12n
0501
(K12n
0501
)
A knot diagram
1
Linearized knot diagam
3 6 12 8 2 11 10 3 6 7 4 9
Solving Sequence
6,11 3,7
2 1 5 10 8 4 9 12
c
6
c
2
c
1
c
5
c
10
c
7
c
4
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−4u
32
+ 16u
31
+ ··· + 4b + 5, 5u
32
− 16u
31
+ ··· + 4a + 6, u
33
− 4u
32
+ ··· − 6u + 1i
I
u
2
= h−au + b, 2u
2
a + a
2
+ au + 2u
2
+ 3a + u + 4, u
3
+ u
2
+ 2u + 1i
I
u
3
= hu
2
+ b + u + 1, −u
2
+ a − 1, u
3
+ u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 42 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−4u
32
+16u
31
+· · ·+4b+5, 5u
32
−16u
31
+· · ·+4a+6, u
33
−4u
32
+· · ·−6u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
−
5
4
u
32
+ 4u
31
+ ··· +
7
4
u −
3
2
u
32
− 4u
31
+ ··· + 9u −
5
4
a
7
=
1
−u
2
a
2
=
−
1
4
u
32
−
5
2
u
30
+ ··· +
43
4
u −
11
4
u
32
− 4u
31
+ ··· + 9u −
5
4
a
1
=
−
11
4
u
32
+
41
4
u
31
+ ··· − 23u +
11
2
3
4
u
32
− 3u
31
+ ··· +
63
4
u −
15
4
a
5
=
−
1
4
u
32
+
3
4
u
31
+ ··· − 4u +
1
4
1
4
u
32
− u
31
+ ··· +
13
4
u −
1
2
a
10
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
4
=
−
1
2
u
32
+
7
4
u
31
+ ··· −
29
4
u +
3
4
1
4
u
32
− u
31
+ ··· +
17
4
u −
3
4
a
9
=
−u
3
− 2u
u
3
+ u
a
12
=
−
9
4
u
32
+
33
4
u
31
+ ··· −
37
2
u +
19
4
3
4
u
32
−
11
4
u
31
+ ··· + 13u −
13
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
9
2
u
32
+
35
2
u
31
+ ··· −
111
2
u +
23
4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
33
+ 46u
32
+ ··· + 42u + 1
c
2
, c
5
u
33
+ 4u
32
+ ··· + 4u − 1
c
3
, c
11
u
33
− 4u
32
+ ··· + 10u − 1
c
4
u
33
− 3u
32
+ ··· − 31624u − 99623
c
6
, c
7
, c
10
u
33
+ 4u
32
+ ··· − 6u − 1
c
8
u
33
+ u
32
+ ··· + 128u
2
− 512
c
9
u
33
− 4u
32
+ ··· − 744u − 137
c
12
u
33
+ 26u
31
+ ··· + 1410u − 2071
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
33
− 114y
32
+ ··· + 1162y −1
c
2
, c
5
y
33
− 46y
32
+ ··· + 42y −1
c
3
, c
11
y
33
+ 26y
32
+ ··· + 42y −1
c
4
y
33
+ 39y
32
+ ··· − 19254673246y −9924742129
c
6
, c
7
, c
10
y
33
+ 34y
32
+ ··· − 6y −1
c
8
y
33
+ 49y
32
+ ··· + 131072y −262144
c
9
y
33
+ 26y
32
+ ··· − 461086y −18769
c
12
y
33
+ 52y
32
+ ··· − 8631988y −4289041
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.783955 + 0.550569I
a = 1.50759 − 1.19961I
b = −1.84235 + 0.11041I
−10.85950 + 2.60786I −1.64483 − 2.62552I
u = 0.783955 − 0.550569I
a = 1.50759 + 1.19961I
b = −1.84235 − 0.11041I
−10.85950 − 2.60786I −1.64483 + 2.62552I
u = 0.745301 + 0.601492I
a = −1.40874 + 1.23496I
b = 1.79275 − 0.07307I
−6.97586 − 2.52993I 1.181503 + 0.379753I
u = 0.745301 − 0.601492I
a = −1.40874 − 1.23496I
b = 1.79275 + 0.07307I
−6.97586 + 2.52993I 1.181503 − 0.379753I
u = 0.799754 + 0.494935I
a = −1.61442 + 1.19019I
b = 1.88021 − 0.15283I
−6.63229 + 7.68298I 1.86058 − 5.38110I
u = 0.799754 − 0.494935I
a = −1.61442 − 1.19019I
b = 1.88021 + 0.15283I
−6.63229 − 7.68298I 1.86058 + 5.38110I
u = −0.071136 + 1.190150I
a = 0.658204 − 0.211530I
b = −0.204931 − 0.798410I
1.03666 − 2.07532I 3.43845 + 3.26136I
u = −0.071136 − 1.190150I
a = 0.658204 + 0.211530I
b = −0.204931 + 0.798410I
1.03666 + 2.07532I 3.43845 − 3.26136I
u = −0.688247 + 0.161266I
a = 0.110580 + 0.396771I
b = 0.140092 + 0.255244I
3.61024 − 0.82218I 5.86040 + 0.81952I
u = −0.688247 − 0.161266I
a = 0.110580 − 0.396771I
b = 0.140092 − 0.255244I
3.61024 + 0.82218I 5.86040 − 0.81952I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.182052 + 1.337320I
a = −0.217943 − 0.164939I
b = −0.260253 + 0.261432I
−3.43336 − 2.41737I 0
u = −0.182052 − 1.337320I
a = −0.217943 + 0.164939I
b = −0.260253 − 0.261432I
−3.43336 + 2.41737I 0
u = −0.294839 + 1.343030I
a = −0.011030 + 0.323726I
b = 0.431523 + 0.110260I
−1.11961 − 4.40985I 0
u = −0.294839 − 1.343030I
a = −0.011030 − 0.323726I
b = 0.431523 − 0.110260I
−1.11961 + 4.40985I 0
u = −0.212327 + 0.555507I
a = −0.012738 − 0.990392I
b = −0.552874 − 0.203211I
1.64764 − 2.26500I 1.52805 + 4.62369I
u = −0.212327 − 0.555507I
a = −0.012738 + 0.990392I
b = −0.552874 + 0.203211I
1.64764 + 2.26500I 1.52805 − 4.62369I
u = 0.09222 + 1.42628I
a = −0.711100 − 0.954177I
b = −1.29535 + 1.10223I
−1.79243 + 4.81413I 0
u = 0.09222 − 1.42628I
a = −0.711100 + 0.954177I
b = −1.29535 − 1.10223I
−1.79243 − 4.81413I 0
u = 0.02669 + 1.47035I
a = 0.484236 + 0.756490I
b = 1.099380 − 0.732184I
−7.20920 + 1.09488I 0
u = 0.02669 − 1.47035I
a = 0.484236 − 0.756490I
b = 1.099380 + 0.732184I
−7.20920 − 1.09488I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.06717 + 1.48459I
a = −0.340624 − 0.578506I
b = −0.881727 + 0.466828I
−4.89325 − 3.34061I 0
u = −0.06717 − 1.48459I
a = −0.340624 + 0.578506I
b = −0.881727 − 0.466828I
−4.89325 + 3.34061I 0
u = −0.497661
a = −0.331095
b = −0.164773
0.854180 12.3760
u = 0.29075 + 1.52597I
a = −0.056410 + 1.324390I
b = 2.03738 − 0.29899I
−13.1973 + 11.6833I 0
u = 0.29075 − 1.52597I
a = −0.056410 − 1.324390I
b = 2.03738 + 0.29899I
−13.1973 − 11.6833I 0
u = 0.27105 + 1.54914I
a = 0.016042 − 1.254950I
b = −1.94844 + 0.31531I
−17.7298 + 6.4903I 0
u = 0.27105 − 1.54914I
a = 0.016042 + 1.254950I
b = −1.94844 − 0.31531I
−17.7298 − 6.4903I 0
u = 0.23795 + 1.56175I
a = 0.049432 + 1.196740I
b = 1.85725 − 0.36197I
−14.1120 + 1.0765I 0
u = 0.23795 − 1.56175I
a = 0.049432 − 1.196740I
b = 1.85725 + 0.36197I
−14.1120 − 1.0765I 0
u = 0.362886 + 0.190995I
a = 0.83437 − 3.02007I
b = −0.879597 + 0.936581I
3.51710 + 3.29019I −0.48267 − 6.06797I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.362886 − 0.190995I
a = 0.83437 + 3.02007I
b = −0.879597 − 0.936581I
3.51710 − 3.29019I −0.48267 + 6.06797I
u = 0.154044 + 0.322324I
a = −0.12190 + 2.14236I
b = 0.709313 − 0.290729I
−1.241050 + 0.560760I −5.36805 − 2.52603I
u = 0.154044 − 0.322324I
a = −0.12190 − 2.14236I
b = 0.709313 + 0.290729I
−1.241050 − 0.560760I −5.36805 + 2.52603I
8
II. I
u
2
= h−au + b, 2u
2
a + a
2
+ au + 2u
2
+ 3a + u + 4, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
a
au
a
7
=
1
−u
2
a
2
=
au + a
au
a
1
=
−u
2
a − 2u
2
− a − u − 4
au + u
2
+ a + 2
a
5
=
−u
2
a − au − a
u
2
+ a + 1
a
10
=
−u
−u
2
− u − 1
a
8
=
u
2
+ 1
−u
2
− u − 1
a
4
=
−u
2
a − au − u
2
− 2a − 2
−au + 2u
2
+ a + 2
a
9
=
u
2
+ 1
−u
2
− u − 1
a
12
=
−u
2
a − au − 2u
2
− a − u − 4
−u
2
a + au + u
2
+ a + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5au + 5u
2
+ 2a + 5u + 8
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
, c
11
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
9
(u
3
+ u
2
− 1)
2
c
3
, c
6
, c
7
(u
3
+ u
2
+ 2u + 1)
2
c
4
u
6
+ u
5
+ 4u
4
+ u
3
+ 2u
2
− 2u + 1
c
5
(u
3
− u
2
+ 1)
2
c
8
u
6
c
12
(u
3
− u − 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
10
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
, c
9
(y
3
− y
2
+ 2y −1)
2
c
4
y
6
+ 7y
5
+ 18y
4
+ 21y
3
+ 16y
2
+ 1
c
8
y
6
c
12
(y
3
− 2y
2
+ y −1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = 0.447279 − 0.744862I
b = 0.877439 + 0.744862I
− 5.65624I 3.89456 + 5.95889I
u = −0.215080 + 1.307140I
a = 0.092519 + 0.562280I
b = −0.754878
−4.13758 − 2.82812I −4.97655 + 4.84887I
u = −0.215080 − 1.307140I
a = 0.447279 + 0.744862I
b = 0.877439 − 0.744862I
5.65624I 3.89456 − 5.95889I
u = −0.215080 − 1.307140I
a = 0.092519 − 0.562280I
b = −0.754878
−4.13758 + 2.82812I −4.97655 − 4.84887I
u = −0.569840
a = −1.53980 + 1.30714I
b = 0.877439 − 0.744862I
4.13758 + 2.82812I 8.08199 − 1.11003I
u = −0.569840
a = −1.53980 − 1.30714I
b = 0.877439 + 0.744862I
4.13758 − 2.82812I 8.08199 + 1.11003I
12
III. I
u
3
= hu
2
+ b + u + 1, −u
2
+ a − 1, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
u
2
+ 1
−u
2
− u − 1
a
7
=
1
−u
2
a
2
=
−u
−u
2
− u − 1
a
1
=
−u
2
− 2u − 1
−u
2
a
5
=
u + 2
u
a
10
=
−u
−u
2
− u − 1
a
8
=
u
2
+ 1
−u
2
− u − 1
a
4
=
1
−u
2
a
9
=
u
2
+ 1
−u
2
− u − 1
a
12
=
−u
−u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
, c
11
c
12
u
3
− u
2
+ 2u − 1
c
2
, c
9
u
3
+ u
2
− 1
c
3
, c
6
, c
7
u
3
+ u
2
+ 2u + 1
c
4
u
3
− 3u
2
+ 2u + 1
c
5
u
3
− u
2
+ 1
c
8
u
3
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
10
, c
11
c
12
y
3
+ 3y
2
+ 2y −1
c
2
, c
5
, c
9
y
3
− y
2
+ 2y −1
c
4
y
3
− 5y
2
+ 10y −1
c
8
y
3
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −0.662359 − 0.562280I
b = 0.877439 − 0.744862I
0 0
u = −0.215080 − 1.307140I
a = −0.662359 + 0.562280I
b = 0.877439 + 0.744862I
0 0
u = −0.569840
a = 1.32472
b = −0.754878
0 0
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
3
)(u
33
+ 46u
32
+ ··· + 42u + 1)
c
2
((u
3
+ u
2
− 1)
3
)(u
33
+ 4u
32
+ ··· + 4u − 1)
c
3
((u
3
+ u
2
+ 2u + 1)
3
)(u
33
− 4u
32
+ ··· + 10u − 1)
c
4
(u
3
− 3u
2
+ 2u + 1)(u
6
+ u
5
+ 4u
4
+ u
3
+ 2u
2
− 2u + 1)
· (u
33
− 3u
32
+ ··· − 31624u − 99623)
c
5
((u
3
− u
2
+ 1)
3
)(u
33
+ 4u
32
+ ··· + 4u − 1)
c
6
, c
7
((u
3
+ u
2
+ 2u + 1)
3
)(u
33
+ 4u
32
+ ··· − 6u − 1)
c
8
u
9
(u
33
+ u
32
+ ··· + 128u
2
− 512)
c
9
((u
3
+ u
2
− 1)
3
)(u
33
− 4u
32
+ ··· − 744u − 137)
c
10
((u
3
− u
2
+ 2u − 1)
3
)(u
33
+ 4u
32
+ ··· − 6u − 1)
c
11
((u
3
− u
2
+ 2u − 1)
3
)(u
33
− 4u
32
+ ··· + 10u − 1)
c
12
((u
3
− u − 1)
2
)(u
3
− u
2
+ 2u − 1)(u
33
+ 26u
31
+ ··· + 1410u − 2071)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
+ 3y
2
+ 2y −1)
3
)(y
33
− 114y
32
+ ··· + 1162y −1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
3
)(y
33
− 46y
32
+ ··· + 42y −1)
c
3
, c
11
((y
3
+ 3y
2
+ 2y −1)
3
)(y
33
+ 26y
32
+ ··· + 42y −1)
c
4
(y
3
− 5y
2
+ 10y −1)(y
6
+ 7y
5
+ 18y
4
+ 21y
3
+ 16y
2
+ 1)
· (y
33
+ 39y
32
+ ··· − 19254673246y −9924742129)
c
6
, c
7
, c
10
((y
3
+ 3y
2
+ 2y −1)
3
)(y
33
+ 34y
32
+ ··· − 6y −1)
c
8
y
9
(y
33
+ 49y
32
+ ··· + 131072y −262144)
c
9
((y
3
− y
2
+ 2y −1)
3
)(y
33
+ 26y
32
+ ··· − 461086y −18769)
c
12
(y
3
− 2y
2
+ y −1)
2
(y
3
+ 3y
2
+ 2y −1)
· (y
33
+ 52y
32
+ ··· − 8631988y −4289041)
18
