12n
0439
(K12n
0439
)
A knot diagram
1
Linearized knot diagam
3 6 12 11 2 4 11 12 3 6 8 10
Solving Sequence
8,12 4,9
3 10 1 11 5 7 6 2
c
8
c
3
c
9
c
12
c
11
c
4
c
7
c
6
c
2
c
1
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−51u
8
+ 4u
7
+ 487u
6
+ 424u
5
− 1094u
4
− 281u
3
+ 1891u
2
+ 551b + 812u + 165,
267u
8
+ 919u
7
+ 173u
6
− 2317u
5
+ 185u
4
+ 5555u
3
+ 2060u
2
+ 1102a − 2436u − 799,
u
9
+ 5u
8
+ 7u
7
− 3u
6
− 7u
5
+ 17u
4
+ 32u
3
+ 18u
2
+ 7u + 2i
I
u
2
= hu
7
− 2u
6
− u
5
+ 5u
4
− 2u
3
− 5u
2
+ b + 2u + 2, −u
7
+ 3u
6
− u
5
− 4u
4
+ 3u
3
+ 3u
2
+ 3a − u − 4,
u
8
− 3u
7
+ u
6
+ 7u
5
− 9u
4
− 3u
3
+ 10u
2
− 2u − 3i
I
u
3
= hb + 2a − 1, 4a
2
− 6a + 7, u − 2i
I
v
1
= ha, b + v, v
2
− v + 1i
* 4 irreducible components of dim
C
= 0, with total 21 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−51u
8
+ 4u
7
+ · · · + 551b + 165, 267u
8
+ 919u
7
+ · · · + 1102a −
799, u
9
+ 5u
8
+ · · · + 7u + 2i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
−0.242287u
8
− 0.833938u
7
+ ··· + 2.21053u + 0.725045
0.0925590u
8
− 0.00725953u
7
+ ··· − 1.47368u − 0.299456
a
9
=
1
−u
2
a
3
=
−0.242287u
8
− 0.833938u
7
+ ··· + 2.21053u + 0.725045
0.441016u
8
+ 1.25953u
7
+ ··· + 0.684211u + 0.455535
a
10
=
0.0735027u
8
+ 0.376588u
7
+ ··· − 0.0526316u + 0.409256
−0.0326679u
8
− 0.0562613u
7
+ ··· + 0.578947u − 0.0707804
a
1
=
−0.0408348u
8
− 0.320327u
7
+ ··· − 0.526316u − 0.338475
−0.0907441u
8
+ 0.399274u
7
+ ··· + 2.05263u + 0.470054
a
11
=
−u
u
a
5
=
0.227768u
8
+ 0.697822u
7
+ ··· + 3.15789u + 0.910163
−0.377495u
8
− 1.53902u
7
+ ··· − 2.42105u − 0.484574
a
7
=
−u
2
+ 1
u
2
a
6
=
0.0825771u
8
+ 0.336661u
7
+ ··· − 1.15789u + 0.262250
−0.00907441u
8
+ 0.0399274u
7
+ ··· + 1.10526u + 0.147005
a
2
=
−0.0735027u
8
− 0.376588u
7
+ ··· + 1.05263u + 0.590744
−0.0762250u
8
− 0.464610u
7
+ ··· − 0.315789u − 0.165154
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
1779
551
u
8
+
8482
551
u
7
+
9752
551
u
6
−
10058
551
u
5
−
11137
551
u
4
+
36963
551
u
3
+
46798
551
u
2
+
379
19
u +
5232
551
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
+ 6u
8
+ ··· + 1105u + 16
c
2
, c
5
u
9
+ 6u
8
+ 15u
7
+ 19u
6
+ 39u
5
+ 96u
4
+ 137u
3
+ 93u
2
+ 19u − 4
c
3
, c
6
u
9
− u
8
+ 6u
7
+ 6u
6
+ 39u
5
+ 60u
4
+ 40u
3
+ 13u
2
+ u − 1
c
4
, c
9
u
9
+ u
8
+ 7u
7
+ 9u
6
+ 14u
5
+ 11u
4
+ 8u
3
+ u
2
− u − 1
c
7
, c
8
, c
11
u
9
− 5u
8
+ 7u
7
+ 3u
6
− 7u
5
− 17u
4
+ 32u
3
− 18u
2
+ 7u − 2
c
10
, c
12
u
9
+ 6u
8
+ 14u
7
− 5u
6
+ 66u
5
− 202u
4
+ 132u
3
+ u
2
− 10u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
+ 114y
8
+ ··· + 1135425y −256
c
2
, c
5
y
9
− 6y
8
+ ··· + 1105y −16
c
3
, c
6
y
9
+ 11y
8
+ ··· + 27y −1
c
4
, c
9
y
9
+ 13y
8
+ 59y
7
+ 109y
6
+ 106y
5
+ 73y
4
+ 32y
3
+ 5y
2
+ 3y −1
c
7
, c
8
, c
11
y
9
− 11y
8
+ ··· − 23y −4
c
10
, c
12
y
9
− 8y
8
+ ··· + 102y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.18169 + 0.81660I
a = −1.13356 − 0.92427I
b = 0.538744 + 0.723261I
4.72372 − 1.44557I 2.79413 + 0.05589I
u = 1.18169 − 0.81660I
a = −1.13356 + 0.92427I
b = 0.538744 − 0.723261I
4.72372 + 1.44557I 2.79413 − 0.05589I
u = −0.557684
a = −0.337455
b = −0.425472
0.803810 12.4850
u = −1.42962 + 0.20941I
a = 0.119321 + 0.349200I
b = −0.023607 − 1.115360I
3.18435 − 3.04209I 1.27965 + 3.40109I
u = −1.42962 − 0.20941I
a = 0.119321 − 0.349200I
b = −0.023607 + 1.115360I
3.18435 + 3.04209I 1.27965 − 3.40109I
u = −0.058623 + 0.424215I
a = 0.74163 + 1.41569I
b = 0.476502 − 0.460767I
−1.48784 + 0.34537I −5.00907 − 1.24859I
u = −0.058623 − 0.424215I
a = 0.74163 − 1.41569I
b = 0.476502 + 0.460767I
−1.48784 − 0.34537I −5.00907 + 1.24859I
u = −1.91461 + 0.93499I
a = −0.308657 − 1.376960I
b = −0.27890 + 2.28182I
13.7395 − 8.2460I 2.69287 + 2.56767I
u = −1.91461 − 0.93499I
a = −0.308657 + 1.376960I
b = −0.27890 − 2.28182I
13.7395 + 8.2460I 2.69287 − 2.56767I
5
II. I
u
2
= hu
7
− 2u
6
− u
5
+ 5u
4
− 2u
3
− 5u
2
+ b + 2u + 2, −u
7
+ 3u
6
+ · · · +
3a − 4, u
8
− 3u
7
+ · · · − 2u − 3i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
1
3
u
7
− u
6
+ ··· +
1
3
u +
4
3
−u
7
+ 2u
6
+ u
5
− 5u
4
+ 2u
3
+ 5u
2
− 2u − 2
a
9
=
1
−u
2
a
3
=
1
3
u
7
− u
6
+ ··· +
1
3
u +
4
3
−u
7
+ 3u
6
− u
5
− 5u
4
+ 5u
3
+ 3u
2
− 3u − 2
a
10
=
2
3
u
7
− u
6
+ ··· +
2
3
u +
8
3
−u
7
+ u
6
+ 2u
5
− 4u
4
− u
3
+ 5u
2
+ u − 1
a
1
=
1
3
u
7
−
2
3
u
5
+ ··· −
5
3
u −
5
3
2u
7
− 5u
6
− 2u
5
+ 14u
4
− 8u
3
− 15u
2
+ 11u + 7
a
11
=
−u
u
a
5
=
−
2
3
u
7
+ u
6
+ ··· −
11
3
u −
5
3
−u
5
+ 2u
4
− 3u
2
+ 2u + 1
a
7
=
−u
2
+ 1
u
2
a
6
=
−
1
3
u
7
+ u
6
+ ··· −
7
3
u +
2
3
u
7
− 2u
6
− u
5
+ 6u
4
− 3u
3
− 5u
2
+ 3u + 2
a
2
=
2
3
u
7
− u
6
+ ··· +
5
3
u +
5
3
−u
3
+ u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
7
− 7u
6
+ 3u
5
+ 10u
4
− 12u
3
− 2u
2
+ 3u + 3
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
− 10u
7
+ 39u
6
− 81u
5
+ 117u
4
− 122u
3
+ 63u
2
− 11u + 1
c
2
u
8
+ 4u
7
+ 3u
6
− 7u
5
− 13u
4
− 4u
3
+ 7u
2
+ 5u + 1
c
3
, c
6
u
8
+ 2u
7
− 2u
5
− 3u
4
− 3u
3
+ 3u
2
+ 4u + 1
c
4
, c
9
u
8
+ 7u
6
+ 10u
4
− 5u
3
− u
2
− 2u − 1
c
5
u
8
− 4u
7
+ 3u
6
+ 7u
5
− 13u
4
+ 4u
3
+ 7u
2
− 5u + 1
c
7
, c
8
u
8
− 3u
7
+ u
6
+ 7u
5
− 9u
4
− 3u
3
+ 10u
2
− 2u − 3
c
10
, c
12
u
8
+ u
7
+ 3u
6
+ 2u
5
− 6u
4
+ 6u
3
− 8u
2
+ 5u − 1
c
11
u
8
+ 3u
7
+ u
6
− 7u
5
− 9u
4
+ 3u
3
+ 10u
2
+ 2u − 3
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
− 22y
7
+ 135y
6
+ 251y
5
− 1379y
4
− 1846y
3
+ 1519y
2
+ 5y + 1
c
2
, c
5
y
8
− 10y
7
+ 39y
6
− 81y
5
+ 117y
4
− 122y
3
+ 63y
2
− 11y + 1
c
3
, c
6
y
8
− 4y
7
+ 2y
6
+ 14y
5
− 17y
4
− 11y
3
+ 27y
2
− 10y + 1
c
4
, c
9
y
8
+ 14y
7
+ 69y
6
+ 138y
5
+ 84y
4
− 59y
3
− 39y
2
− 2y + 1
c
7
, c
8
, c
11
y
8
− 7y
7
+ 25y
6
− 65y
5
+ 125y
4
− 167y
3
+ 142y
2
− 64y + 9
c
10
, c
12
y
8
+ 5y
7
− 7y
6
− 68y
5
− 48y
4
+ 34y
3
+ 16y
2
− 9y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.143550 + 0.105994I
a = 0.027839 + 1.059490I
b = −0.039265 − 0.565787I
4.97207 − 3.05412I 5.35335 + 5.43549I
u = −1.143550 − 0.105994I
a = 0.027839 − 1.059490I
b = −0.039265 + 0.565787I
4.97207 + 3.05412I 5.35335 − 5.43549I
u = 1.084730 + 0.492548I
a = 0.753232 − 0.582528I
b = 0.13996 + 2.26543I
−12.04710 + 1.95234I −1.37368 − 3.45942I
u = 1.084730 − 0.492548I
a = 0.753232 + 0.582528I
b = 0.13996 − 2.26543I
−12.04710 − 1.95234I −1.37368 + 3.45942I
u = 1.03265 + 1.04538I
a = −0.527623 + 0.897663I
b = −0.31449 − 1.42869I
−5.86984 + 3.80835I 2.98023 − 3.30420I
u = 1.03265 − 1.04538I
a = −0.527623 − 0.897663I
b = −0.31449 + 1.42869I
−5.86984 − 3.80835I 2.98023 + 3.30420I
u = −0.483343
a = 1.10069
b = −0.358657
−0.371792 2.79670
u = 1.53567
a = −0.274253
b = 0.786231
6.52231 9.28350
9
III. I
u
3
= hb + 2a − 1, 4a
2
− 6a + 7, u − 2i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
2
a
4
=
a
−2a + 1
a
9
=
1
−4
a
3
=
a
−6a + 1
a
10
=
2a −
5
2
−10a + 16
a
1
=
−8a −
3
2
54a − 8
a
11
=
−2
2
a
5
=
−3a + 4
2a − 3
a
7
=
−3
4
a
6
=
0.5
−2a
a
2
=
a −
3
2
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u + 9)
2
c
2
, c
5
(u − 3)
2
c
3
, c
6
u
2
− 3u + 7
c
4
, c
9
u
2
− u + 5
c
7
, c
8
, c
11
(u + 2)
2
c
10
, c
12
u
2
− 4u + 23
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y −81)
2
c
2
, c
5
(y −9)
2
c
3
, c
6
y
2
+ 5y + 49
c
4
, c
9
y
2
+ 9y + 25
c
7
, c
8
, c
11
(y −4)
2
c
10
, c
12
y
2
+ 30y + 529
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 2.00000
a = 0.750000 + 1.089730I
b = −0.50000 − 2.17945I
−9.86960 3.00000
u = 2.00000
a = 0.750000 − 1.089730I
b = −0.50000 + 2.17945I
−9.86960 3.00000
13
IV. I
v
1
= ha, b + v, v
2
− v + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
v
0
a
4
=
0
−v
a
9
=
1
0
a
3
=
1
−v
a
10
=
v + 1
−v + 1
a
1
=
−1
v − 1
a
11
=
v
0
a
5
=
1
−v
a
7
=
1
0
a
6
=
1
−v + 1
a
2
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
10
c
12
(u − 1)
2
c
3
, c
4
, c
6
c
9
u
2
+ u + 1
c
5
(u + 1)
2
c
7
, c
8
, c
11
u
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
10
, c
12
(y − 1)
2
c
3
, c
4
, c
6
c
9
y
2
+ y + 1
c
7
, c
8
, c
11
y
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.500000 + 0.866025I
a = 0
b = −0.500000 − 0.866025I
0 3.00000
v = 0.500000 − 0.866025I
a = 0
b = −0.500000 + 0.866025I
0 3.00000
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
(u + 9)
2
· (u
8
− 10u
7
+ 39u
6
− 81u
5
+ 117u
4
− 122u
3
+ 63u
2
− 11u + 1)
· (u
9
+ 6u
8
+ ··· + 1105u + 16)
c
2
((u − 3)
2
)(u − 1)
2
(u
8
+ 4u
7
+ ··· + 5u + 1)
· (u
9
+ 6u
8
+ 15u
7
+ 19u
6
+ 39u
5
+ 96u
4
+ 137u
3
+ 93u
2
+ 19u − 4)
c
3
, c
6
(u
2
− 3u + 7)(u
2
+ u + 1)(u
8
+ 2u
7
+ ··· + 4u + 1)
· (u
9
− u
8
+ 6u
7
+ 6u
6
+ 39u
5
+ 60u
4
+ 40u
3
+ 13u
2
+ u − 1)
c
4
, c
9
(u
2
− u + 5)(u
2
+ u + 1)(u
8
+ 7u
6
+ 10u
4
− 5u
3
− u
2
− 2u − 1)
· (u
9
+ u
8
+ 7u
7
+ 9u
6
+ 14u
5
+ 11u
4
+ 8u
3
+ u
2
− u − 1)
c
5
((u − 3)
2
)(u + 1)
2
(u
8
− 4u
7
+ ··· − 5u + 1)
· (u
9
+ 6u
8
+ 15u
7
+ 19u
6
+ 39u
5
+ 96u
4
+ 137u
3
+ 93u
2
+ 19u − 4)
c
7
, c
8
u
2
(u + 2)
2
(u
8
− 3u
7
+ u
6
+ 7u
5
− 9u
4
− 3u
3
+ 10u
2
− 2u − 3)
· (u
9
− 5u
8
+ 7u
7
+ 3u
6
− 7u
5
− 17u
4
+ 32u
3
− 18u
2
+ 7u − 2)
c
10
, c
12
((u − 1)
2
)(u
2
− 4u + 23)(u
8
+ u
7
+ ··· + 5u − 1)
· (u
9
+ 6u
8
+ 14u
7
− 5u
6
+ 66u
5
− 202u
4
+ 132u
3
+ u
2
− 10u − 1)
c
11
u
2
(u + 2)
2
(u
8
+ 3u
7
+ u
6
− 7u
5
− 9u
4
+ 3u
3
+ 10u
2
+ 2u − 3)
· (u
9
− 5u
8
+ 7u
7
+ 3u
6
− 7u
5
− 17u
4
+ 32u
3
− 18u
2
+ 7u − 2)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y − 81)
2
(y − 1)
2
· (y
8
− 22y
7
+ 135y
6
+ 251y
5
− 1379y
4
− 1846y
3
+ 1519y
2
+ 5y + 1)
· (y
9
+ 114y
8
+ ··· + 1135425y − 256)
c
2
, c
5
(y − 9)
2
(y − 1)
2
· (y
8
− 10y
7
+ 39y
6
− 81y
5
+ 117y
4
− 122y
3
+ 63y
2
− 11y + 1)
· (y
9
− 6y
8
+ ··· + 1105y − 16)
c
3
, c
6
(y
2
+ y + 1)(y
2
+ 5y + 49)
· (y
8
− 4y
7
+ 2y
6
+ 14y
5
− 17y
4
− 11y
3
+ 27y
2
− 10y + 1)
· (y
9
+ 11y
8
+ ··· + 27y − 1)
c
4
, c
9
(y
2
+ y + 1)(y
2
+ 9y + 25)
· (y
8
+ 14y
7
+ 69y
6
+ 138y
5
+ 84y
4
− 59y
3
− 39y
2
− 2y + 1)
· (y
9
+ 13y
8
+ 59y
7
+ 109y
6
+ 106y
5
+ 73y
4
+ 32y
3
+ 5y
2
+ 3y − 1)
c
7
, c
8
, c
11
y
2
(y − 4)
2
· (y
8
− 7y
7
+ 25y
6
− 65y
5
+ 125y
4
− 167y
3
+ 142y
2
− 64y + 9)
· (y
9
− 11y
8
+ ··· − 23y − 4)
c
10
, c
12
(y − 1)
2
(y
2
+ 30y + 529)
· (y
8
+ 5y
7
− 7y
6
− 68y
5
− 48y
4
+ 34y
3
+ 16y
2
− 9y + 1)
· (y
9
− 8y
8
+ ··· + 102y − 1)
19
