12n
0006
(K12n
0006
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 9 11 4 12 7 9 11
Solving Sequence
5,8
4
9,11
12 7 6 3 2 1 10
c
4
c
8
c
11
c
7
c
6
c
3
c
2
c
1
c
10
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.91035 × 10
42
u
35
− 5.59036 × 10
42
u
34
+ ··· + 5.65992 × 10
43
b − 1.39033 × 10
44
,
− 3.14927 × 10
42
u
35
− 2.52651 × 10
43
u
34
+ ··· + 9.05587 × 10
44
a − 2.23223 × 10
44
,
u
36
+ 2u
35
+ ··· − 80u + 16i
I
u
2
= hu
3
+ b + u + 1, a, u
4
+ u
2
+ u + 1i
I
u
3
= hu
5
− u
4
+ 2u
3
− 2u
2
+ b + 2u − 2, a, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
I
v
1
= ha, 5v
3
+ 16v
2
+ 8b + 40v + 15, v
4
+ 3v
3
+ 8v
2
+ 3v + 1i
* 4 irreducible components of dim
C
= 0, with total 50 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−1.91 × 10
42
u
35
− 5.59 × 10
42
u
34
+ · · · + 5.66 × 10
43
b − 1.39 ×
10
44
, −3.15 × 10
42
u
35
− 2.53 × 10
43
u
34
+ · · · + 9.06 × 10
44
a − 2.23 ×
10
44
, u
36
+ 2u
35
+ · · · − 80u + 16i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
0.00347761u
35
+ 0.0278991u
34
+ ··· − 5.05646u + 0.246496
0.0337523u
35
+ 0.0987710u
34
+ ··· − 8.01212u + 2.45644
a
12
=
0.0174509u
35
+ 0.0681602u
34
+ ··· − 5.53487u + 0.386666
0.0549946u
35
+ 0.150360u
34
+ ··· − 7.72896u + 2.39958
a
7
=
−0.0689309u
35
− 0.141959u
34
+ ··· − 11.1126u + 2.81049
0.0411861u
35
+ 0.104236u
34
+ ··· − 0.312163u + 0.732592
a
6
=
−0.0905615u
35
− 0.196881u
34
+ ··· − 11.3120u + 2.51944
0.0271025u
35
+ 0.0698189u
34
+ ··· − 1.09845u + 0.628132
a
3
=
−0.0217183u
35
− 0.0278113u
34
+ ··· − 4.98975u + 2.48305
0.0154821u
35
+ 0.0215115u
34
+ ··· + 5.72787u − 0.965440
a
2
=
−0.0372004u
35
− 0.0493228u
34
+ ··· − 10.7176u + 3.44849
0.0154821u
35
+ 0.0215115u
34
+ ··· + 5.72787u − 0.965440
a
1
=
−0.103417u
35
− 0.225024u
34
+ ··· − 10.0253u + 2.14344
−0.0128556u
35
− 0.0281432u
34
+ ··· + 1.28677u − 0.376001
a
10
=
0.0909240u
35
+ 0.190594u
34
+ ··· + 8.47685u − 2.12683
−0.0276885u
35
− 0.0503355u
34
+ ··· − 10.4945u + 2.31821
(ii) Obstruction class = −1
(iii) Cusp Shapes = −0.512287u
35
− 1.02092u
34
+ ··· − 70.5736u − 4.04563
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
36
+ 20u
35
+ ··· − 86u + 1
c
2
, c
5
u
36
+ 4u
35
+ ··· − 6u + 1
c
3
u
36
− 4u
35
+ ··· − 276u + 36
c
4
, c
8
u
36
− 2u
35
+ ··· + 80u + 16
c
6
u
36
− 4u
35
+ ··· − 4u + 1
c
7
, c
10
u
36
+ 3u
35
+ ··· + 2048u − 1024
c
9
, c
11
u
36
− 13u
35
+ ··· + 17u − 1
c
12
u
36
+ 59u
35
+ ··· + 9u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
36
− 4y
35
+ ··· − 8550y + 1
c
2
, c
5
y
36
+ 20y
35
+ ··· − 86y + 1
c
3
y
36
− 28y
35
+ ··· − 99144y + 1296
c
4
, c
8
y
36
+ 30y
35
+ ··· + 1408y + 256
c
6
y
36
− 80y
35
+ ··· − 26y + 1
c
7
, c
10
y
36
− 69y
35
+ ··· + 7864320y + 1048576
c
9
, c
11
y
36
− 59y
35
+ ··· − 9y + 1
c
12
y
36
− 151y
35
+ ··· − 6173y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.386520 + 1.057590I
a = −0.503327 + 0.379742I
b = 0.690277 + 0.591486I
−0.77161 − 2.27001I −1.34133 + 2.22423I
u = −0.386520 − 1.057590I
a = −0.503327 − 0.379742I
b = 0.690277 − 0.591486I
−0.77161 + 2.27001I −1.34133 − 2.22423I
u = −0.807607 + 0.285041I
a = 0.827740 − 0.668125I
b = −0.25011 − 2.14554I
−2.91712 − 2.06171I −8.82963 + 1.43740I
u = −0.807607 − 0.285041I
a = 0.827740 + 0.668125I
b = −0.25011 + 2.14554I
−2.91712 + 2.06171I −8.82963 − 1.43740I
u = 0.583843 + 0.511757I
a = −0.884356 + 0.778254I
b = −0.381315 + 0.236790I
−1.74932 − 0.04789I −9.54847 + 0.41128I
u = 0.583843 − 0.511757I
a = −0.884356 − 0.778254I
b = −0.381315 − 0.236790I
−1.74932 + 0.04789I −9.54847 − 0.41128I
u = −0.502707 + 0.522986I
a = −0.379878 + 0.383729I
b = −0.011979 + 0.723756I
0.74759 − 1.37712I 2.57358 + 4.27221I
u = −0.502707 − 0.522986I
a = −0.379878 − 0.383729I
b = −0.011979 − 0.723756I
0.74759 + 1.37712I 2.57358 − 4.27221I
u = 0.127839 + 1.278020I
a = 0.646852 + 0.410189I
b = −1.053190 + 0.114026I
−4.72217 − 1.11094I −8.05050 + 1.28077I
u = 0.127839 − 1.278020I
a = 0.646852 − 0.410189I
b = −1.053190 − 0.114026I
−4.72217 + 1.11094I −8.05050 − 1.28077I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.499062 + 1.208100I
a = 0.516361 + 0.301634I
b = −0.886666 + 0.792695I
−3.38469 + 7.01546I −4.45254 − 4.63795I
u = 0.499062 − 1.208100I
a = 0.516361 − 0.301634I
b = −0.886666 − 0.792695I
−3.38469 − 7.01546I −4.45254 + 4.63795I
u = 0.625800 + 0.176987I
a = 0.515428 + 0.332782I
b = 0.664166 + 0.803195I
−0.30087 − 2.59940I 0.94853 + 4.22855I
u = 0.625800 − 0.176987I
a = 0.515428 − 0.332782I
b = 0.664166 − 0.803195I
−0.30087 + 2.59940I 0.94853 − 4.22855I
u = 1.35690
a = 1.66154
b = 3.31123
−12.0538 −5.89580
u = 0.10948 + 1.41069I
a = −0.05528 + 1.62617I
b = −0.522174 − 0.242877I
−12.85970 + 3.05068I −8.36572 − 2.61847I
u = 0.10948 − 1.41069I
a = −0.05528 − 1.62617I
b = −0.522174 + 0.242877I
−12.85970 − 3.05068I −8.36572 + 2.61847I
u = 0.13985 + 1.42281I
a = 0.715759 − 0.915325I
b = −0.770423 + 0.258625I
−5.54338 + 1.88156I −7.87001 − 1.20785I
u = 0.13985 − 1.42281I
a = 0.715759 + 0.915325I
b = −0.770423 − 0.258625I
−5.54338 − 1.88156I −7.87001 + 1.20785I
u = 0.253629 + 0.486794I
a = 1.45456 − 2.43782I
b = 0.399439 + 0.873101I
−9.11143 − 1.69402I −16.0216 − 6.5848I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.253629 − 0.486794I
a = 1.45456 + 2.43782I
b = 0.399439 − 0.873101I
−9.11143 + 1.69402I −16.0216 + 6.5848I
u = −1.54854 + 0.24882I
a = −1.54790 − 0.09130I
b = −3.97196 + 1.02080I
−15.7283 + 4.6602I 0
u = −1.54854 − 0.24882I
a = −1.54790 + 0.09130I
b = −3.97196 − 1.02080I
−15.7283 − 4.6602I 0
u = 0.016490 + 0.425398I
a = 0.317088 − 1.129550I
b = 0.15143 − 2.98728I
−1.27554 + 2.18577I −31.5174 − 1.0818I
u = 0.016490 − 0.425398I
a = 0.317088 + 1.129550I
b = 0.15143 + 2.98728I
−1.27554 − 2.18577I −31.5174 + 1.0818I
u = 0.11625 + 1.61362I
a = −0.740706 − 0.914136I
b = 2.00964 + 0.28151I
−9.56302 + 2.68337I 0
u = 0.11625 − 1.61362I
a = −0.740706 + 0.914136I
b = 2.00964 − 0.28151I
−9.56302 − 2.68337I 0
u = −0.36830 + 1.57756I
a = −0.709122 − 0.932633I
b = 0.246782 + 1.199400I
−9.08039 − 6.82329I 0
u = −0.36830 − 1.57756I
a = −0.709122 + 0.932633I
b = 0.246782 − 1.199400I
−9.08039 + 6.82329I 0
u = 0.67064 + 1.51850I
a = −0.19618 + 1.43098I
b = −3.05517 − 0.48149I
−16.7576 + 7.1899I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.67064 − 1.51850I
a = −0.19618 − 1.43098I
b = −3.05517 + 0.48149I
−16.7576 − 7.1899I 0
u = 0.324270
a = −1.75817
b = 0.383485
−1.11333 −8.97030
u = −0.82187 + 1.51160I
a = 0.206115 + 1.375210I
b = 3.63575 − 0.26624I
−19.6617 − 12.9119I 0
u = −0.82187 − 1.51160I
a = 0.206115 − 1.375210I
b = 3.63575 + 0.26624I
−19.6617 + 12.9119I 0
u = −0.54791 + 1.74851I
a = 0.11518 + 1.43966I
b = 2.75814 − 1.63321I
17.2769 − 3.0755I 0
u = −0.54791 − 1.74851I
a = 0.11518 − 1.43966I
b = 2.75814 + 1.63321I
17.2769 + 3.0755I 0
8
II. I
u
2
= hu
3
+ b + u + 1, a, u
4
+ u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
0
−u
3
− u − 1
a
12
=
−u
−2u
3
− 2u − 1
a
7
=
0
u
a
6
=
u
3
−u
2
a
3
=
u
3
+ u
2
+ 1
−u
a
2
=
u
3
+ u
2
+ u + 1
−u
a
1
=
−u
−u
3
− u
a
10
=
0
−u
3
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
3
− 5u
2
− u − 9
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
4
− 2u
3
+ 3u
2
− u + 1
c
2
, c
4
u
4
+ u
2
+ u + 1
c
3
u
4
+ 3u
3
+ 4u
2
+ 3u + 2
c
5
, c
8
u
4
+ u
2
− u + 1
c
7
, c
10
u
4
c
9
(u − 1)
4
c
11
, c
12
(u + 1)
4
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
4
+ 2y
3
+ 7y
2
+ 5y + 1
c
2
, c
4
, c
5
c
8
y
4
+ 2y
3
+ 3y
2
+ y + 1
c
3
y
4
− y
3
+ 2y
2
+ 7y + 4
c
7
, c
10
y
4
c
9
, c
11
, c
12
(y −1)
4
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.547424 + 0.585652I
a = 0
b = −0.851808 − 0.911292I
−0.66484 − 1.39709I −7.03830 + 3.59727I
u = −0.547424 − 0.585652I
a = 0
b = −0.851808 + 0.911292I
−0.66484 + 1.39709I −7.03830 − 3.59727I
u = 0.547424 + 1.120870I
a = 0
b = 0.351808 − 0.720342I
−4.26996 + 7.64338I −10.46170 − 8.45840I
u = 0.547424 − 1.120870I
a = 0
b = 0.351808 + 0.720342I
−4.26996 − 7.64338I −10.46170 + 8.45840I
12
III.
I
u
3
= hu
5
− u
4
+ 2u
3
− 2u
2
+ b + 2u − 2, a, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
0
−u
5
+ u
4
− 2u
3
+ 2u
2
− 2u + 2
a
12
=
−u
−u
5
+ u
4
− 3u
3
+ 2u
2
− 3u + 2
a
7
=
0
u
a
6
=
u
3
u
5
+ u
3
+ u
a
3
=
−u
5
+ u
4
− 2u
3
+ 2u
2
− 2u + 2
−u
5
− 2u
3
+ u
2
− u + 1
a
2
=
u
4
+ u
2
− u + 1
−u
5
− 2u
3
+ u
2
− u + 1
a
1
=
−u
−u
3
− u
a
10
=
0
−u
5
+ u
4
− 2u
3
+ 2u
2
− 2u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
5
+ u
4
+ u
2
+ u − 8
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1
c
2
, c
4
u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1
c
3
(u
3
− u
2
+ 1)
2
c
5
, c
8
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
7
, c
10
u
6
c
9
(u − 1)
6
c
11
, c
12
(u + 1)
6
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
c
2
, c
4
, c
5
c
8
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
3
(y
3
− y
2
+ 2y −1)
2
c
7
, c
10
y
6
c
9
, c
11
, c
12
(y −1)
6
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.498832 + 1.001300I
a = 0
b = −0.398606 − 0.800120I
−1.91067 − 2.82812I −7.09522 + 3.87141I
u = −0.498832 − 1.001300I
a = 0
b = −0.398606 + 0.800120I
−1.91067 + 2.82812I −7.09522 − 3.87141I
u = 0.284920 + 1.115140I
a = 0
b = 0.215080 − 0.841795I
−6.04826 −11.76463 − 0.99756I
u = 0.284920 − 1.115140I
a = 0
b = 0.215080 + 0.841795I
−6.04826 −11.76463 + 0.99756I
u = 0.713912 + 0.305839I
a = 0
b = 1.183530 − 0.507021I
−1.91067 − 2.82812I −6.64015 + 0.59776I
u = 0.713912 − 0.305839I
a = 0
b = 1.183530 + 0.507021I
−1.91067 + 2.82812I −6.64015 − 0.59776I
16
IV. I
v
1
= ha, 5v
3
+ 16v
2
+ 8b + 40v + 15, v
4
+ 3v
3
+ 8v
2
+ 3v + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
v
0
a
4
=
1
0
a
9
=
v
0
a
11
=
0
−
5
8
v
3
− 2v
2
− 5v −
15
8
a
12
=
−
3
8
v
3
− v
2
− v −
1
8
−
5
8
v
3
− 2v
2
− 5v −
15
8
a
7
=
v
3
8
v
3
+ v
2
+ 3v +
9
8
a
6
=
3
8
v
3
+ v
2
+ v +
1
8
3
8
v
3
+ v
2
+ 3v +
9
8
a
3
=
−
1
4
v
3
+
5
4
−
3
8
v
3
− v
2
− 3v −
1
8
a
2
=
1
8
v
3
+ v
2
+ 3v +
11
8
−
3
8
v
3
− v
2
− 3v −
1
8
a
1
=
−
3
8
v
3
− v
2
− v −
1
8
−
3
8
v
3
− v
2
− 3v −
9
8
a
10
=
3
8
v
3
+ v
2
+ v +
1
8
−
3
8
v
3
− v
2
− 3v −
9
8
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
3
2
v
3
− 5v
2
− 9v −
13
2
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
2
c
2
(u
2
+ u + 1)
2
c
4
, c
8
u
4
c
6
(u
2
− 3u + 1)
2
c
7
, c
9
(u
2
+ u − 1)
2
c
10
, c
11
(u
2
− u − 1)
2
c
12
(u
2
+ 3u + 1)
2
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
2
c
4
, c
8
y
4
c
6
, c
12
(y
2
− 7y + 1)
2
c
7
, c
9
, c
10
c
11
(y
2
− 3y + 1)
2
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.190983 + 0.330792I
a = 0
b = −0.80902 − 1.40126I
−0.98696 − 2.02988I −4.50000 − 2.34537I
v = −0.190983 − 0.330792I
a = 0
b = −0.80902 + 1.40126I
−0.98696 + 2.02988I −4.50000 + 2.34537I
v = −1.30902 + 2.26728I
a = 0
b = 0.309017 + 0.535233I
−8.88264 − 2.02988I −4.50000 + 9.27358I
v = −1.30902 − 2.26728I
a = 0
b = 0.309017 − 0.535233I
−8.88264 + 2.02988I −4.50000 − 9.27358I
20
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
2
(u
4
− 2u
3
+ 3u
2
− u + 1)(u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1)
· (u
36
+ 20u
35
+ ··· − 86u + 1)
c
2
(u
2
+ u + 1)
2
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
36
+ 4u
35
+ ··· − 6u + 1)
c
3
(u
2
− u + 1)
2
(u
3
− u
2
+ 1)
2
(u
4
+ 3u
3
+ 4u
2
+ 3u + 2)
· (u
36
− 4u
35
+ ··· − 276u + 36)
c
4
u
4
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
36
− 2u
35
+ ··· + 80u + 16)
c
5
(u
2
− u + 1)
2
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
36
+ 4u
35
+ ··· − 6u + 1)
c
6
(u
2
− 3u + 1)
2
(u
4
− 2u
3
+ 3u
2
− u + 1)(u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1)
· (u
36
− 4u
35
+ ··· − 4u + 1)
c
7
u
10
(u
2
+ u − 1)
2
(u
36
+ 3u
35
+ ··· + 2048u − 1024)
c
8
u
4
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
36
− 2u
35
+ ··· + 80u + 16)
c
9
((u − 1)
10
)(u
2
+ u − 1)
2
(u
36
− 13u
35
+ ··· + 17u − 1)
c
10
u
10
(u
2
− u − 1)
2
(u
36
+ 3u
35
+ ··· + 2048u − 1024)
c
11
((u + 1)
10
)(u
2
− u − 1)
2
(u
36
− 13u
35
+ ··· + 17u − 1)
c
12
((u + 1)
10
)(u
2
+ 3u + 1)
2
(u
36
+ 59u
35
+ ··· + 9u + 1)
21
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
2
)(y
4
+ 2y
3
+ ··· + 5y + 1)(y
6
− y
5
+ ··· + 8y
2
+ 1)
· (y
36
− 4y
35
+ ··· − 8550y + 1)
c
2
, c
5
(y
2
+ y + 1)
2
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
36
+ 20y
35
+ ··· − 86y + 1)
c
3
(y
2
+ y + 1)
2
(y
3
− y
2
+ 2y − 1)
2
(y
4
− y
3
+ 2y
2
+ 7y + 4)
· (y
36
− 28y
35
+ ··· − 99144y + 1296)
c
4
, c
8
y
4
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
36
+ 30y
35
+ ··· + 1408y + 256)
c
6
((y
2
− 7y + 1)
2
)(y
4
+ 2y
3
+ ··· + 5y + 1)(y
6
− y
5
+ ··· + 8y
2
+ 1)
· (y
36
− 80y
35
+ ··· − 26y + 1)
c
7
, c
10
y
10
(y
2
− 3y + 1)
2
(y
36
− 69y
35
+ ··· + 7864320y + 1048576)
c
9
, c
11
((y − 1)
10
)(y
2
− 3y + 1)
2
(y
36
− 59y
35
+ ··· − 9y + 1)
c
12
((y − 1)
10
)(y
2
− 7y + 1)
2
(y
36
− 151y
35
+ ··· − 6173y + 1)
22
