12a
0022
(K12a
0022
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 10 12 11 4 1 8 7
Solving Sequence
3,5
2 6
4,10
7 1 11 9 8 12
c
2
c
5
c
3
c
6
c
1
c
10
c
9
c
8
c
12
c
4
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−40u
81
+ 145u
80
+ ··· + 8b − 29, −19u
81
+ 107u
80
+ ··· + 4a + 41, u
82
− 5u
81
+ ··· − 5u + 1i
I
u
2
= hb
4
− b
3
u − b
3
+ b
2
u − u − 1, a, u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 90 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−40u
81
+ 145u
80
+ · · · + 8b − 29, −19u
81
+ 107u
80
+ · · · + 4a +
41, u
82
− 5u
81
+ · · · − 5u + 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
4
=
u
4
+ u
2
+ 1
u
6
+ 2u
4
+ u
2
a
10
=
4.75000u
81
− 26.7500u
80
+ ··· + 42.2500u − 10.2500
5u
81
−
145
8
u
80
+ ··· − 9u +
29
8
a
7
=
u
12
+ 3u
10
+ 5u
8
− 2u
7
+ 4u
6
− 4u
5
+ 2u
4
− 4u
3
+ u
2
+ 1
−
1
8
u
81
+
5
8
u
80
+ ··· −
5
8
u +
1
8
a
1
=
u
2
+ 1
u
2
a
11
=
6.87500u
81
− 28.1250u
80
+ ··· + 13.6250u − 4.12500
18u
81
− 74u
80
+ ··· +
5
2
u +
7
2
a
9
=
23
4
u
81
−
103
4
u
80
+ ··· +
69
4
u −
17
4
61
4
u
81
−
509
8
u
80
+ ··· +
9
4
u +
29
8
a
8
=
11
2
u
81
−
105
4
u
80
+ ··· +
81
4
u −
5
2
13
2
u
81
−
251
8
u
80
+ ··· + 18u −
25
8
a
12
=
3
8
u
81
−
3
2
u
80
+ ··· +
3
8
u + 1
−
1
2
u
81
+
9
2
u
80
+ ··· −
43
4
u +
5
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
109
8
u
81
−
281
4
u
80
+ ··· +
271
8
u +
21
4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
82
+ 41u
81
+ ··· + 5u + 1
c
2
, c
5
u
82
+ 5u
81
+ ··· + 5u + 1
c
3
u
82
− 5u
81
+ ··· + 4087u + 1321
c
4
, c
9
u
82
+ u
81
+ ··· + 128u + 256
c
6
u
82
+ 3u
81
+ ··· + 81583u + 8329
c
7
, c
8
, c
11
c
12
u
82
+ 3u
81
+ ··· + 5u + 1
c
10
u
82
− 23u
81
+ ··· − 63989u + 3971
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
82
+ 5y
81
+ ··· + 41y + 1
c
2
, c
5
y
82
+ 41y
81
+ ··· + 5y + 1
c
3
y
82
− 31y
81
+ ··· − 31850155y + 1745041
c
4
, c
9
y
82
+ 45y
81
+ ··· + 1327104y + 65536
c
6
y
82
− 37y
81
+ ··· − 2124260175y + 69372241
c
7
, c
8
, c
11
c
12
y
82
+ 95y
81
+ ··· + y + 1
c
10
y
82
− 17y
81
+ ··· + 101206189y + 15768841
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.704314 + 0.732833I
a = −0.039392 − 0.613136I
b = −0.383636 + 0.526485I
−0.38177 − 5.18756I 0
u = −0.704314 − 0.732833I
a = −0.039392 + 0.613136I
b = −0.383636 − 0.526485I
−0.38177 + 5.18756I 0
u = −0.105779 + 0.955141I
a = −1.38658 − 0.34528I
b = −0.737097 + 0.693464I
−8.72790 − 2.31871I 0
u = −0.105779 − 0.955141I
a = −1.38658 + 0.34528I
b = −0.737097 − 0.693464I
−8.72790 + 2.31871I 0
u = −0.741465 + 0.744493I
a = 0.296277 + 0.595555I
b = 0.501466 − 0.980876I
−8.14095 − 7.21306I 0
u = −0.741465 − 0.744493I
a = 0.296277 − 0.595555I
b = 0.501466 + 0.980876I
−8.14095 + 7.21306I 0
u = −0.625049 + 0.701797I
a = −0.378967 + 0.585059I
b = 0.198631 + 0.083439I
1.02352 − 2.11165I 0
u = −0.625049 − 0.701797I
a = −0.378967 − 0.585059I
b = 0.198631 − 0.083439I
1.02352 + 2.11165I 0
u = −0.567368 + 0.920415I
a = −0.378006 + 0.357257I
b = −0.244076 + 0.661524I
0.39375 − 2.56563I 0
u = −0.567368 − 0.920415I
a = −0.378006 − 0.357257I
b = −0.244076 − 0.661524I
0.39375 + 2.56563I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.659941 + 0.872179I
a = 0.277339 − 0.046151I
b = 0.879179 + 0.019524I
−0.794028 − 0.020453I 0
u = −0.659941 − 0.872179I
a = 0.277339 + 0.046151I
b = 0.879179 − 0.019524I
−0.794028 + 0.020453I 0
u = 0.854489 + 0.292104I
a = −2.34167 + 0.05715I
b = −2.00466 − 0.97235I
−10.7779 − 10.0786I 0
u = 0.854489 − 0.292104I
a = −2.34167 − 0.05715I
b = −2.00466 + 0.97235I
−10.7779 + 10.0786I 0
u = 0.410592 + 1.038090I
a = −0.128468 − 0.117863I
b = −1.282850 + 0.198070I
−7.98722 − 0.83968I 0
u = 0.410592 − 1.038090I
a = −0.128468 + 0.117863I
b = −1.282850 − 0.198070I
−7.98722 + 0.83968I 0
u = 0.829105 + 0.286958I
a = 2.10320 − 0.00102I
b = 1.63515 + 0.78199I
−2.85556 − 7.68645I 0
u = 0.829105 − 0.286958I
a = 2.10320 + 0.00102I
b = 1.63515 − 0.78199I
−2.85556 + 7.68645I 0
u = −0.701766 + 0.880022I
a = −0.179014 − 0.153824I
b = −1.302540 − 0.384385I
−8.54157 + 1.77332I 0
u = −0.701766 − 0.880022I
a = −0.179014 + 0.153824I
b = −1.302540 + 0.384385I
−8.54157 − 1.77332I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.835427 + 0.163941I
a = −1.43377 + 0.88175I
b = −1.58005 + 0.86107I
−12.74480 + 0.10746I −3.09571 + 0.I
u = 0.835427 − 0.163941I
a = −1.43377 − 0.88175I
b = −1.58005 − 0.86107I
−12.74480 − 0.10746I −3.09571 + 0.I
u = −0.228543 + 0.813377I
a = 1.064610 + 0.363456I
b = 0.610969 − 0.498748I
−1.37873 − 1.65649I −3.71571 + 4.84793I
u = −0.228543 − 0.813377I
a = 1.064610 − 0.363456I
b = 0.610969 + 0.498748I
−1.37873 + 1.65649I −3.71571 − 4.84793I
u = 0.447742 + 1.064870I
a = 0.026648 + 0.300164I
b = 0.936468 + 0.133542I
−1.27806 + 1.55480I 0
u = 0.447742 − 1.064870I
a = 0.026648 − 0.300164I
b = 0.936468 − 0.133542I
−1.27806 − 1.55480I 0
u = −0.401903 + 1.086010I
a = −0.50943 − 1.80938I
b = −1.94745 − 0.21375I
−3.10543 − 0.73820I 0
u = −0.401903 − 1.086010I
a = −0.50943 + 1.80938I
b = −1.94745 + 0.21375I
−3.10543 + 0.73820I 0
u = 0.792544 + 0.272348I
a = −1.77905 − 0.02879I
b = −1.224390 − 0.466760I
−1.18892 − 3.99153I 4.00000 + 2.29977I
u = 0.792544 − 0.272348I
a = −1.77905 + 0.02879I
b = −1.224390 + 0.466760I
−1.18892 + 3.99153I 4.00000 − 2.29977I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.646960 + 0.529804I
a = 1.00756 − 1.18930I
b = −0.436228 − 1.048150I
−4.34258 − 1.45718I 4.00000 + 3.03293I
u = −0.646960 − 0.529804I
a = 1.00756 + 1.18930I
b = −0.436228 + 1.048150I
−4.34258 + 1.45718I 4.00000 − 3.03293I
u = −0.585550 + 1.006890I
a = 0.847118 − 0.654790I
b = 0.14547 − 1.74184I
−5.72135 − 3.35594I 0
u = −0.585550 − 1.006890I
a = 0.847118 + 0.654790I
b = 0.14547 + 1.74184I
−5.72135 + 3.35594I 0
u = −0.473787 + 1.064880I
a = −0.18710 + 1.59396I
b = 1.60049 + 1.10893I
−0.74392 − 3.35669I 0
u = −0.473787 − 1.064880I
a = −0.18710 − 1.59396I
b = 1.60049 − 1.10893I
−0.74392 + 3.35669I 0
u = 0.480590 + 1.080350I
a = −0.007942 − 0.520219I
b = −0.542127 − 0.479128I
−0.99744 + 5.35792I 0
u = 0.480590 − 1.080350I
a = −0.007942 + 0.520219I
b = −0.542127 + 0.479128I
−0.99744 − 5.35792I 0
u = 0.786259 + 0.197156I
a = 1.39080 − 0.38724I
b = 1.192400 − 0.326068I
−4.31599 − 1.20763I −1.64666 − 0.57039I
u = 0.786259 − 0.197156I
a = 1.39080 + 0.38724I
b = 1.192400 + 0.326068I
−4.31599 + 1.20763I −1.64666 + 0.57039I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.387160 + 1.125240I
a = 0.74985 + 2.17217I
b = 2.37416 − 0.11109I
−11.12530 + 0.93298I 0
u = −0.387160 − 1.125240I
a = 0.74985 − 2.17217I
b = 2.37416 + 0.11109I
−11.12530 − 0.93298I 0
u = 0.520504 + 1.092360I
a = 0.095680 + 0.832445I
b = −0.021566 + 1.043640I
−7.02432 + 7.67620I 0
u = 0.520504 − 1.092360I
a = 0.095680 − 0.832445I
b = −0.021566 − 1.043640I
−7.02432 − 7.67620I 0
u = −0.489951 + 1.107000I
a = 0.44869 − 2.01555I
b = −2.19473 − 1.55021I
−2.46183 − 6.60369I 0
u = −0.489951 − 1.107000I
a = 0.44869 + 2.01555I
b = −2.19473 + 1.55021I
−2.46183 + 6.60369I 0
u = 0.278575 + 1.179870I
a = −0.079195 − 1.106280I
b = 1.35992 − 0.65097I
−5.69304 − 0.73230I 0
u = 0.278575 − 1.179870I
a = −0.079195 + 1.106280I
b = 1.35992 + 0.65097I
−5.69304 + 0.73230I 0
u = 0.252443 + 1.203480I
a = −0.05183 + 1.44329I
b = −1.47746 + 0.48712I
−7.63888 − 4.38566I 0
u = 0.252443 − 1.203480I
a = −0.05183 − 1.44329I
b = −1.47746 − 0.48712I
−7.63888 + 4.38566I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.494792 + 1.131070I
a = −0.56672 + 2.31046I
b = 2.65647 + 1.74358I
−10.36760 − 8.71428I 0
u = −0.494792 − 1.131070I
a = −0.56672 − 2.31046I
b = 2.65647 − 1.74358I
−10.36760 + 8.71428I 0
u = 0.322106 + 1.195250I
a = 0.527321 + 0.957086I
b = −1.34997 + 0.92937I
−8.57034 + 2.40449I 0
u = 0.322106 − 1.195250I
a = 0.527321 − 0.957086I
b = −1.34997 − 0.92937I
−8.57034 − 2.40449I 0
u = 0.241449 + 1.225680I
a = 0.08124 − 1.72667I
b = 1.60146 − 0.35341I
−15.7352 − 6.6947I 0
u = 0.241449 − 1.225680I
a = 0.08124 + 1.72667I
b = 1.60146 + 0.35341I
−15.7352 + 6.6947I 0
u = 0.633300 + 0.364129I
a = 1.146130 + 0.757063I
b = −0.206752 + 0.243475I
−4.89815 − 3.14508I 2.29655 + 3.69057I
u = 0.633300 − 0.364129I
a = 1.146130 − 0.757063I
b = −0.206752 − 0.243475I
−4.89815 + 3.14508I 2.29655 − 3.69057I
u = 0.338253 + 1.225240I
a = −0.917777 − 1.065290I
b = 1.49348 − 1.21156I
−17.0811 + 4.0269I 0
u = 0.338253 − 1.225240I
a = −0.917777 + 1.065290I
b = 1.49348 + 1.21156I
−17.0811 − 4.0269I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.529169 + 1.165730I
a = −0.55607 + 1.33364I
b = −1.49781 + 0.01915I
−7.15078 + 6.07252I 0
u = 0.529169 − 1.165730I
a = −0.55607 − 1.33364I
b = −1.49781 − 0.01915I
−7.15078 − 6.07252I 0
u = 0.556937 + 1.154960I
a = 0.23026 − 1.59190I
b = 1.89941 − 0.80869I
−3.79190 + 9.02671I 0
u = 0.556937 − 1.154960I
a = 0.23026 + 1.59190I
b = 1.89941 + 0.80869I
−3.79190 − 9.02671I 0
u = 0.571315 + 1.163160I
a = −0.20043 + 1.83707I
b = −2.42638 + 0.96617I
−5.46594 + 12.87520I 0
u = 0.571315 − 1.163160I
a = −0.20043 − 1.83707I
b = −2.42638 − 0.96617I
−5.46594 − 12.87520I 0
u = 0.358574 + 0.605548I
a = −0.725871 − 0.684487I
b = 0.181111 + 1.173260I
−6.57846 + 4.17153I −0.218037 + 0.553666I
u = 0.358574 − 0.605548I
a = −0.725871 + 0.684487I
b = 0.181111 − 1.173260I
−6.57846 − 4.17153I −0.218037 − 0.553666I
u = 0.522003 + 1.192470I
a = 0.95599 − 1.38115I
b = 1.72988 + 0.74459I
−15.8138 + 4.8468I 0
u = 0.522003 − 1.192470I
a = 0.95599 + 1.38115I
b = 1.72988 − 0.74459I
−15.8138 − 4.8468I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.580577 + 1.170980I
a = 0.20698 − 2.03262I
b = 2.86387 − 1.01322I
−13.4120 + 15.3715I 0
u = 0.580577 − 1.170980I
a = 0.20698 + 2.03262I
b = 2.86387 + 1.01322I
−13.4120 − 15.3715I 0
u = −0.633105 + 0.172755I
a = −2.82368 + 0.73281I
b = −1.72041 + 1.20662I
−7.70310 + 4.34280I 1.05723 − 2.52323I
u = −0.633105 − 0.172755I
a = −2.82368 − 0.73281I
b = −1.72041 − 1.20662I
−7.70310 − 4.34280I 1.05723 + 2.52323I
u = −0.469905 + 0.400587I
a = −1.70527 + 0.49698I
b = −0.460855 + 0.893450I
1.183240 − 0.630202I 8.42558 + 2.80238I
u = −0.469905 − 0.400587I
a = −1.70527 − 0.49698I
b = −0.460855 − 0.893450I
1.183240 + 0.630202I 8.42558 − 2.80238I
u = 0.353527 + 0.489455I
a = 0.831523 + 0.783639I
b = −0.196486 − 0.779621I
0.57076 + 2.04372I 3.58253 − 2.28115I
u = 0.353527 − 0.489455I
a = 0.831523 − 0.783639I
b = −0.196486 + 0.779621I
0.57076 − 2.04372I 3.58253 + 2.28115I
u = −0.550775 + 0.221389I
a = 2.42931 − 0.59762I
b = 1.16401 − 1.01324I
−0.02868 + 2.38938I 3.67522 − 4.58945I
u = −0.550775 − 0.221389I
a = 2.42931 + 0.59762I
b = 1.16401 + 1.01324I
−0.02868 − 2.38938I 3.67522 + 4.58945I
12
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.472632 + 0.351594I
a = −0.840285 − 0.800430I
b = 0.213524 + 0.348569I
1.10262 − 1.31646I 6.13864 + 5.00378I
u = 0.472632 − 0.351594I
a = −0.840285 + 0.800430I
b = 0.213524 − 0.348569I
1.10262 + 1.31646I 6.13864 − 5.00378I
13
II. I
u
2
= hb
4
− b
3
u − b
3
+ b
2
u − u − 1, a, u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
−u − 1
a
6
=
u
u + 1
a
4
=
0
u
a
10
=
0
b
a
7
=
u
b
2
u + u + 1
a
1
=
−u
−u − 1
a
11
=
bu + b
2b
a
9
=
0
b
a
8
=
−b
3
u
−2b
3
u − 2b
3
+ b
a
12
=
−b
2
− u
−b
3
u − b
3
+ 2b
2
u − 2u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5b
2
u − 3b
2
+ 3bu − b + 5u + 6
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
4
, c
9
u
8
c
6
, c
10
(u
4
+ u
3
+ u
2
+ 1)
2
c
7
, c
8
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
11
, 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
3
c
5
(y
2
+ y + 1)
4
c
4
, c
9
y
8
c
6
, c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
7
, c
8
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0
b = 0.447930 − 0.664845I
0.21101 − 3.44499I 1.64912 + 8.49900I
u = −0.500000 + 0.866025I
a = 0
b = −0.799738 + 0.055496I
0.211005 − 0.614778I 4.65255 − 0.59814I
u = −0.500000 + 0.866025I
a = 0
b = −0.363298 + 1.193330I
−6.79074 − 5.19385I −1.80063 + 6.43123I
u = −0.500000 + 0.866025I
a = 0
b = 1.215110 + 0.282041I
−6.79074 + 1.13408I 1.99896 + 0.39034I
u = −0.500000 − 0.866025I
a = 0
b = 0.447930 + 0.664845I
0.21101 + 3.44499I 1.64912 − 8.49900I
u = −0.500000 − 0.866025I
a = 0
b = −0.799738 − 0.055496I
0.211005 + 0.614778I 4.65255 + 0.59814I
u = −0.500000 − 0.866025I
a = 0
b = −0.363298 − 1.193330I
−6.79074 + 5.19385I −1.80063 − 6.43123I
u = −0.500000 − 0.866025I
a = 0
b = 1.215110 − 0.282041I
−6.79074 − 1.13408I 1.99896 − 0.39034I
17
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
82
+ 41u
81
+ ··· + 5u + 1)
c
2
((u
2
+ u + 1)
4
)(u
82
+ 5u
81
+ ··· + 5u + 1)
c
3
((u
2
− u + 1)
4
)(u
82
− 5u
81
+ ··· + 4087u + 1321)
c
4
, c
9
u
8
(u
82
+ u
81
+ ··· + 128u + 256)
c
5
((u
2
− u + 1)
4
)(u
82
+ 5u
81
+ ··· + 5u + 1)
c
6
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
82
+ 3u
81
+ ··· + 81583u + 8329)
c
7
, c
8
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
82
+ 3u
81
+ ··· + 5u + 1)
c
10
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
82
− 23u
81
+ ··· − 63989u + 3971)
c
11
, c
12
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
82
+ 3u
81
+ ··· + 5u + 1)
18
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
82
+ 5y
81
+ ··· + 41y + 1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
82
+ 41y
81
+ ··· + 5y + 1)
c
3
((y
2
+ y + 1)
4
)(y
82
− 31y
81
+ ··· − 3.18502 × 10
7
y + 1745041)
c
4
, c
9
y
8
(y
82
+ 45y
81
+ ··· + 1327104y + 65536)
c
6
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
82
− 37y
81
+ ··· − 2124260175y + 69372241)
c
7
, c
8
, c
11
c
12
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
82
+ 95y
81
+ ··· + y + 1)
c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
82
− 17y
81
+ ··· + 101206189y + 15768841)
19
