12n
0559
(K12n
0559
)
A knot diagram
1
Linearized knot diagam
3 7 12 8 11 2 10 12 3 5 6 10
Solving Sequence
5,10
11 6 12
1,3
2 9 8 4 7
c
10
c
5
c
11
c
12
c
1
c
9
c
8
c
4
c
7
c
2
, c
3
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= h−2.74895 × 10
16
u
37
+ 2.53324 × 10
16
u
36
+ ··· + 4.11981 × 10
16
b − 1.42189 × 10
17
,
− 1.62383 × 10
17
u
37
+ 1.68076 × 10
17
u
36
+ ··· + 8.23961 × 10
16
a − 1.06430 × 10
18
, u
38
− u
37
+ ··· + 7u − 1i
I
u
2
= h−u
5
+ 2u
3
+ b − u, −u
5
− 2u
3
a + 2u
4
+ u
2
a + 2u
3
+ a
2
+ 2au − 3u
2
− a − u, u
6
− 3u
4
+ 2u
2
+ 1i
* 2 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−2.75×10
16
u
37
+2.53×10
16
u
36
+· · ·+4.12×10
16
b−1.42×10
17
, −1.62×
10
17
u
37
+1.68×10
17
u
36
+· · ·+8.24×10
16
a−1.06×10
18
, u
38
−u
37
+· · ·+7u−1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
3
=
1.97076u
37
− 2.03985u
36
+ ··· − 21.0299u + 12.9168
0.667251u
37
− 0.614893u
36
+ ··· − 9.05648u + 3.45135
a
2
=
2.02554u
37
− 1.32040u
36
+ ··· − 28.4281u + 5.80356
−0.659268u
37
+ 0.328623u
36
+ ··· + 6.36982u − 3.31602
a
9
=
3.06399u
37
− 2.12216u
36
+ ··· − 40.8760u + 12.2142
0.0456579u
37
+ 0.419048u
36
+ ··· − 0.102231u − 1.50203
a
8
=
3.45135u
37
− 2.78410u
36
+ ··· − 45.4240u + 15.1030
0.121448u
37
+ 0.408171u
36
+ ··· − 0.340899u − 1.30351
a
4
=
1.50203u
37
− 1.54769u
36
+ ··· − 13.7795u + 10.6164
0.477120u
37
− 0.620718u
36
+ ··· − 7.41211u + 3.01833
a
7
=
3.57280u
37
− 2.37593u
36
+ ··· − 45.7649u + 13.7995
0.121448u
37
+ 0.408171u
36
+ ··· − 0.340899u − 1.30351
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
106268405583162365
41198072891018797
u
37
+
98605685292409883
41198072891018797
u
36
+···+
62037259094258982
3169082530078369
u−
858011531178729634
41198072891018797
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
38
+ 25u
37
+ ··· + 101u + 25
c
2
, c
6
u
38
− u
37
+ ··· + 11u − 5
c
3
u
38
− 5u
37
+ ··· − 3u + 1
c
4
, c
9
u
38
− u
37
+ ··· + 23u − 1
c
5
, c
10
, c
11
u
38
− u
37
+ ··· + 7u − 1
c
7
u
38
− 5u
37
+ ··· − 40603u + 13213
c
8
u
38
− 3u
37
+ ··· − u + 5
c
12
u
38
+ 3u
37
+ ··· − 17u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
38
− 17y
37
+ ··· − 41901y + 625
c
2
, c
6
y
38
− 25y
37
+ ··· − 101y + 25
c
3
y
38
− 65y
37
+ ··· + 205y + 1
c
4
, c
9
y
38
+ 53y
37
+ ··· − 195y + 1
c
5
, c
10
, c
11
y
38
− 39y
37
+ ··· − 39y + 1
c
7
y
38
− 49y
37
+ ··· − 3299858645y + 174583369
c
8
y
38
+ 59y
37
+ ··· − 201y + 25
c
12
y
38
+ 57y
37
+ ··· − 183y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.644745 + 0.752188I
a = 0.801740 − 0.434056I
b = 0.11351 − 1.82495I
−13.38780 − 2.90157I −9.03118 + 0.30817I
u = −0.644745 − 0.752188I
a = 0.801740 + 0.434056I
b = 0.11351 + 1.82495I
−13.38780 + 2.90157I −9.03118 − 0.30817I
u = −0.478467 + 0.830062I
a = 1.281060 − 0.364339I
b = −0.25094 − 1.78475I
−12.8639 + 8.1852I −8.10422 − 5.25532I
u = −0.478467 − 0.830062I
a = 1.281060 + 0.364339I
b = −0.25094 + 1.78475I
−12.8639 − 8.1852I −8.10422 + 5.25532I
u = 0.531918 + 0.751960I
a = −1.088960 − 0.565285I
b = 0.07319 − 1.72540I
−8.90525 − 2.50092I −5.86634 + 2.56670I
u = 0.531918 − 0.751960I
a = −1.088960 + 0.565285I
b = 0.07319 + 1.72540I
−8.90525 + 2.50092I −5.86634 − 2.56670I
u = 0.547239 + 0.502053I
a = 1.267080 + 0.591264I
b = −0.676584 + 0.904931I
−3.49790 − 4.12703I −9.00602 + 6.42337I
u = 0.547239 − 0.502053I
a = 1.267080 − 0.591264I
b = −0.676584 − 0.904931I
−3.49790 + 4.12703I −9.00602 − 6.42337I
u = 0.219443 + 0.699987I
a = −0.037483 − 0.669697I
b = 0.415341 + 1.142040I
−2.33417 + 0.42500I −7.90152 + 0.53407I
u = 0.219443 − 0.699987I
a = −0.037483 + 0.669697I
b = 0.415341 − 1.142040I
−2.33417 − 0.42500I −7.90152 − 0.53407I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.263470 + 0.115418I
a = 0.155909 + 0.360863I
b = −0.480768 + 0.017588I
−2.25857 + 0.57001I −2.81061 + 0.I
u = −1.263470 − 0.115418I
a = 0.155909 − 0.360863I
b = −0.480768 − 0.017588I
−2.25857 − 0.57001I −2.81061 + 0.I
u = 1.280750 + 0.149729I
a = 0.851613 + 1.095860I
b = −0.202776 + 1.036480I
−5.14131 − 2.83687I −11.03494 + 3.31873I
u = 1.280750 − 0.149729I
a = 0.851613 − 1.095860I
b = −0.202776 − 1.036480I
−5.14131 + 2.83687I −11.03494 − 3.31873I
u = 1.300970 + 0.201402I
a = −0.050909 + 0.890884I
b = 0.1131480 + 0.0174250I
−3.00626 − 4.83883I −4.00000 + 6.35067I
u = 1.300970 − 0.201402I
a = −0.050909 − 0.890884I
b = 0.1131480 − 0.0174250I
−3.00626 + 4.83883I −4.00000 − 6.35067I
u = −0.061251 + 0.594784I
a = −0.417558 + 1.055680I
b = 0.192238 − 0.029719I
1.20430 + 1.95319I 2.65095 − 4.43332I
u = −0.061251 − 0.594784I
a = −0.417558 − 1.055680I
b = 0.192238 + 0.029719I
1.20430 − 1.95319I 2.65095 + 4.43332I
u = −1.365440 + 0.327458I
a = −0.843246 + 0.605113I
b = −0.369076 + 1.327770I
−7.31262 + 3.36514I 0
u = −1.365440 − 0.327458I
a = −0.843246 − 0.605113I
b = −0.369076 − 1.327770I
−7.31262 − 3.36514I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.408360 + 0.089637I
a = 0.48516 + 1.53710I
b = −0.507713 + 1.049310I
−5.54125 − 2.92041I 0
u = 1.408360 − 0.089637I
a = 0.48516 − 1.53710I
b = −0.507713 − 1.049310I
−5.54125 + 2.92041I 0
u = −1.45723 + 0.02361I
a = −0.27899 − 2.06538I
b = 0.43890 − 1.34446I
−7.45168 + 2.35722I 0
u = −1.45723 − 0.02361I
a = −0.27899 + 2.06538I
b = 0.43890 + 1.34446I
−7.45168 − 2.35722I 0
u = −0.528921
a = 0.210332
b = −0.547836
−1.28259 −8.18110
u = 1.48529
a = 0.507416
b = 1.07105
−7.77665 0
u = −0.269231 + 0.414079I
a = −1.058820 + 0.163044I
b = 0.294017 + 0.615100I
−0.254814 + 1.145580I −3.55444 − 5.99940I
u = −0.269231 − 0.414079I
a = −1.058820 − 0.163044I
b = 0.294017 − 0.615100I
−0.254814 − 1.145580I −3.55444 + 5.99940I
u = −1.51707 + 0.17151I
a = −0.106102 + 1.396060I
b = 0.998525 + 0.969650I
−10.27140 + 6.63615I 0
u = −1.51707 − 0.17151I
a = −0.106102 − 1.396060I
b = 0.998525 − 0.969650I
−10.27140 − 6.63615I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.52395 + 0.30793I
a = −1.14837 − 1.87775I
b = 0.37644 − 1.80124I
−19.3511 − 12.3559I 0
u = 1.52395 − 0.30793I
a = −1.14837 + 1.87775I
b = 0.37644 + 1.80124I
−19.3511 + 12.3559I 0
u = −1.53293 + 0.26223I
a = 1.00320 − 2.03551I
b = −0.19546 − 1.80459I
−15.6450 + 6.2232I 0
u = −1.53293 − 0.26223I
a = 1.00320 + 2.03551I
b = −0.19546 + 1.80459I
−15.6450 − 6.2232I 0
u = 1.58427 + 0.22493I
a = −0.76018 − 1.95860I
b = 0.01038 − 1.96358I
18.6754 − 0.6900I 0
u = 1.58427 − 0.22493I
a = −0.76018 + 1.95860I
b = 0.01038 + 1.96358I
18.6754 + 0.6900I 0
u = 0.214741 + 0.039322I
a = 4.08599 − 2.69357I
b = −0.103979 − 1.039010I
−1.75783 − 2.05331I −11.26159 + 3.08731I
u = 0.214741 − 0.039322I
a = 4.08599 + 2.69357I
b = −0.103979 + 1.039010I
−1.75783 + 2.05331I −11.26159 − 3.08731I
8
II. I
u
2
= h−u
5
+ 2u
3
+ b − u, −u
5
+ 2u
4
+ · · · + a
2
− a, u
6
− 3u
4
+ 2u
2
+ 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
3
=
a
u
5
− 2u
3
+ u
a
2
=
−u
3
− au + u
2
+ a + u
−u
3
a + au + u
2
− 1
a
9
=
u
5
a − 2u
3
a + au + 1
−1
a
8
=
u
4
− 2u
2
+ 1
−u
4
+ au + u
2
− 1
a
4
=
−u
5
+ 3u
3
− 2u
−u
4
a + 2u
5
+ u
2
a − 4u
3
+ a + 2u
a
7
=
au − u
2
−u
4
+ au + u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
a + 4u
4
− 8u
2
a − 8u
2
+ 4u − 8
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
6
c
2
, c
6
, c
8
(u
4
− u
2
+ 1)
3
c
3
u
12
+ 6u
11
+ ··· − 2u + 1
c
4
, c
9
(u
2
+ 1)
6
c
5
, c
10
, c
11
(u
6
− 3u
4
+ 2u
2
+ 1)
2
c
7
u
12
− 12u
11
+ ··· − 2u + 1
c
12
(u
3
+ u
2
− 1)
4
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)
6
c
2
, c
6
, c
8
(y
2
− y + 1)
6
c
3
y
12
+ 10y
11
+ ··· − 40y + 1
c
4
, c
9
(y + 1)
12
c
5
, c
10
, c
11
(y
3
− 3y
2
+ 2y + 1)
4
c
7
y
12
− 14y
11
+ ··· + 14y + 1
c
12
(y
3
− y
2
+ 2y −1)
4
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.307140 + 0.215080I
a = 0.900631 + 0.022679I
b = 1.000000I
−4.66906 − 4.85801I −9.50976 + 6.44355I
u = 1.307140 + 0.215080I
a = −0.073266 + 1.169920I
b = 1.000000I
−4.66906 − 0.79824I −9.50976 − 0.48465I
u = 1.307140 − 0.215080I
a = 0.900631 − 0.022679I
b = − 1.000000I
−4.66906 + 4.85801I −9.50976 − 6.44355I
u = 1.307140 − 0.215080I
a = −0.073266 − 1.169920I
b = − 1.000000I
−4.66906 + 0.79824I −9.50976 + 0.48465I
u = −1.307140 + 0.215080I
a = −1.56299 + 0.58496I
b = 1.000000I
−4.66906 + 0.79824I −9.50976 + 0.48465I
u = −1.307140 + 0.215080I
a = −0.58909 + 1.73220I
b = 1.000000I
−4.66906 + 4.85801I −9.50976 − 6.44355I
u = −1.307140 − 0.215080I
a = −1.56299 − 0.58496I
b = − 1.000000I
−4.66906 − 0.79824I −9.50976 − 0.48465I
u = −1.307140 − 0.215080I
a = −0.58909 − 1.73220I
b = − 1.000000I
−4.66906 − 4.85801I −9.50976 + 6.44355I
u = 0.569840I
a = 0.662359 + 0.392362I
b = 1.000000I
−0.53148 − 2.02988I −2.98049 + 3.46410I
u = 0.569840I
a = 0.66236 − 1.90212I
b = 1.000000I
−0.53148 + 2.02988I −2.98049 − 3.46410I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = − 0.569840I
a = 0.662359 − 0.392362I
b = − 1.000000I
−0.53148 + 2.02988I −2.98049 − 3.46410I
u = − 0.569840I
a = 0.66236 + 1.90212I
b = − 1.000000I
−0.53148 − 2.02988I −2.98049 + 3.46410I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
6
)(u
38
+ 25u
37
+ ··· + 101u + 25)
c
2
, c
6
((u
4
− u
2
+ 1)
3
)(u
38
− u
37
+ ··· + 11u − 5)
c
3
(u
12
+ 6u
11
+ ··· − 2u + 1)(u
38
− 5u
37
+ ··· − 3u + 1)
c
4
, c
9
((u
2
+ 1)
6
)(u
38
− u
37
+ ··· + 23u − 1)
c
5
, c
10
, c
11
((u
6
− 3u
4
+ 2u
2
+ 1)
2
)(u
38
− u
37
+ ··· + 7u − 1)
c
7
(u
12
− 12u
11
+ ··· − 2u + 1)(u
38
− 5u
37
+ ··· − 40603u + 13213)
c
8
((u
4
− u
2
+ 1)
3
)(u
38
− 3u
37
+ ··· − u + 5)
c
12
((u
3
+ u
2
− 1)
4
)(u
38
+ 3u
37
+ ··· − 17u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
38
− 17y
37
+ ··· − 41901y + 625)
c
2
, c
6
((y
2
− y + 1)
6
)(y
38
− 25y
37
+ ··· − 101y + 25)
c
3
(y
12
+ 10y
11
+ ··· − 40y + 1)(y
38
− 65y
37
+ ··· + 205y + 1)
c
4
, c
9
((y + 1)
12
)(y
38
+ 53y
37
+ ··· − 195y + 1)
c
5
, c
10
, c
11
((y
3
− 3y
2
+ 2y + 1)
4
)(y
38
− 39y
37
+ ··· − 39y + 1)
c
7
(y
12
− 14y
11
+ ··· + 14y + 1)
· (y
38
− 49y
37
+ ··· − 3299858645y + 174583369)
c
8
((y
2
− y + 1)
6
)(y
38
+ 59y
37
+ ··· − 201y + 25)
c
12
((y
3
− y
2
+ 2y −1)
4
)(y
38
+ 57y
37
+ ··· − 183y + 1)
15
