12a
1233
(K12a
1233
)
A knot diagram
1
Linearized knot diagam
5 6 9 10 2 11 12 1 4 3 7 8
Solving Sequence
6,11
7 12
3,8
2 5 1 10 4 9
c
6
c
11
c
7
c
2
c
5
c
1
c
10
c
4
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h55915836184337u
43
+ 64179161931292u
42
+ ··· + 45302578214423b + 117930377236025,
− 212034836392771u
43
− 365740891628043u
42
+ ··· + 90605156428846a − 1349574182090028,
u
44
+ 2u
43
+ ··· + 10u + 1i
I
u
2
= hb + 1, a, u
2
+ u − 1i
I
u
3
= hb − 1, a
2
+ 2u − 4, u
2
− u − 1i
* 3 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
= h5.59 × 10
13
u
43
+ 6.42 × 10
13
u
42
+ · · · + 4.53 × 10
13
b + 1.18 ×
10
14
, −2.12 × 10
14
u
43
− 3.66 × 10
14
u
42
+ · · · + 9.06 × 10
13
a − 1.35 ×
10
15
, u
44
+ 2u
43
+ · · · + 10u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
12
=
−u
−u
3
+ u
a
3
=
2.34021u
43
+ 4.03665u
42
+ ··· + 34.7587u + 14.8951
−1.23427u
43
− 1.41668u
42
+ ··· − 12.4742u − 2.60317
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
1.10593u
43
+ 2.61997u
42
+ ··· + 22.2845u + 12.2919
−1.23427u
43
− 1.41668u
42
+ ··· − 12.4742u − 2.60317
a
5
=
1.85429u
43
+ 3.86193u
42
+ ··· + 41.9220u + 15.2508
−0.560033u
43
− 0.357346u
42
+ ··· + 0.835350u − 0.740724
a
1
=
u
3
− 2u
u
5
− 3u
3
+ u
a
10
=
2.83815u
43
+ 5.23204u
42
+ ··· + 53.7766u + 20.3839
−0.609358u
43
− 0.379534u
42
+ ··· − 8.15167u − 2.41432
a
4
=
−1.91530u
43
− 3.28109u
42
+ ··· − 28.9627u − 12.6975
0.723413u
43
+ 1.18474u
42
+ ··· + 12.5122u + 2.70679
a
9
=
u
4
− 3u
2
+ 1
u
6
− 4u
4
+ 3u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
18072105995455
45302578214423
u
43
−
25950739447177
45302578214423
u
42
+ ··· −
71920607974749
45302578214423
u +
76480599133623
45302578214423
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
44
+ 3u
43
+ ··· − 11u − 1
c
3
, c
4
, c
9
u
44
− u
43
+ ··· + 4u + 4
c
6
, c
7
, c
8
c
11
, c
12
u
44
− 2u
43
+ ··· − 10u + 1
c
10
u
44
+ 3u
43
+ ··· − 540u − 68
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
44
− 45y
43
+ ··· − 251y + 1
c
3
, c
4
, c
9
y
44
− 39y
43
+ ··· − 176y + 16
c
6
, c
7
, c
8
c
11
, c
12
y
44
− 60y
43
+ ··· − 66y + 1
c
10
y
44
+ 21y
43
+ ··· − 261680y + 4624
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.998544 + 0.111218I
a = 0.351759 − 0.614033I
b = −0.801508 + 0.547302I
−0.663058 − 0.817762I −8.32759 + 0.71545I
u = 0.998544 − 0.111218I
a = 0.351759 + 0.614033I
b = −0.801508 − 0.547302I
−0.663058 + 0.817762I −8.32759 − 0.71545I
u = −1.001850 + 0.127057I
a = 0.120314 + 0.945722I
b = 0.516618 − 0.643752I
−3.83529 + 2.20768I −11.83990 − 4.73522I
u = −1.001850 − 0.127057I
a = 0.120314 − 0.945722I
b = 0.516618 + 0.643752I
−3.83529 − 2.20768I −11.83990 + 4.73522I
u = −0.954649
a = −2.19836
b = −1.29146
0.461481 −10.4800
u = 1.009480 + 0.273056I
a = −0.335081 + 1.217640I
b = −0.390450 − 0.804941I
0.61880 − 5.62694I −5.80251 + 6.42762I
u = 1.009480 − 0.273056I
a = −0.335081 − 1.217640I
b = −0.390450 + 0.804941I
0.61880 + 5.62694I −5.80251 − 6.42762I
u = 1.065250 + 0.397438I
a = 0.04875 − 1.63500I
b = 1.49141 + 0.28927I
−5.49263 − 9.59429I −8.85592 + 6.59284I
u = 1.065250 − 0.397438I
a = 0.04875 + 1.63500I
b = 1.49141 − 0.28927I
−5.49263 + 9.59429I −8.85592 − 6.59284I
u = −1.131940 + 0.326750I
a = −0.162725 − 1.267720I
b = −1.50777 + 0.18978I
−10.46760 + 5.17349I −13.11043 − 4.56372I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.131940 − 0.326750I
a = −0.162725 + 1.267720I
b = −1.50777 − 0.18978I
−10.46760 − 5.17349I −13.11043 + 4.56372I
u = 1.193530 + 0.183455I
a = 0.388564 − 0.688390I
b = 1.47060 + 0.06865I
−7.96364 − 0.55193I −10.90446 + 0.I
u = 1.193530 − 0.183455I
a = 0.388564 + 0.688390I
b = 1.47060 − 0.06865I
−7.96364 + 0.55193I −10.90446 + 0.I
u = −0.534767 + 0.564063I
a = −1.168730 + 0.715602I
b = 1.42904 + 0.11240I
−2.23516 − 1.94324I −7.08189 − 0.05666I
u = −0.534767 − 0.564063I
a = −1.168730 − 0.715602I
b = 1.42904 − 0.11240I
−2.23516 + 1.94324I −7.08189 + 0.05666I
u = 0.767069
a = −0.450820
b = −0.373478
−1.47485 −4.57540
u = 0.374316 + 0.607175I
a = 1.31959 + 1.13185I
b = −1.43997 − 0.06548I
−5.73016 − 1.98434I −10.23333 + 3.59268I
u = 0.374316 − 0.607175I
a = 1.31959 − 1.13185I
b = −1.43997 + 0.06548I
−5.73016 + 1.98434I −10.23333 − 3.59268I
u = −0.257658 + 0.663265I
a = −1.53428 + 1.37125I
b = 1.44829 − 0.20795I
−1.39014 + 5.99441I −5.18773 − 5.53945I
u = −0.257658 − 0.663265I
a = −1.53428 − 1.37125I
b = 1.44829 + 0.20795I
−1.39014 − 5.99441I −5.18773 + 5.53945I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.475748 + 0.282175I
a = 1.74537 + 0.58249I
b = −0.390691 − 0.314994I
3.52033 − 0.22404I −1.41771 − 1.59086I
u = −0.475748 − 0.282175I
a = 1.74537 − 0.58249I
b = −0.390691 + 0.314994I
3.52033 + 0.22404I −1.41771 + 1.59086I
u = −0.209850 + 0.494994I
a = 0.58968 − 2.01314I
b = −0.337217 + 0.618314I
4.39585 + 3.01078I 0.72066 − 6.01437I
u = −0.209850 − 0.494994I
a = 0.58968 + 2.01314I
b = −0.337217 − 0.618314I
4.39585 − 3.01078I 0.72066 + 6.01437I
u = 1.50823
a = −0.100172
b = 1.32001
−8.29422 0
u = −0.387467
a = 0.781408
b = 1.09644
−2.22865 3.06750
u = 1.62350
a = 0.906030
b = 0.0651186
−3.95658 0
u = −1.64624
a = −0.216085
b = −0.708554
−9.99566 0
u = 0.189239 + 0.289124I
a = −0.78500 − 1.22940I
b = 0.260740 + 0.297980I
−0.172531 − 0.795475I −4.76256 + 8.61865I
u = 0.189239 − 0.289124I
a = −0.78500 + 1.22940I
b = 0.260740 − 0.297980I
−0.172531 + 0.795475I −4.76256 − 8.61865I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.72050
a = −1.48319
b = −1.43451
−9.17931 0
u = −1.72348 + 0.02231I
a = 0.015179 + 0.587021I
b = −0.834304 − 0.720855I
−10.41810 + 1.32296I 0
u = −1.72348 − 0.02231I
a = 0.015179 − 0.587021I
b = −0.834304 + 0.720855I
−10.41810 − 1.32296I 0
u = −1.72606 + 0.06803I
a = −0.285786 − 0.879713I
b = −0.421727 + 0.935386I
−9.13626 + 6.99467I 0
u = −1.72606 − 0.06803I
a = −0.285786 + 0.879713I
b = −0.421727 − 0.935386I
−9.13626 − 6.99467I 0
u = 1.72925 + 0.02883I
a = 0.161648 − 0.753573I
b = 0.595671 + 0.841277I
−13.66280 − 2.81563I 0
u = 1.72925 − 0.02883I
a = 0.161648 + 0.753573I
b = 0.595671 − 0.841277I
−13.66280 + 2.81563I 0
u = −1.74076 + 0.10823I
a = 0.423301 + 1.179640I
b = 1.52903 − 0.35516I
−15.4403 + 11.7022I 0
u = −1.74076 − 0.10823I
a = 0.423301 − 1.179640I
b = 1.52903 + 0.35516I
−15.4403 − 11.7022I 0
u = 1.75924 + 0.08406I
a = −0.481703 + 0.904253I
b = −1.57902 − 0.27351I
18.6454 − 6.9174I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.75924 − 0.08406I
a = −0.481703 − 0.904253I
b = −1.57902 + 0.27351I
18.6454 + 6.9174I 0
u = −1.76489 + 0.04721I
a = 0.643456 + 0.550041I
b = 1.58894 − 0.15433I
−18.6224 + 1.5527I 0
u = −1.76489 − 0.04721I
a = 0.643456 − 0.550041I
b = 1.58894 + 0.15433I
−18.6224 − 1.5527I 0
u = −0.134595
a = 9.65258
b = −0.928944
3.24469 2.21600
9
II. I
u
2
= hb + 1, a, u
2
+ u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u + 1
a
12
=
−u
−u + 1
a
3
=
0
−1
a
8
=
u
u
a
2
=
−1
−1
a
5
=
0
−1
a
1
=
−1
0
a
10
=
0
u
a
4
=
0
−1
a
9
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −18
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
9
c
10
u
2
c
5
(u + 1)
2
c
6
, c
7
, c
8
u
2
+ u − 1
c
11
, c
12
u
2
− u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
2
c
3
, c
4
, c
9
c
10
y
2
c
6
, c
7
, c
8
c
11
, c
12
y
2
− 3y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0
b = −1.00000
−2.63189 −18.0000
u = −1.61803
a = 0
b = −1.00000
−10.5276 −18.0000
13
III. I
u
3
= hb − 1, a
2
+ 2u − 4, u
2
− u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u + 1
a
12
=
−u
−u − 1
a
3
=
a
1
a
8
=
−u
−u
a
2
=
a + 1
1
a
5
=
−a
−1
a
1
=
1
0
a
10
=
−2u + 2
−au + u
a
4
=
a
−au − a + 1
a
9
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u + 1)
4
c
3
, c
4
, c
9
c
10
(u
2
− 2)
2
c
5
(u − 1)
4
c
6
, c
7
, c
8
(u
2
− u − 1)
2
c
11
, c
12
(u
2
+ u − 1)
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
9
c
10
(y −2)
4
c
6
, c
7
, c
8
c
11
, c
12
(y
2
− 3y + 1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 2.28825
b = 1.00000
2.30291 −8.00000
u = −0.618034
a = −2.28825
b = 1.00000
2.30291 −8.00000
u = 1.61803
a = −0.874032
b = 1.00000
−5.59278 −8.00000
u = 1.61803
a = 0.874032
b = 1.00000
−5.59278 −8.00000
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
2
)(u + 1)
4
(u
44
+ 3u
43
+ ··· − 11u − 1)
c
3
, c
4
, c
9
u
2
(u
2
− 2)
2
(u
44
− u
43
+ ··· + 4u + 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
44
+ 3u
43
+ ··· − 11u − 1)
c
6
, c
7
, c
8
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
44
− 2u
43
+ ··· − 10u + 1)
c
10
u
2
(u
2
− 2)
2
(u
44
+ 3u
43
+ ··· − 540u − 68)
c
11
, c
12
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
44
− 2u
43
+ ··· − 10u + 1)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
((y −1)
6
)(y
44
− 45y
43
+ ··· − 251y + 1)
c
3
, c
4
, c
9
y
2
(y −2)
4
(y
44
− 39y
43
+ ··· − 176y + 16)
c
6
, c
7
, c
8
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
44
− 60y
43
+ ··· − 66y + 1)
c
10
y
2
(y −2)
4
(y
44
+ 21y
43
+ ··· − 261680y + 4624)
19
