12a
1243
(K12a
1243
)
A knot diagram
1
Linearized knot diagam
5 6 10 9 2 11 12 1 4 3 8 7
Solving Sequence
8,11
12 7 1 9
3,6
2 5 10 4
c
11
c
7
c
12
c
8
c
6
c
2
c
5
c
10
c
3
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h5.47710 × 10
15
u
50
+ 1.14235 × 10
16
u
49
+ ··· + 3.64380 × 10
16
b − 9.84918 × 10
15
,
1.63458 × 10
16
u
50
+ 1.13811 × 10
16
u
49
+ ··· + 1.09314 × 10
17
a − 5.47544 × 10
16
, u
51
+ 2u
50
+ ··· − 3u − 3i
I
u
2
= h−au − u
2
+ b + u − 1, 2u
2
a + a
2
+ 5u
2
+ 2a − 3u + 8, u
3
− u
2
+ 2u − 1i
I
u
3
= hb, u
2
+ a + 1, u
3
+ u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 60 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
= h5.48 × 10
15
u
50
+ 1.14 × 10
16
u
49
+ · · · + 3.64 × 10
16
b − 9.85 × 10
15
, 1.63 ×
10
16
u
50
+1.14×10
16
u
49
+· · ·+1.09×10
17
a−5.48×10
16
, u
51
+2u
50
+· · ·−3u−3i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
4
+ 2u
2
a
9
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
3
=
−0.149530u
50
− 0.104114u
49
+ ··· + 4.89363u + 0.500892
−0.150313u
50
− 0.313505u
49
+ ··· + 0.630080u + 0.270300
a
6
=
u
3
+ 2u
u
3
+ u
a
2
=
−0.00864236u
50
− 0.150838u
49
+ ··· + 4.04860u + 0.105532
−0.210594u
50
− 0.927821u
49
+ ··· + 0.757201u + 0.320503
a
5
=
−0.106834u
50
− 0.00307520u
49
+ ··· + 3.67017u − 0.436697
0.133553u
50
− 0.167686u
49
+ ··· − 0.0796052u + 0.0259271
a
10
=
0.440680u
50
+ 0.784124u
49
+ ··· − 2.10894u + 0.0232897
−0.114496u
50
− 0.354107u
49
+ ··· + 1.06838u + 0.0329283
a
4
=
−0.0918724u
50
+ 0.0332380u
49
+ ··· + 3.96964u − 0.414486
0.0811430u
50
− 0.353168u
49
+ ··· + 0.231766u + 0.517111
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
7943644690210290
18218984011570829
u
50
−
5242160220330023
18218984011570829
u
49
+ ···−
74345173899570890
18218984011570829
u −
57601482949200522
18218984011570829
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
51
+ 4u
50
+ ··· + 44u − 17
c
3
, c
4
, c
9
c
10
u
51
+ u
50
+ ··· − 124u
3
− 8
c
6
, c
8
u
51
− 2u
50
+ ··· − 231u − 87
c
7
, c
11
, c
12
u
51
+ 2u
50
+ ··· − 3u − 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
51
− 56y
50
+ ··· + 3092y −289
c
3
, c
4
, c
9
c
10
y
51
+ 65y
50
+ ··· + 4352y
2
− 64
c
6
, c
8
y
51
− 46y
50
+ ··· − 73311y −7569
c
7
, c
11
, c
12
y
51
+ 42y
50
+ ··· + 57y −9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.907928 + 0.131274I
a = −0.85159 + 2.57553I
b = 0.17354 + 1.71169I
18.9001 + 8.0570I −13.49826 − 3.74079I
u = −0.907928 − 0.131274I
a = −0.85159 − 2.57553I
b = 0.17354 − 1.71169I
18.9001 − 8.0570I −13.49826 + 3.74079I
u = 0.049806 + 1.119400I
a = −1.65276 − 0.63158I
b = 0.14103 − 1.43819I
−4.53961 − 0.47932I −9.08978 − 0.54111I
u = 0.049806 − 1.119400I
a = −1.65276 + 0.63158I
b = 0.14103 + 1.43819I
−4.53961 + 0.47932I −9.08978 + 0.54111I
u = 0.870747 + 0.070256I
a = −0.10989 − 1.48062I
b = 0.608525 − 0.958375I
−11.35660 − 4.92570I −12.66557 + 4.02206I
u = 0.870747 − 0.070256I
a = −0.10989 + 1.48062I
b = 0.608525 + 0.958375I
−11.35660 + 4.92570I −12.66557 − 4.02206I
u = −0.866790 + 0.044839I
a = 0.44024 − 3.22463I
b = −0.05531 − 1.67384I
−13.05010 + 3.20345I −12.09771 − 2.59432I
u = −0.866790 − 0.044839I
a = 0.44024 + 3.22463I
b = −0.05531 + 1.67384I
−13.05010 − 3.20345I −12.09771 + 2.59432I
u = −0.853858
a = 0.332469
b = 0.869508
−8.45310 −10.4770
u = 0.645538 + 0.554017I
a = 1.09788 + 1.43143I
b = −0.02763 + 1.69086I
−14.2712 − 2.2782I −11.91209 + 2.93097I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.645538 − 0.554017I
a = 1.09788 − 1.43143I
b = −0.02763 − 1.69086I
−14.2712 + 2.2782I −11.91209 − 2.93097I
u = 0.098307 + 1.231160I
a = −1.087560 + 0.607902I
b = 0.422786 + 0.380273I
1.39334 − 1.58061I −4.00000 + 0.I
u = 0.098307 − 1.231160I
a = −1.087560 − 0.607902I
b = 0.422786 − 0.380273I
1.39334 + 1.58061I −4.00000 + 0.I
u = 0.761889 + 0.030162I
a = 0.14348 + 1.84705I
b = −0.227333 + 0.836438I
−4.19077 − 2.14878I −11.40445 + 4.59039I
u = 0.761889 − 0.030162I
a = 0.14348 − 1.84705I
b = −0.227333 − 0.836438I
−4.19077 + 2.14878I −11.40445 − 4.59039I
u = −0.488015 + 1.149550I
a = −0.480142 + 1.284100I
b = −0.14196 + 1.72564I
−17.4585 − 3.1002I 0
u = −0.488015 − 1.149550I
a = −0.480142 − 1.284100I
b = −0.14196 − 1.72564I
−17.4585 + 3.1002I 0
u = 0.303333 + 1.240150I
a = 0.481754 + 0.904540I
b = 0.082101 + 0.850538I
−0.47130 − 1.69507I 0
u = 0.303333 − 1.240150I
a = 0.481754 − 0.904540I
b = 0.082101 − 0.850538I
−0.47130 + 1.69507I 0
u = 0.418137 + 1.207070I
a = −0.181479 − 0.248186I
b = −0.556081 − 1.021520I
−7.85706 + 0.30900I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.418137 − 1.207070I
a = −0.181479 + 0.248186I
b = −0.556081 + 1.021520I
−7.85706 − 0.30900I 0
u = −0.052017 + 1.294910I
a = 0.953642 − 0.051833I
b = −0.447473 − 0.419214I
4.49060 + 1.57458I 0
u = −0.052017 − 1.294910I
a = 0.953642 + 0.051833I
b = −0.447473 + 0.419214I
4.49060 − 1.57458I 0
u = −0.410167 + 1.234410I
a = 0.96089 − 1.81167I
b = 0.01537 − 1.67225I
−9.37724 + 1.37166I 0
u = −0.410167 − 1.234410I
a = 0.96089 + 1.81167I
b = 0.01537 + 1.67225I
−9.37724 − 1.37166I 0
u = −0.259573 + 1.277840I
a = −0.313273 + 0.355431I
b = 0.450243 − 0.039258I
2.27192 + 3.32252I 0
u = −0.259573 − 1.277840I
a = −0.313273 − 0.355431I
b = 0.450243 + 0.039258I
2.27192 − 3.32252I 0
u = 0.332771 + 1.284440I
a = −1.05807 − 1.05291I
b = 0.338435 − 0.828395I
−0.09882 − 6.10554I 0
u = 0.332771 − 1.284440I
a = −1.05807 + 1.05291I
b = 0.338435 + 0.828395I
−0.09882 + 6.10554I 0
u = −0.393305 + 1.273120I
a = 0.441102 − 0.573712I
b = −0.865536 + 0.073675I
−4.49930 + 4.47398I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.393305 − 1.273120I
a = 0.441102 + 0.573712I
b = −0.865536 − 0.073675I
−4.49930 − 4.47398I 0
u = −0.660643
a = −0.232662
b = −0.401553
−1.71027 −3.43300
u = 0.148769 + 1.331880I
a = 1.213820 − 0.003770I
b = −0.05720 + 1.47291I
−1.60931 − 3.17982I 0
u = 0.148769 − 1.331880I
a = 1.213820 + 0.003770I
b = −0.05720 − 1.47291I
−1.60931 + 3.17982I 0
u = −0.487213 + 0.407368I
a = 0.952102 − 0.113234I
b = −0.144630 − 0.903505I
−5.05412 + 1.66912I −10.97373 − 4.62737I
u = −0.487213 − 0.407368I
a = 0.952102 + 0.113234I
b = −0.144630 + 0.903505I
−5.05412 − 1.66912I −10.97373 + 4.62737I
u = −0.397349 + 1.308030I
a = −1.65562 + 1.61052I
b = 0.08962 + 1.66718I
−8.82655 + 7.73768I 0
u = −0.397349 − 1.308030I
a = −1.65562 − 1.61052I
b = 0.08962 − 1.66718I
−8.82655 − 7.73768I 0
u = 0.396413 + 1.325070I
a = 1.20491 + 0.91251I
b = −0.640552 + 0.900828I
−6.99006 − 9.47255I 0
u = 0.396413 − 1.325070I
a = 1.20491 − 0.91251I
b = −0.640552 − 0.900828I
−6.99006 + 9.47255I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.153785 + 1.383640I
a = −0.828540 − 0.587282I
b = 0.236607 + 0.683213I
0.58476 + 3.87146I 0
u = −0.153785 − 1.383640I
a = −0.828540 + 0.587282I
b = 0.236607 − 0.683213I
0.58476 − 3.87146I 0
u = −0.40573 + 1.36865I
a = 1.84188 − 1.07241I
b = −0.19235 − 1.69170I
−15.8596 + 12.7667I 0
u = −0.40573 − 1.36865I
a = 1.84188 + 1.07241I
b = −0.19235 + 1.69170I
−15.8596 − 12.7667I 0
u = 0.15668 + 1.46609I
a = −0.952170 + 0.297878I
b = 0.06047 − 1.64270I
−7.63888 − 4.94907I 0
u = 0.15668 − 1.46609I
a = −0.952170 − 0.297878I
b = 0.06047 + 1.64270I
−7.63888 + 4.94907I 0
u = 0.424316 + 0.265115I
a = −1.41012 − 2.43053I
b = −0.00144 − 1.49759I
−6.57052 − 1.14577I −8.04300 + 6.02810I
u = 0.424316 − 0.265115I
a = −1.41012 + 2.43053I
b = −0.00144 + 1.49759I
−6.57052 + 1.14577I −8.04300 − 6.02810I
u = 0.351513
a = 2.15459
b = −0.342179
−2.23327 1.33350
u = −0.203339 + 0.264461I
a = −0.777669 + 0.662838I
b = 0.175873 + 0.407856I
−0.158036 + 0.749712I −4.84555 − 9.27744I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.203339 − 0.264461I
a = −0.777669 − 0.662838I
b = 0.175873 − 0.407856I
−0.158036 − 0.749712I −4.84555 + 9.27744I
10
II.
I
u
2
= h−au − u
2
+ b + u − 1, 2u
2
a + a
2
+ 5u
2
+ 2a − 3u + 8, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
7
=
u
u
2
− u + 1
a
1
=
u
2
+ 1
u
2
− u + 1
a
9
=
−1
0
a
3
=
a
au + u
2
− u + 1
a
6
=
u
2
+ 1
u
2
− u + 1
a
2
=
u
2
+ a + 1
au + 2u
2
− 2u + 2
a
5
=
−a
−au − u
2
+ u − 1
a
10
=
−u
2
a + au − 2u
2
− a + 2u − 4
−2
a
4
=
au + u
2
− a − u + 1
−au − u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
+ 4u − 16
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u + 1)
6
c
3
, c
4
, c
9
c
10
(u
2
+ 2)
3
c
5
(u − 1)
6
c
6
, c
8
(u
3
− u
2
+ 1)
2
c
7
(u
3
+ u
2
+ 2u + 1)
2
c
11
, c
12
(u
3
− u
2
+ 2u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
9
c
10
(y + 2)
6
c
6
, c
8
(y
3
− y
2
+ 2y −1)
2
c
7
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −0.391035 − 0.735607I
b = − 1.414210I
−3.55561 − 2.82812I −8.49024 + 2.97945I
u = 0.215080 + 1.307140I
a = 1.71575 − 0.38895I
b = 1.414210I
−3.55561 − 2.82812I −8.49024 + 2.97945I
u = 0.215080 − 1.307140I
a = −0.391035 + 0.735607I
b = 1.414210I
−3.55561 + 2.82812I −8.49024 − 2.97945I
u = 0.215080 − 1.307140I
a = 1.71575 + 0.38895I
b = − 1.414210I
−3.55561 + 2.82812I −8.49024 − 2.97945I
u = 0.569840
a = −1.32472 + 2.48177I
b = 1.414210I
−7.69319 −15.0200
u = 0.569840
a = −1.32472 − 2.48177I
b = − 1.414210I
−7.69319 −15.0200
14
III. I
u
3
= hb, u
2
+ a + 1, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
7
=
u
−u
2
− u − 1
a
1
=
u
2
+ 1
u
2
+ u + 1
a
9
=
1
0
a
3
=
−u
2
− 1
0
a
6
=
−u
2
− 1
−u
2
− u − 1
a
2
=
0
u
2
+ u + 1
a
5
=
−u
2
− 1
0
a
10
=
1
0
a
4
=
−u
2
− 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −6u
2
− 4u − 16
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
9
c
10
u
3
c
5
(u + 1)
3
c
6
, c
8
u
3
+ u
2
− 1
c
7
u
3
− u
2
+ 2u − 1
c
11
, c
12
u
3
+ u
2
+ 2u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
3
c
3
, c
4
, c
9
c
10
y
3
c
6
, c
8
y
3
− y
2
+ 2y −1
c
7
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
17
(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
1.37919 + 2.82812I −5.16553 − 1.85489I
u = −0.215080 − 1.307140I
a = 0.662359 − 0.562280I
b = 0
1.37919 − 2.82812I −5.16553 + 1.85489I
u = −0.569840
a = −1.32472
b = 0
−2.75839 −15.6690
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
3
)(u + 1)
6
(u
51
+ 4u
50
+ ··· + 44u − 17)
c
3
, c
4
, c
9
c
10
u
3
(u
2
+ 2)
3
(u
51
+ u
50
+ ··· − 124u
3
− 8)
c
5
((u − 1)
6
)(u + 1)
3
(u
51
+ 4u
50
+ ··· + 44u − 17)
c
6
, c
8
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
51
− 2u
50
+ ··· − 231u − 87)
c
7
(u
3
− u
2
+ 2u − 1)(u
3
+ u
2
+ 2u + 1)
2
(u
51
+ 2u
50
+ ··· − 3u − 3)
c
11
, c
12
((u
3
− u
2
+ 2u − 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
51
+ 2u
50
+ ··· − 3u − 3)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
((y −1)
9
)(y
51
− 56y
50
+ ··· + 3092y −289)
c
3
, c
4
, c
9
c
10
y
3
(y + 2)
6
(y
51
+ 65y
50
+ ··· + 4352y
2
− 64)
c
6
, c
8
((y
3
− y
2
+ 2y −1)
3
)(y
51
− 46y
50
+ ··· − 73311y −7569)
c
7
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
51
+ 42y
50
+ ··· + 57y −9)
20
