12a
0790
(K12a
0790
)
A knot diagram
1
Linearized knot diagam
3 8 11 12 9 10 2 7 6 1 5 4
Solving Sequence
5,9
6 10
7,12
4 1 8 11 3 2
c
5
c
9
c
6
c
4
c
12
c
8
c
11
c
3
c
2
c
1
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
73
+ 3u
72
+ ··· + 4b − 1, −u
73
+ 3u
72
+ ··· + 4a + 3, u
74
− 4u
73
+ ··· + u − 1i
I
u
2
= hb − a − 1, a
3
+ 2a
2
+ 3a + 1, u − 1i
* 2 irreducible components of dim
C
= 0, with total 77 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−u
73
+3u
72
+· · ·+4b−1, −u
73
+3u
72
+· · ·+4a+3, u
74
−4u
73
+· · ·+u−1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
6
=
1
u
2
a
10
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
12
=
1
4
u
73
−
3
4
u
72
+ ··· − 6u −
3
4
1
4
u
73
−
3
4
u
72
+ ··· −
15
4
u
2
+
1
4
a
4
=
17
4
u
73
−
49
4
u
72
+ ··· −
1
2
u +
19
4
5u
73
−
29
2
u
72
+ ··· −
1
2
u +
9
2
a
1
=
21
2
u
73
−
57
2
u
72
+ ··· − 7u +
15
2
77
4
u
73
−
213
4
u
72
+ ··· −
5
2
u +
63
4
a
8
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
11
=
−u
18
+ 7u
16
+ ··· − 6u − 1
1
4
u
73
−
3
4
u
72
+ ··· −
15
4
u
2
+
1
4
a
3
=
−
25
2
u
73
+
67
2
u
72
+ ··· − 2u −
19
2
59
4
u
73
−
163
4
u
72
+ ··· −
5
2
u +
49
4
a
2
=
−
1
2
u
73
+
5
2
u
72
+ ··· − 5u −
3
2
117
4
u
73
−
325
4
u
72
+ ··· −
9
2
u +
95
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = 63u
73
−
343
2
u
72
+ ··· −
3
2
u +
101
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
74
+ 21u
73
+ ··· + 848u + 64
c
2
, c
7
u
74
+ u
73
+ ··· + 12u + 8
c
3
u
74
− 2u
73
+ ··· − 60u − 9
c
4
, c
11
, c
12
u
74
+ 2u
73
+ ··· − 6u − 1
c
5
, c
6
, c
9
u
74
− 4u
73
+ ··· + u − 1
c
10
u
74
+ 18u
73
+ ··· + 560u + 49
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
74
+ 59y
73
+ ··· − 36096y + 4096
c
2
, c
7
y
74
− 21y
73
+ ··· − 848y + 64
c
3
y
74
− 6y
73
+ ··· + 162y + 81
c
4
, c
11
, c
12
y
74
+ 66y
73
+ ··· − 14y + 1
c
5
, c
6
, c
9
y
74
− 60y
73
+ ··· + 17y + 1
c
10
y
74
+ 6y
73
+ ··· + 49882y + 2401
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.982931 + 0.351670I
a = 1.29013 − 1.44788I
b = −0.114226 − 1.346610I
−5.40881 + 1.48488I 0
u = −0.982931 − 0.351670I
a = 1.29013 + 1.44788I
b = −0.114226 + 1.346610I
−5.40881 − 1.48488I 0
u = −0.114338 + 0.877618I
a = 1.79001 − 0.30234I
b = −0.29499 − 1.39816I
0.40081 + 10.70940I 0
u = −0.114338 − 0.877618I
a = 1.79001 + 0.30234I
b = −0.29499 + 1.39816I
0.40081 − 10.70940I 0
u = −0.096801 + 0.866328I
a = 1.151770 − 0.473352I
b = −0.736562 − 0.242206I
5.61875 + 6.96802I 0. − 6.74295I
u = −0.096801 − 0.866328I
a = 1.151770 + 0.473352I
b = −0.736562 + 0.242206I
5.61875 − 6.96802I 0. + 6.74295I
u = −0.841922
a = 0.589364
b = −0.325299
−1.16434 −11.0700
u = −0.071633 + 0.837460I
a = 0.325108 − 0.396010I
b = −0.289957 + 0.808580I
3.72265 + 3.07550I 0. − 1.96212I
u = −0.071633 − 0.837460I
a = 0.325108 + 0.396010I
b = −0.289957 − 0.808580I
3.72265 − 3.07550I 0. + 1.96212I
u = −0.032962 + 0.823375I
a = −0.347239 − 0.544285I
b = 0.256055 + 0.912970I
3.87638 + 2.84826I 0. − 3.67608I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.032962 − 0.823375I
a = −0.347239 + 0.544285I
b = 0.256055 − 0.912970I
3.87638 − 2.84826I 0. + 3.67608I
u = 0.011012 + 0.810021I
a = −1.250280 − 0.486422I
b = 0.733701 − 0.206836I
6.08567 − 0.95765I 3.69454 + 1.25612I
u = 0.011012 − 0.810021I
a = −1.250280 + 0.486422I
b = 0.733701 + 0.206836I
6.08567 + 0.95765I 3.69454 − 1.25612I
u = 0.038683 + 0.799560I
a = −1.97178 − 0.13620I
b = 0.294669 − 1.379250I
1.05738 − 4.68103I −0.94722 + 2.51196I
u = 0.038683 − 0.799560I
a = −1.97178 + 0.13620I
b = 0.294669 + 1.379250I
1.05738 + 4.68103I −0.94722 − 2.51196I
u = −0.623280 + 0.490874I
a = −0.097681 + 1.280110I
b = −0.185972 + 1.388820I
−6.53161 − 2.00010I −10.45886 + 0.I
u = −0.623280 − 0.490874I
a = −0.097681 − 1.280110I
b = −0.185972 − 1.388820I
−6.53161 + 2.00010I −10.45886 + 0.I
u = −0.138267 + 0.766818I
a = −0.172017 + 0.580214I
b = −0.051081 + 1.397250I
−2.82199 + 2.49718I −4.67442 − 3.51648I
u = −0.138267 − 0.766818I
a = −0.172017 − 0.580214I
b = −0.051081 − 1.397250I
−2.82199 − 2.49718I −4.67442 + 3.51648I
u = 1.229430 + 0.060313I
a = −0.812751 − 0.979382I
b = 0.261875 − 1.235300I
−5.67724 − 3.54075I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.229430 − 0.060313I
a = −0.812751 + 0.979382I
b = 0.261875 + 1.235300I
−5.67724 + 3.54075I 0
u = 1.23207
a = −0.416426
b = 0.710023
−1.89563 0
u = −1.156710 + 0.449194I
a = 0.368328 + 1.040150I
b = −0.28561 + 1.38677I
−2.79362 − 5.97206I 0
u = −1.156710 − 0.449194I
a = 0.368328 − 1.040150I
b = −0.28561 − 1.38677I
−2.79362 + 5.97206I 0
u = −0.447414 + 0.610883I
a = 1.87368 − 1.20772I
b = −0.22452 − 1.39679I
−5.97032 + 6.07295I −8.10796 − 7.73862I
u = −0.447414 − 0.610883I
a = 1.87368 + 1.20772I
b = −0.22452 + 1.39679I
−5.97032 − 6.07295I −8.10796 + 7.73862I
u = −1.174400 + 0.427018I
a = 0.406561 − 0.005470I
b = −0.716781 + 0.220171I
2.31230 − 2.33683I 0
u = −1.174400 − 0.427018I
a = 0.406561 + 0.005470I
b = −0.716781 − 0.220171I
2.31230 + 2.33683I 0
u = −1.250100 + 0.050558I
a = 0.58611 + 1.34682I
b = 0.502125 + 0.290673I
−2.76802 + 1.42671I 0
u = −1.250100 − 0.050558I
a = 0.58611 − 1.34682I
b = 0.502125 − 0.290673I
−2.76802 − 1.42671I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.207660 + 0.387161I
a = 0.908549 − 0.496550I
b = −0.200242 − 0.878527I
0.234639 + 1.326060I 0
u = −1.207660 − 0.387161I
a = 0.908549 + 0.496550I
b = −0.200242 + 0.878527I
0.234639 − 1.326060I 0
u = −1.249840 + 0.252900I
a = 1.62462 − 2.52638I
b = 0.080915 − 1.377220I
−5.96096 + 0.85034I 0
u = −1.249840 − 0.252900I
a = 1.62462 + 2.52638I
b = 0.080915 + 1.377220I
−5.96096 − 0.85034I 0
u = 1.237920 + 0.345248I
a = −0.454616 + 1.014780I
b = 0.303590 + 1.360390I
−2.63807 + 0.55572I 0
u = 1.237920 − 0.345248I
a = −0.454616 − 1.014780I
b = 0.303590 − 1.360390I
−2.63807 − 0.55572I 0
u = −1.234510 + 0.365040I
a = 0.701470 − 0.198406I
b = 0.225501 − 0.802739I
0.16896 + 1.43086I 0
u = −1.234510 − 0.365040I
a = 0.701470 + 0.198406I
b = 0.225501 + 0.802739I
0.16896 − 1.43086I 0
u = −1.307020 + 0.054439I
a = −1.03660 + 4.23394I
b = 0.206167 + 1.397270I
−8.12854 + 4.09193I 0
u = −1.307020 − 0.054439I
a = −1.03660 − 4.23394I
b = 0.206167 − 1.397270I
−8.12854 − 4.09193I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.260470 + 0.357448I
a = −0.411022 − 0.003429I
b = 0.747744 + 0.176905I
2.21416 − 3.24351I 0
u = 1.260470 − 0.357448I
a = −0.411022 + 0.003429I
b = 0.747744 − 0.176905I
2.21416 + 3.24351I 0
u = −1.277440 + 0.359296I
a = −0.58395 + 1.32299I
b = 0.721895 + 0.236951I
2.07943 + 5.16591I 0
u = −1.277440 − 0.359296I
a = −0.58395 − 1.32299I
b = 0.721895 − 0.236951I
2.07943 − 5.16591I 0
u = −0.394203 + 0.545208I
a = 0.829429 − 0.695537I
b = −0.569066 − 0.273917I
−0.65164 + 3.14368I −3.24051 − 8.93750I
u = −0.394203 − 0.545208I
a = 0.829429 + 0.695537I
b = −0.569066 + 0.273917I
−0.65164 − 3.14368I −3.24051 + 8.93750I
u = 1.290360 + 0.368627I
a = −0.878129 − 0.593703I
b = 0.286921 − 0.976043I
−0.24573 − 7.13569I 0
u = 1.290360 − 0.368627I
a = −0.878129 + 0.593703I
b = 0.286921 + 0.976043I
−0.24573 + 7.13569I 0
u = −1.296910 + 0.352312I
a = −2.56859 + 2.04684I
b = 0.28811 + 1.39495I
−3.11175 + 8.83167I 0
u = −1.296910 − 0.352312I
a = −2.56859 − 2.04684I
b = 0.28811 − 1.39495I
−3.11175 − 8.83167I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.365930 + 0.104815I
a = −0.439263 − 0.721972I
b = −0.463262 − 0.494084I
−6.91013 − 1.79711I 0
u = 1.365930 − 0.104815I
a = −0.439263 + 0.721972I
b = −0.463262 + 0.494084I
−6.91013 + 1.79711I 0
u = 1.322090 + 0.373208I
a = −0.504347 − 0.222253I
b = −0.351613 − 0.778711I
−0.64388 − 7.42880I 0
u = 1.322090 − 0.373208I
a = −0.504347 + 0.222253I
b = −0.351613 + 0.778711I
−0.64388 + 7.42880I 0
u = 1.343220 + 0.336289I
a = −1.02775 − 2.21581I
b = −0.05949 − 1.43435I
−7.46754 − 6.50190I 0
u = 1.343220 − 0.336289I
a = −1.02775 + 2.21581I
b = −0.05949 + 1.43435I
−7.46754 + 6.50190I 0
u = 1.376590 + 0.153168I
a = −0.007356 + 1.170020I
b = −0.632070 + 0.344676I
−6.25872 − 5.47972I 0
u = 1.376590 − 0.153168I
a = −0.007356 − 1.170020I
b = −0.632070 − 0.344676I
−6.25872 + 5.47972I 0
u = −0.501713 + 0.351989I
a = 0.250347 + 0.020930I
b = −0.377876 + 0.306739I
−1.234330 + 0.285895I −7.26902 − 0.13078I
u = −0.501713 − 0.351989I
a = 0.250347 − 0.020930I
b = −0.377876 − 0.306739I
−1.234330 − 0.285895I −7.26902 + 0.13078I
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.338950 + 0.387128I
a = 0.548552 + 1.167370I
b = −0.746647 + 0.261104I
1.11467 − 11.46740I 0
u = 1.338950 − 0.387128I
a = 0.548552 − 1.167370I
b = −0.746647 − 0.261104I
1.11467 + 11.46740I 0
u = 1.350930 + 0.390369I
a = 2.24091 + 1.86991I
b = −0.29851 + 1.40822I
−4.2026 − 15.2588I 0
u = 1.350930 − 0.390369I
a = 2.24091 − 1.86991I
b = −0.29851 − 1.40822I
−4.2026 + 15.2588I 0
u = 1.409670 + 0.083415I
a = −0.11722 − 3.16504I
b = −0.16218 − 1.43303I
−12.99120 + 0.43188I 0
u = 1.409670 − 0.083415I
a = −0.11722 + 3.16504I
b = −0.16218 + 1.43303I
−12.99120 − 0.43188I 0
u = 1.40588 + 0.16480I
a = 1.49179 + 2.95937I
b = −0.23906 + 1.42598I
−11.9190 − 8.6567I 0
u = 1.40588 − 0.16480I
a = 1.49179 − 2.95937I
b = −0.23906 − 1.42598I
−11.9190 + 8.6567I 0
u = 0.011339 + 0.462213I
a = 1.055710 − 0.545353I
b = 0.125209 + 1.263110I
−2.24513 + 1.95677I −0.38158 − 3.64483I
u = 0.011339 − 0.462213I
a = 1.055710 + 0.545353I
b = 0.125209 − 1.263110I
−2.24513 − 1.95677I −0.38158 + 3.64483I
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.260606 + 0.183987I
a = −2.74543 − 3.45998I
b = 0.215232 − 1.341580I
−3.38918 − 3.25880I 0.46002 + 3.22263I
u = 0.260606 − 0.183987I
a = −2.74543 + 3.45998I
b = 0.215232 + 1.341580I
−3.38918 + 3.25880I 0.46002 − 3.22263I
u = 0.109980 + 0.232584I
a = −0.60353 − 1.96816I
b = 0.557633 − 0.099372I
1.189920 − 0.442233I 6.78124 + 1.28723I
u = 0.109980 − 0.232584I
a = −0.60353 + 1.96816I
b = 0.557633 + 0.099372I
1.189920 + 0.442233I 6.78124 − 1.28723I
12
II. I
u
2
= hb − a − 1, a
3
+ 2a
2
+ 3a + 1, u − 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
1
a
6
=
1
1
a
10
=
−1
0
a
7
=
0
1
a
12
=
a
a + 1
a
4
=
a
2
+ a + 1
a
2
+ 2a + 1
a
1
=
0
a
2
+ a + 1
a
8
=
0
1
a
11
=
−1
a + 1
a
3
=
0
a
2
+ a + 1
a
2
=
0
a
2
+ a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5a
2
+ 6a + 5
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
8
u
3
c
3
u
3
+ u
2
− 1
c
4
u
3
− u
2
+ 2u − 1
c
5
, c
6
(u − 1)
3
c
9
(u + 1)
3
c
10
u
3
− u
2
+ 1
c
11
, c
12
u
3
+ u
2
+ 2u + 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
c
8
y
3
c
3
, c
10
y
3
− y
2
+ 2y − 1
c
4
, c
11
, c
12
y
3
+ 3y
2
+ 2y − 1
c
5
, c
6
, c
9
(y − 1)
3
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.78492 + 1.30714I
b = 0.215080 + 1.307140I
−4.66906 + 2.82812I −5.17211 − 2.41717I
u = 1.00000
a = −0.78492 − 1.30714I
b = 0.215080 − 1.307140I
−4.66906 − 2.82812I −5.17211 + 2.41717I
u = 1.00000
a = −0.430160
b = 0.569840
−0.531480 3.34420
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
8
u
3
(u
74
+ 21u
73
+ ··· + 848u + 64)
c
2
, c
7
u
3
(u
74
+ u
73
+ ··· + 12u + 8)
c
3
(u
3
+ u
2
− 1)(u
74
− 2u
73
+ ··· − 60u − 9)
c
4
(u
3
− u
2
+ 2u − 1)(u
74
+ 2u
73
+ ··· − 6u − 1)
c
5
, c
6
((u − 1)
3
)(u
74
− 4u
73
+ ··· + u − 1)
c
9
((u + 1)
3
)(u
74
− 4u
73
+ ··· + u − 1)
c
10
(u
3
− u
2
+ 1)(u
74
+ 18u
73
+ ··· + 560u + 49)
c
11
, c
12
(u
3
+ u
2
+ 2u + 1)(u
74
+ 2u
73
+ ··· − 6u − 1)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
3
(y
74
+ 59y
73
+ ··· − 36096y + 4096)
c
2
, c
7
y
3
(y
74
− 21y
73
+ ··· − 848y + 64)
c
3
(y
3
− y
2
+ 2y − 1)(y
74
− 6y
73
+ ··· + 162y + 81)
c
4
, c
11
, c
12
(y
3
+ 3y
2
+ 2y − 1)(y
74
+ 66y
73
+ ··· − 14y + 1)
c
5
, c
6
, c
9
((y − 1)
3
)(y
74
− 60y
73
+ ··· + 17y + 1)
c
10
(y
3
− y
2
+ 2y − 1)(y
74
+ 6y
73
+ ··· + 49882y + 2401)
18
