 2.25291I 0
u = 0.259476 + 0.549278I
a = 0.69911 − 1.56843I
b = −0.640976 − 0.684812I
2.10455 + 0.34368I 5.70795 − 0.43243I
u = 0.259476 − 0.549278I
a = 0.69911 + 1.56843I
b = −0.640976 + 0.684812I
2.10455 − 0.34368I 5.70795 + 0.43243I
u = 0.150083 + 0.588539I
a = −0.81656 + 1.59215I
b = 0.461872 + 0.688311I
0.54630 + 5.24919I 2.36877 − 5.53828I
u = 0.150083 − 0.588539I
a = −0.81656 − 1.59215I
b = 0.461872 − 0.688311I
0.54630 − 5.24919I 2.36877 + 5.53828I
u = 0.577939
a = 0.191919
b = −0.448397
0.745390 14.0380
u = −0.122739 + 0.348425I
a = −0.75582 + 1.74701I
b = 0.302481 + 0.442001I
−1.73126 − 0.78637I −2.82494 + 1.32713I
u = −0.122739 − 0.348425I
a = −0.75582 − 1.74701I
b = 0.302481 − 0.442001I
−1.73126 + 0.78637I −2.82494 − 1.32713I
11
II. I
u
2
= h−u
4
+ u
3
+ b − 1, −u
4
+ u
3
+ a − u − 1, u
5
− u
4
+ u
2
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
4
=
u
4
− u
3
+ u + 1
u
4
− u
3
+ 1
a
7
=
u
−u
3
+ u
a
8
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
2
a
3
=
u
4
− u
3
+ u + 1
u
4
− u
3
+ 1
a
5
=
−u
3
u
4
− u
3
− u
2
+ 1
a
2
=
u
4
+ u + 1
u
2
a
9
=
u
4
− u
2
+ 1
u
4
a
1
=
u
3
−u
4
+ u
3
+ u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
3
− 3u
2
+ u + 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
7
u
5
c
4
(u + 1)
5
c
5
, c
8
, c
9
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
6
u
5
+ u
4
− u
2
+ u + 1
c
10
u
5
− u
4
+ u
2
+ u − 1
c
11
, c
12
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
7
y
5
c
5
, c
8
, c
9
c
11
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
c
6
, c
10
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.758138 + 0.584034I
a = −0.827780 − 0.637683I
b = −0.069642 − 1.221720I
−3.46474 − 2.21397I 0.88087 + 4.04855I
u = −0.758138 − 0.584034I
a = −0.827780 + 0.637683I
b = −0.069642 + 1.221720I
−3.46474 + 2.21397I 0.88087 − 4.04855I
u = 0.935538 + 0.903908I
a = 0.552827 − 0.534136I
b = −0.38271 − 1.43804I
−12.60320 + 3.33174I 1.28666 − 2.53508I
u = 0.935538 − 0.903908I
a = 0.552827 + 0.534136I
b = −0.38271 + 1.43804I
−12.60320 − 3.33174I 1.28666 + 2.53508I
u = 0.645200
a = 1.54991
b = 0.904706
−0.762751 1.66490
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
70
+ 34u
69
+ ··· + 90u + 1)
c
2
((u − 1)
5
)(u
70
− 6u
69
+ ··· + 10u − 1)
c
3
, c
7
u
5
(u
70
− u
69
+ ··· − 160u + 32)
c
4
((u + 1)
5
)(u
70
− 6u
69
+ ··· + 10u − 1)
c
5
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
70
+ 2u
69
+ ··· + 1156u − 809)
c
6
(u
5
+ u
4
− u
2
+ u + 1)(u
70
+ 2u
69
+ ··· + 7u
3
− 1)
c
8
, c
9
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
70
− 14u
69
+ ··· − 6u
2
+ 1)
c
10
(u
5
− u
4
+ u
2
+ u − 1)(u
70
+ 2u
69
+ ··· + 7u
3
− 1)
c
11
, c
12
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
70
− 14u
69
+ ··· − 6u
2
+ 1)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
70
+ 10y
69
+ ··· − 6198y + 1)
c
2
, c
4
((y −1)
5
)(y
70
− 34y
69
+ ··· − 90y + 1)
c
3
, c
7
y
5
(y
70
− 33y
69
+ ··· − 19968y + 1024)
c
5
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
· (y
70
+ 2y
69
+ ··· − 21431896y + 654481)
c
6
, c
10
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
70
− 14y
69
+ ··· − 6y
2
+ 1)
c
8
, c
9
, c
11
c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
70
+ 86y
69
+ ··· − 12y + 1)
17
