12a
0669
(K12a
0669
)
A knot diagram
1
Linearized knot diagam
3 7 11 9 8 2 6 5 12 1 4 10
Solving Sequence
4,9 5,12
10 1 8 6 7 11 3 2
c
4
c
9
c
12
c
8
c
5
c
7
c
11
c
3
c
2
c
1
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−25220615118u
37
− 125516232099u
36
+ ··· + 42034358527b − 24647116036,
60529476279u
37
+ 353049488728u
36
+ ··· + 42034358527a + 595728741584,
u
38
+ 6u
37
+ ··· + 16u + 1i
I
u
2
= hb, −u
4
+ u
3
− 4u
2
+ a + 3u − 3, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
* 2 irreducible components of dim
C
= 0, with total 43 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−2.52 × 10
10
u
37
− 1.26 × 10
11
u
36
+ · · · + 4.20 × 10
10
b − 2.46 ×
10
10
, 6.05 × 10
10
u
37
+ 3.53 × 10
11
u
36
+ · · · + 4.20 × 10
10
a + 5.96 ×
10
11
, u
38
+ 6u
37
+ · · · + 16u + 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
5
=
1
u
2
a
12
=
−1.44000u
37
− 8.39907u
36
+ ··· − 114.524u − 14.1724
0.600000u
37
+ 2.98604u
36
+ ··· + 1.86170u + 0.586356
a
10
=
−1.24000u
37
− 7.40372u
36
+ ··· − 102.904u − 12.3103
0.400000u
37
+ 2.01396u
36
+ ··· + 2.13830u + 0.413644
a
1
=
−0.600000u
37
− 3.00931u
36
+ ··· − 19.7589u − 3.27576
0.400000u
37
+ 2.01396u
36
+ ··· + 2.13830u + 0.413644
a
8
=
u
u
3
+ u
a
6
=
u
2
+ 1
u
4
+ 2u
2
a
7
=
u
3
+ 2u
u
5
+ 3u
3
+ u
a
11
=
−2.04000u
37
− 11.3851u
36
+ ··· − 116.386u − 14.7588
0.600000u
37
+ 2.98604u
36
+ ··· + 1.86170u + 0.586356
a
3
=
−1.00434u
37
− 5.62602u
36
+ ··· − 39.9468u − 5.08000
0.590693u
37
+ 3.54416u
36
+ ··· + 6.32424u + 0.600000
a
2
=
−1.00000u
37
− 5.60444u
36
+ ··· − 30.7482u − 4.28010
0.400000u
37
+ 2.59513u
36
+ ··· + 10.9894u + 1.00434
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
323207095835
42034358527
u
37
+
1803051253389
42034358527
u
36
+ ··· +
12006740049581
42034358527
u +
1316852383934
42034358527
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
7
, c
8
u
38
+ 6u
37
+ ··· + 16u + 1
c
2
, c
6
u
38
− 2u
37
+ ··· + 4u − 1
c
3
, c
11
u
38
− u
37
+ ··· − 32u + 32
c
9
, c
10
, c
12
u
38
+ 6u
37
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
7
, c
8
y
38
+ 54y
37
+ ··· − 12y + 1
c
2
, c
6
y
38
− 6y
37
+ ··· − 16y + 1
c
3
, c
11
y
38
− 33y
37
+ ··· − 3584y + 1024
c
9
, c
10
, c
12
y
38
− 42y
37
+ ··· + 6y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.713718 + 0.696674I
a = 0.98664 + 1.13350I
b = 1.355910 + 0.254773I
6.26346 + 5.04642I 0. − 6.40435I
u = −0.713718 − 0.696674I
a = 0.98664 − 1.13350I
b = 1.355910 − 0.254773I
6.26346 − 5.04642I 0. + 6.40435I
u = −0.205081 + 0.964197I
a = −0.002810 + 0.336869I
b = −0.045418 + 0.545182I
2.26768 + 2.34844I 0. − 4.01424I
u = −0.205081 − 0.964197I
a = −0.002810 − 0.336869I
b = −0.045418 − 0.545182I
2.26768 − 2.34844I 0. + 4.01424I
u = −0.861163
a = 1.70659
b = 1.30561
4.20147 −0.424720
u = −0.032507 + 1.171160I
a = −0.645677 − 0.518349I
b = 1.192870 − 0.112572I
6.07449 + 0.14915I 0
u = −0.032507 − 1.171160I
a = −0.645677 + 0.518349I
b = 1.192870 + 0.112572I
6.07449 − 0.14915I 0
u = 0.190395 + 1.187080I
a = −0.038520 + 1.334050I
b = −1.55601 + 0.43836I
13.51030 − 3.29880I 0
u = 0.190395 − 1.187080I
a = −0.038520 − 1.334050I
b = −1.55601 − 0.43836I
13.51030 + 3.29880I 0
u = −0.119477 + 1.235160I
a = −0.083865 − 1.148000I
b = 0.070500 − 1.218200I
8.02065 + 2.76566I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.119477 − 1.235160I
a = −0.083865 + 1.148000I
b = 0.070500 + 1.218200I
8.02065 − 2.76566I 0
u = −0.223356 + 1.231840I
a = 0.531548 − 0.533329I
b = −1.185810 − 0.238140I
5.79827 + 5.29458I 0
u = −0.223356 − 1.231840I
a = 0.531548 + 0.533329I
b = −1.185810 + 0.238140I
5.79827 − 5.29458I 0
u = −0.454460 + 0.526349I
a = 0.225842 − 0.858062I
b = −0.810125 − 0.280595I
0.14659 + 2.93637I −1.27878 − 9.48259I
u = −0.454460 − 0.526349I
a = 0.225842 + 0.858062I
b = −0.810125 + 0.280595I
0.14659 − 2.93637I −1.27878 + 9.48259I
u = −0.383220 + 1.338060I
a = 0.143279 + 1.122790I
b = 1.48427 + 0.49972I
12.7262 + 8.9717I 0
u = −0.383220 − 1.338060I
a = 0.143279 − 1.122790I
b = 1.48427 − 0.49972I
12.7262 − 8.9717I 0
u = −0.242924 + 0.500540I
a = −0.77914 − 2.00057I
b = 0.217646 − 0.715322I
2.34679 + 1.50207I 1.29303 − 3.53914I
u = −0.242924 − 0.500540I
a = −0.77914 + 2.00057I
b = 0.217646 + 0.715322I
2.34679 − 1.50207I 1.29303 + 3.53914I
u = 0.390857 + 0.382528I
a = −1.94196 + 2.07024I
b = −1.47608 + 0.13954I
8.43405 − 1.32281I 8.28038 + 0.36190I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.390857 − 0.382528I
a = −1.94196 − 2.07024I
b = −1.47608 − 0.13954I
8.43405 + 1.32281I 8.28038 − 0.36190I
u = −0.468337 + 0.127698I
a = −0.182428 + 0.431853I
b = −0.370032 + 0.319434I
−1.073810 + 0.172026I −9.19796 − 0.41861I
u = −0.468337 − 0.127698I
a = −0.182428 − 0.431853I
b = −0.370032 − 0.319434I
−1.073810 − 0.172026I −9.19796 + 0.41861I
u = −0.05162 + 1.71260I
a = 0.001886 + 0.320518I
b = −0.001031 + 0.619536I
11.82160 + 3.35648I 0
u = −0.05162 − 1.71260I
a = 0.001886 − 0.320518I
b = −0.001031 − 0.619536I
11.82160 − 3.35648I 0
u = −0.031423 + 0.280285I
a = −1.24282 − 2.10298I
b = 0.728609 + 0.020401I
1.222660 − 0.148523I 6.95502 − 0.36222I
u = −0.031423 − 0.280285I
a = −1.24282 + 2.10298I
b = 0.728609 − 0.020401I
1.222660 + 0.148523I 6.95502 + 0.36222I
u = −0.00837 + 1.78126I
a = −0.528064 − 0.356522I
b = 1.45222 − 0.21060I
16.9093 + 0.3307I 0
u = −0.00837 − 1.78126I
a = −0.528064 + 0.356522I
b = 1.45222 + 0.21060I
16.9093 − 0.3307I 0
u = 0.04878 + 1.78515I
a = 0.151260 + 0.916668I
b = −1.62783 + 0.66338I
−15.0990 − 4.3588I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.04878 − 1.78515I
a = 0.151260 − 0.916668I
b = −1.62783 − 0.66338I
−15.0990 + 4.3588I 0
u = −0.05643 + 1.79298I
a = 0.514785 − 0.362719I
b = −1.44825 − 0.23847I
16.8561 + 6.5503I 0
u = −0.05643 − 1.79298I
a = 0.514785 + 0.362719I
b = −1.44825 + 0.23847I
16.8561 − 6.5503I 0
u = −0.03067 + 1.79505I
a = −0.010921 − 0.875007I
b = 0.01525 − 1.51868I
19.1652 + 3.4491I 0
u = −0.03067 − 1.79505I
a = −0.010921 + 0.875007I
b = 0.01525 + 1.51868I
19.1652 − 3.4491I 0
u = −0.10251 + 1.82008I
a = −0.126790 + 0.897308I
b = 1.60864 + 0.67923I
−15.2648 + 11.2768I 0
u = −0.10251 − 1.82008I
a = −0.126790 − 0.897308I
b = 1.60864 − 0.67923I
−15.2648 − 11.2768I 0
u = −0.150697
a = −5.65109
b = 0.483736
1.16378 11.6280
8
II. I
u
2
= hb, −u
4
+ u
3
− 4u
2
+ a + 3u − 3, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
5
=
1
u
2
a
12
=
u
4
− u
3
+ 4u
2
− 3u + 3
0
a
10
=
u
4
− u
3
+ 4u
2
− 3u + 3
u
a
1
=
0
−u
a
8
=
u
u
3
+ u
a
6
=
u
2
+ 1
u
4
+ 2u
2
a
7
=
u
3
+ 2u
u
4
− u
3
+ 3u
2
− 2u + 1
a
11
=
u
4
− u
3
+ 4u
2
− 3u + 3
0
a
3
=
1
0
a
2
=
u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7u
4
+ 6u
3
− 28u
2
+ 17u − 12
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
2
u
5
− u
4
+ u
2
+ u − 1
c
3
, c
11
u
5
c
6
u
5
+ u
4
− u
2
+ u + 1
c
7
, c
8
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
9
, c
10
(u + 1)
5
c
12
(u − 1)
5
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
7
, c
8
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
c
2
, c
6
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
3
, c
11
y
5
c
9
, c
10
, c
12
(y −1)
5
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = 0.278580 − 1.055720I
b = 0
3.46474 − 2.21397I 6.65223 + 4.39723I
u = 0.233677 − 0.885557I
a = 0.278580 + 1.055720I
b = 0
3.46474 + 2.21397I 6.65223 − 4.39723I
u = 0.416284
a = 2.40221
b = 0
0.762751 −9.55270
u = 0.05818 + 1.69128I
a = 0.020316 − 0.590570I
b = 0
12.60320 − 3.33174I 9.12414 + 2.18947I
u = 0.05818 − 1.69128I
a = 0.020316 + 0.590570I
b = 0
12.60320 + 3.33174I 9.12414 − 2.18947I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
38
+ 6u
37
+ ··· + 16u + 1)
c
2
(u
5
− u
4
+ u
2
+ u − 1)(u
38
− 2u
37
+ ··· + 4u − 1)
c
3
, c
11
u
5
(u
38
− u
37
+ ··· − 32u + 32)
c
6
(u
5
+ u
4
− u
2
+ u + 1)(u
38
− 2u
37
+ ··· + 4u − 1)
c
7
, c
8
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
38
+ 6u
37
+ ··· + 16u + 1)
c
9
, c
10
((u + 1)
5
)(u
38
+ 6u
37
+ ··· − 2u − 1)
c
12
((u − 1)
5
)(u
38
+ 6u
37
+ ··· − 2u − 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
7
, c
8
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
38
+ 54y
37
+ ··· − 12y + 1)
c
2
, c
6
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
38
− 6y
37
+ ··· − 16y + 1)
c
3
, c
11
y
5
(y
38
− 33y
37
+ ··· − 3584y + 1024)
c
9
, c
10
, c
12
((y −1)
5
)(y
38
− 42y
37
+ ··· + 6y + 1)
14
