12n
0145
(K12n
0145
)
A knot diagram
1
Linearized knot diagam
3 4 11 2 10 3 12 4 6 9 8 7
Solving Sequence
4,8 9,11
12 3 2 5 1 7 6 10
c
8
c
11
c
3
c
2
c
4
c
1
c
7
c
6
c
10
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.89198 × 10
43
u
23
+ 1.93706 × 10
43
u
22
+ ··· + 7.16795 × 10
45
b + 5.20518 × 10
46
,
− 8.64361 × 10
44
u
23
− 3.86826 × 10
43
u
22
+ ··· + 2.95871 × 10
48
a + 6.38636 × 10
47
,
u
24
− 2u
23
+ ··· − 4461u + 2683i
I
u
2
= hu
3
− 3u
2
+ 2b + 3u − 1, −u
3
+ 4u
2
+ 6a − u − 6, u
4
− 4u
3
+ 4u
2
+ 3i
I
u
3
= hb, a
2
+ a + 1, u + 1i
I
u
4
= h−u
3
− 3u
2
+ b − 3u + 4, −3u
3
− 8u
2
+ 3a − 8u + 12, u
4
+ 2u
3
+ u
2
− 6u + 3i
I
u
5
= hb, a − 1, u
2
− u + 1i
* 5 irreducible components of dim
C
= 0, with total 36 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−1.89 × 10
43
u
23
+ 1.94 × 10
43
u
22
+ · · · + 7.17 × 10
45
b + 5.21 ×
10
46
, −8.64 × 10
44
u
23
− 3.87 × 10
43
u
22
+ · · · + 2.96 × 10
48
a + 6.39 ×
10
47
, u
24
− 2u
23
+ · · · − 4461u + 2683i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
11
=
0.000292141u
23
+ 0.0000130742u
22
+ ··· + 0.720325u − 0.215850
0.00263951u
23
− 0.00270240u
22
+ ··· + 4.46888u − 7.26175
a
12
=
0.00293165u
23
− 0.00268932u
22
+ ··· + 5.18921u − 7.47760
0.00263951u
23
− 0.00270240u
22
+ ··· + 4.46888u − 7.26175
a
3
=
0.00111111u
23
− 0.00111164u
22
+ ··· + 2.33618u − 2.85740
0.00144537u
23
− 0.00161867u
22
+ ··· + 2.86920u − 4.75166
a
2
=
0.00111111u
23
− 0.00111164u
22
+ ··· + 2.33618u − 2.85740
0.00274132u
23
− 0.00268026u
22
+ ··· + 4.84242u − 7.73137
a
5
=
0.00192313u
23
− 0.00205057u
22
+ ··· + 3.04993u − 5.46309
0.00295092u
23
− 0.00292037u
22
+ ··· + 5.36339u − 8.37266
a
1
=
0.00361033u
23
− 0.00409191u
22
+ ··· + 6.99968u − 10.2748
0.00380171u
23
− 0.00433836u
22
+ ··· + 6.92476u − 11.2245
a
7
=
−0.00177479u
23
+ 0.00171084u
22
+ ··· − 3.65352u + 4.54167
−0.00354581u
23
+ 0.00380753u
22
+ ··· − 6.59708u + 9.57302
a
6
=
0.000894363u
23
− 0.000991958u
22
+ ··· + 0.953738u − 2.60877
−0.00266810u
23
+ 0.00288557u
22
+ ··· − 5.10145u + 7.77036
a
10
=
0.00191635u
23
− 0.00193496u
22
+ ··· + 3.30821u − 5.87489
0.00372145u
23
− 0.00402551u
22
+ ··· + 5.91212u − 10.7507
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0.00451241u
23
− 0.00553102u
22
+ ··· + 5.54839u − 13.6361
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
24
+ 43u
23
+ ··· + 1172u + 81
c
2
, c
4
u
24
− 3u
23
+ ··· − 68u + 9
c
3
u
24
+ 3u
23
+ ··· − 4u + 3
c
5
, c
9
u
24
− 3u
23
+ ··· + 10u + 3
c
6
u
24
+ 2u
23
+ ··· + 20857u + 9299
c
7
, c
11
, c
12
u
24
+ u
23
+ ··· − 32u + 16
c
8
u
24
− 2u
23
+ ··· − 4461u + 2683
c
10
u
24
+ 19u
23
+ ··· + 68u + 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
− 117y
23
+ ··· − 1128316y + 6561
c
2
, c
4
y
24
+ 43y
23
+ ··· + 1172y + 81
c
3
y
24
+ 3y
23
+ ··· + 68y + 9
c
5
, c
9
y
24
+ 19y
23
+ ··· + 68y + 9
c
6
y
24
+ 82y
23
+ ··· + 1135326279y + 86471401
c
7
, c
11
, c
12
y
24
+ 41y
23
+ ··· + 2048y + 256
c
8
y
24
− 34y
23
+ ··· − 21038113y + 7198489
c
10
y
24
− 21y
23
+ ··· + 5492y + 81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.232862 + 0.947035I
a = −0.649985 − 0.077686I
b = 0.374440 + 0.304409I
0.779807 + 1.048970I 7.25519 − 5.58365I
u = 0.232862 − 0.947035I
a = −0.649985 + 0.077686I
b = 0.374440 − 0.304409I
0.779807 − 1.048970I 7.25519 + 5.58365I
u = 0.943560 + 0.064690I
a = 0.988902 + 0.233923I
b = −0.682289 + 0.802318I
−3.89306 − 0.24557I −2.31984 + 1.35715I
u = 0.943560 − 0.064690I
a = 0.988902 − 0.233923I
b = −0.682289 − 0.802318I
−3.89306 + 0.24557I −2.31984 − 1.35715I
u = −0.959379 + 0.455770I
a = 0.646370 − 0.274793I
b = −0.415041 + 0.335328I
−0.27313 + 3.15044I 2.29434 − 0.19419I
u = −0.959379 − 0.455770I
a = 0.646370 + 0.274793I
b = −0.415041 − 0.335328I
−0.27313 − 3.15044I 2.29434 + 0.19419I
u = −0.502276 + 0.647230I
a = −0.990775 + 0.287392I
b = 0.278387 + 0.380293I
0.422299 + 1.283840I 4.39270 − 6.02370I
u = −0.502276 − 0.647230I
a = −0.990775 − 0.287392I
b = 0.278387 − 0.380293I
0.422299 − 1.283840I 4.39270 + 6.02370I
u = 0.933542 + 0.786148I
a = −0.749432 − 0.148046I
b = 0.084838 − 1.367360I
−4.97420 − 2.33173I −0.59644 + 2.89442I
u = 0.933542 − 0.786148I
a = −0.749432 + 0.148046I
b = 0.084838 + 1.367360I
−4.97420 + 2.33173I −0.59644 − 2.89442I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.607235 + 1.070690I
a = −0.470711 + 0.110611I
b = 0.07764 − 1.45718I
−4.85669 − 2.39093I 0.84102 + 2.37108I
u = 0.607235 − 1.070690I
a = −0.470711 − 0.110611I
b = 0.07764 + 1.45718I
−4.85669 + 2.39093I 0.84102 − 2.37108I
u = 1.21724 + 0.74996I
a = 0.400953 − 1.002140I
b = 0.15154 − 2.05062I
15.8709 − 2.8783I −1.59016 + 0.69856I
u = 1.21724 − 0.74996I
a = 0.400953 + 1.002140I
b = 0.15154 + 2.05062I
15.8709 + 2.8783I −1.59016 − 0.69856I
u = 1.52531 + 0.37916I
a = 0.108996 − 0.705821I
b = 0.138338 − 0.696834I
−1.17892 + 4.17832I −0.72824 − 4.64199I
u = 1.52531 − 0.37916I
a = 0.108996 + 0.705821I
b = 0.138338 + 0.696834I
−1.17892 − 4.17832I −0.72824 + 4.64199I
u = −1.78012 + 0.29967I
a = −0.680725 + 0.483351I
b = 0.08900 + 1.95850I
−18.4389 + 3.9255I 0.72969 − 1.86370I
u = −1.78012 − 0.29967I
a = −0.680725 − 0.483351I
b = 0.08900 − 1.95850I
−18.4389 − 3.9255I 0.72969 + 1.86370I
u = −1.82828 + 0.75998I
a = 0.679108 − 0.172953I
b = −0.71021 − 1.48438I
−10.81740 + 5.66544I −1.94284 − 3.68332I
u = −1.82828 − 0.75998I
a = 0.679108 + 0.172953I
b = −0.71021 + 1.48438I
−10.81740 − 5.66544I −1.94284 + 3.68332I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.85631 + 0.81470I
a = 0.104205 + 0.704147I
b = 0.40009 + 1.66939I
−9.33657 − 1.50490I −1.60303 + 1.18708I
u = −1.85631 − 0.81470I
a = 0.104205 − 0.704147I
b = 0.40009 − 1.66939I
−9.33657 + 1.50490I −1.60303 − 1.18708I
u = 2.46660 + 0.93685I
a = 0.504632 + 0.296006I
b = −0.28674 + 1.91373I
16.9567 − 10.8896I 0. + 4.72097I
u = 2.46660 − 0.93685I
a = 0.504632 − 0.296006I
b = −0.28674 − 1.91373I
16.9567 + 10.8896I 0. − 4.72097I
7
II. I
u
2
= hu
3
− 3u
2
+ 2b + 3u − 1, −u
3
+ 4u
2
+ 6a − u − 6, u
4
− 4u
3
+ 4u
2
+ 3i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
11
=
1
6
u
3
−
2
3
u
2
+
1
6
u + 1
−
1
2
u
3
+
3
2
u
2
−
3
2
u +
1
2
a
12
=
−
1
3
u
3
+
5
6
u
2
−
4
3
u +
3
2
−
1
2
u
3
+
3
2
u
2
−
3
2
u +
1
2
a
3
=
1
6
u
3
−
2
3
u
2
+
7
6
u − 1
1
2
u
3
−
3
2
u
2
+
3
2
u +
1
2
a
2
=
1
6
u
3
−
2
3
u
2
+
7
6
u − 1
u
3
−
5
2
u
2
+ u +
1
2
a
5
=
1
6
u
3
−
2
3
u
2
+
1
6
u + 1
−
1
2
u
3
+
3
2
u
2
−
3
2
u +
1
2
a
1
=
−
1
6
u
3
+
2
3
u
2
−
1
6
u − 1
1
2
u
3
−
3
2
u
2
+
3
2
u −
1
2
a
7
=
−
1
6
u
3
+
7
6
u
2
−
13
6
u −
1
2
−2
a
6
=
−
1
6
u
3
+
2
3
u
2
−
7
6
u
−
1
2
u
3
+
1
2
u
2
−
1
2
u −
1
2
a
10
=
1
6
u
3
−
1
6
u
2
−
5
6
u +
3
2
1
2
u
3
−
3
2
u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 8u
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
(u
2
− u + 1)
2
c
2
, c
9
, c
10
(u
2
+ u + 1)
2
c
6
u
4
+ 4u
3
+ 4u
2
+ 3
c
7
, c
11
, c
12
(u
2
+ 2)
2
c
8
u
4
− 4u
3
+ 4u
2
+ 3
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
9
c
10
(y
2
+ y + 1)
2
c
6
, c
8
y
4
− 8y
3
+ 22y
2
+ 24y + 9
c
7
, c
11
, c
12
(y + 2)
4
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.224745 + 0.707107I
a = 1.316500 + 0.288675I
b = − 1.414210I
−4.93480 + 4.05977I 0. − 6.92820I
u = −0.224745 − 0.707107I
a = 1.316500 − 0.288675I
b = 1.414210I
−4.93480 − 4.05977I 0. + 6.92820I
u = 2.22474 + 0.70711I
a = −0.316497 − 0.288675I
b = − 1.414210I
−4.93480 − 4.05977I 0. + 6.92820I
u = 2.22474 − 0.70711I
a = −0.316497 + 0.288675I
b = 1.414210I
−4.93480 + 4.05977I 0. − 6.92820I
11
III. I
u
3
= hb, a
2
+ a + 1, u + 1i
(i) Arc colorings
a
4
=
0
−1
a
8
=
1
0
a
9
=
1
1
a
11
=
a
0
a
12
=
a
0
a
3
=
a + 1
−1
a
2
=
a + 1
a
a
5
=
−a
0
a
1
=
a
0
a
7
=
1
0
a
6
=
−a
1
a
10
=
0
−a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8a + 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
9
u
2
− u + 1
c
2
, c
3
, c
5
c
10
u
2
+ u + 1
c
6
, c
8
(u + 1)
2
c
7
, c
11
, c
12
u
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
9
c
10
y
2
+ y + 1
c
6
, c
8
(y −1)
2
c
7
, c
11
, c
12
y
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.500000 + 0.866025I
b = 0
4.05977I 6.00000 − 6.92820I
u = −1.00000
a = −0.500000 − 0.866025I
b = 0
− 4.05977I 6.00000 + 6.92820I
15
IV.
I
u
4
= h−u
3
−3u
2
+b −3u+4, −3u
3
−8u
2
+3a −8u+12, u
4
+2u
3
+u
2
−6u +3i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
11
=
u
3
+
8
3
u
2
+
8
3
u − 4
u
3
+ 3u
2
+ 3u − 4
a
12
=
2u
3
+
17
3
u
2
+
17
3
u − 8
u
3
+ 3u
2
+ 3u − 4
a
3
=
−
2
3
u
3
− 2u
2
−
7
3
u + 2
−
1
3
u
3
−
4
3
u
2
− u + 1
a
2
=
−
2
3
u
3
− 2u
2
−
7
3
u + 2
−
2
3
u
3
−
8
3
u
2
− 3u + 3
a
5
=
u
3
+
8
3
u
2
+
8
3
u − 4
u
3
+ 3u
2
+ 3u − 4
a
1
=
−u
3
−
8
3
u
2
−
8
3
u + 4
−u
3
− 3u
2
− 3u + 4
a
7
=
−
1
3
u
3
− u
2
−
5
3
u − 1
−2
a
6
=
1
3
u
3
+
2
3
u
2
+
1
3
u − 3
2
3
u
3
+
2
3
u
2
+ u − 3
a
10
=
5
3
u
3
+
13
3
u
2
+
14
3
u − 6
5
3
u
3
+
14
3
u
2
+ 3u − 5
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
(u
2
− u + 1)
2
c
2
, c
9
, c
10
(u
2
+ u + 1)
2
c
6
u
4
− 2u
3
+ u
2
+ 6u + 3
c
7
, c
11
, c
12
(u
2
+ 2)
2
c
8
u
4
+ 2u
3
+ u
2
− 6u + 3
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
9
c
10
(y
2
+ y + 1)
2
c
6
, c
8
y
4
− 2y
3
+ 31y
2
− 30y + 9
c
7
, c
11
, c
12
(y + 2)
4
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.724745 + 0.158919I
a = −0.408248 + 1.284460I
b = 1.414210I
−4.93480 0
u = 0.724745 − 0.158919I
a = −0.408248 − 1.284460I
b = − 1.414210I
−4.93480 0
u = −1.72474 + 1.57313I
a = 0.408248 − 0.129757I
b = − 1.414210I
−4.93480 0
u = −1.72474 − 1.57313I
a = 0.408248 + 0.129757I
b = 1.414210I
−4.93480 0
19
V. I
u
5
= hb, a − 1, u
2
− u + 1i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u − 1
a
11
=
1
0
a
12
=
1
0
a
3
=
u
u
a
2
=
u
u − 1
a
5
=
−1
0
a
1
=
1
0
a
7
=
1
0
a
6
=
u
u − 1
a
10
=
−u + 2
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
8
, c
9
u
2
− u + 1
c
2
, c
3
, c
5
c
10
u
2
+ u + 1
c
7
, c
11
, c
12
u
2
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
9
, c
10
y
2
+ y + 1
c
7
, c
11
, c
12
y
2
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 1.00000
b = 0
0 0
u = 0.500000 − 0.866025I
a = 1.00000
b = 0
0 0
23
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
6
)(u
24
+ 43u
23
+ ··· + 1172u + 81)
c
2
((u
2
+ u + 1)
6
)(u
24
− 3u
23
+ ··· − 68u + 9)
c
3
((u
2
− u + 1)
4
)(u
2
+ u + 1)
2
(u
24
+ 3u
23
+ ··· − 4u + 3)
c
4
((u
2
− u + 1)
6
)(u
24
− 3u
23
+ ··· − 68u + 9)
c
5
((u
2
− u + 1)
4
)(u
2
+ u + 1)
2
(u
24
− 3u
23
+ ··· + 10u + 3)
c
6
(u + 1)
2
(u
2
− u + 1)(u
4
− 2u
3
+ u
2
+ 6u + 3)(u
4
+ 4u
3
+ 4u
2
+ 3)
· (u
24
+ 2u
23
+ ··· + 20857u + 9299)
c
7
, c
11
, c
12
u
4
(u
2
+ 2)
4
(u
24
+ u
23
+ ··· − 32u + 16)
c
8
(u + 1)
2
(u
2
− u + 1)(u
4
− 4u
3
+ 4u
2
+ 3)(u
4
+ 2u
3
+ u
2
− 6u + 3)
· (u
24
− 2u
23
+ ··· − 4461u + 2683)
c
9
((u
2
− u + 1)
2
)(u
2
+ u + 1)
4
(u
24
− 3u
23
+ ··· + 10u + 3)
c
10
((u
2
+ u + 1)
6
)(u
24
+ 19u
23
+ ··· + 68u + 9)
24
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
24
− 117y
23
+ ··· − 1128316y + 6561)
c
2
, c
4
((y
2
+ y + 1)
6
)(y
24
+ 43y
23
+ ··· + 1172y + 81)
c
3
((y
2
+ y + 1)
6
)(y
24
+ 3y
23
+ ··· + 68y + 9)
c
5
, c
9
((y
2
+ y + 1)
6
)(y
24
+ 19y
23
+ ··· + 68y + 9)
c
6
(y −1)
2
(y
2
+ y + 1)(y
4
− 8y
3
+ 22y
2
+ 24y + 9)
· (y
4
− 2y
3
+ 31y
2
− 30y + 9)
· (y
24
+ 82y
23
+ ··· + 1135326279y + 86471401)
c
7
, c
11
, c
12
y
4
(y + 2)
8
(y
24
+ 41y
23
+ ··· + 2048y + 256)
c
8
(y −1)
2
(y
2
+ y + 1)(y
4
− 8y
3
+ 22y
2
+ 24y + 9)
· (y
4
− 2y
3
+ 31y
2
− 30y + 9)
· (y
24
− 34y
23
+ ··· − 21038113y + 7198489)
c
10
((y
2
+ y + 1)
6
)(y
24
− 21y
23
+ ··· + 5492y + 81)
25
