12a
0826
(K12a
0826
)
A knot diagram
1
Linearized knot diagam
4 5 8 2 10 11 3 1 12 6 7 9
Solving Sequence
6,10
11 7 12
2,5
3 8 4 1 9
c
10
c
6
c
11
c
5
c
2
c
7
c
4
c
1
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
46
− 24u
44
+ ··· + b − 2u, u
49
− u
48
+ ··· + a − 3, u
50
− 2u
49
+ ··· − 3u + 1i
I
u
2
= hu
4
− 2u
2
+ b, −u
5
+ 3u
3
+ a − 2u − 1, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 56 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
= hu
46
−24u
44
+· · ·+b−2u, u
49
−u
48
+· · ·+a−3, u
50
−2u
49
+· · ·−3u+1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
−u
49
+ u
48
+ ··· + u + 3
−u
46
+ 24u
44
+ ··· − u
2
+ 2u
a
5
=
−u
u
a
3
=
u
48
− 25u
46
+ ··· + 3u + 2
−u
49
+ 26u
47
+ ··· − 8u
2
+ 1
a
8
=
u
14
− 7u
12
+ 18u
10
− 21u
8
+ 14u
6
− 10u
4
+ 4u
2
− 1
−u
16
+ 8u
14
− 24u
12
+ 32u
10
− 18u
8
+ 8u
6
− 8u
4
a
4
=
−2u
49
+ u
48
+ ··· − u + 3
u
49
− 26u
47
+ ··· + 5u − 1
a
1
=
−u
10
+ 5u
8
− 8u
6
+ 5u
4
− 3u
2
+ 1
u
12
− 6u
10
+ 12u
8
− 8u
6
+ u
4
− 2u
2
a
9
=
−u
6
+ 3u
4
− 2u
2
+ 1
u
8
− 4u
6
+ 4u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
49
− 7u
48
+ ··· − 34u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
50
− 7u
49
+ ··· + 2u − 1
c
3
, c
7
u
50
− u
49
+ ··· + 224u
2
− 64
c
5
, c
6
, c
10
c
11
u
50
+ 2u
49
+ ··· + 3u + 1
c
8
, c
9
, c
12
u
50
+ 6u
49
+ ··· + 45u − 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
50
− 53y
49
+ ··· + 24y + 1
c
3
, c
7
y
50
− 39y
49
+ ··· − 28672y + 4096
c
5
, c
6
, c
10
c
11
y
50
− 54y
49
+ ··· − 23y + 1
c
8
, c
9
, c
12
y
50
+ 54y
49
+ ··· − 2871y + 81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.553117 + 0.664163I
a = 1.18281 − 1.02165I
b = −0.27217 + 2.65461I
−15.1392 − 9.4868I −4.47770 + 6.36607I
u = −0.553117 − 0.664163I
a = 1.18281 + 1.02165I
b = −0.27217 − 2.65461I
−15.1392 + 9.4868I −4.47770 − 6.36607I
u = −0.521122 + 0.653186I
a = −0.360363 − 0.132250I
b = 0.744618 − 1.077000I
−7.98509 − 5.12361I −2.94810 + 5.86362I
u = −0.521122 − 0.653186I
a = −0.360363 + 0.132250I
b = 0.744618 + 1.077000I
−7.98509 + 5.12361I −2.94810 − 5.86362I
u = −0.832663
a = 1.37585
b = 0.379950
−4.23580 0.843030
u = 0.505395 + 0.660908I
a = 1.86551 + 1.12155I
b = −0.76904 − 2.58054I
−10.25710 + 2.22642I −3.92809 − 3.01901I
u = 0.505395 − 0.660908I
a = 1.86551 − 1.12155I
b = −0.76904 + 2.58054I
−10.25710 − 2.22642I −3.92809 + 3.01901I
u = −0.460008 + 0.686028I
a = 2.07694 − 0.50744I
b = −0.71544 + 1.92454I
−15.4170 + 4.9513I −5.18278 − 0.57560I
u = −0.460008 − 0.686028I
a = 2.07694 + 0.50744I
b = −0.71544 − 1.92454I
−15.4170 − 4.9513I −5.18278 + 0.57560I
u = −0.487371 + 0.659703I
a = −0.906971 + 0.645379I
b = −0.393135 − 0.627704I
−8.08531 + 0.69745I −3.35798 + 0.23540I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.487371 − 0.659703I
a = −0.906971 − 0.645379I
b = −0.393135 + 0.627704I
−8.08531 − 0.69745I −3.35798 − 0.23540I
u = 0.660442 + 0.437938I
a = −1.175670 − 0.632150I
b = −0.27825 + 1.82886I
−6.65255 + 5.49617I −1.92374 − 6.88073I
u = 0.660442 − 0.437938I
a = −1.175670 + 0.632150I
b = −0.27825 − 1.82886I
−6.65255 − 5.49617I −1.92374 + 6.88073I
u = 0.494871 + 0.598464I
a = −0.642916 − 0.366018I
b = 0.316810 + 0.829305I
−3.61220 + 2.04087I 3.66627 − 3.62580I
u = 0.494871 − 0.598464I
a = −0.642916 + 0.366018I
b = 0.316810 − 0.829305I
−3.61220 − 2.04087I 3.66627 + 3.62580I
u = −1.34345
a = 2.34922
b = −1.18992
−3.73496 0
u = 0.539188 + 0.357182I
a = −0.060078 − 0.229300I
b = 0.065780 − 0.818765I
−0.53394 + 3.00224I 1.62141 − 9.45825I
u = 0.539188 − 0.357182I
a = −0.060078 + 0.229300I
b = 0.065780 + 0.818765I
−0.53394 − 3.00224I 1.62141 + 9.45825I
u = 0.197556 + 0.567801I
a = −1.33311 + 0.94398I
b = 0.34749 + 1.38106I
−8.06992 − 2.05324I −6.14708 + 0.44806I
u = 0.197556 − 0.567801I
a = −1.33311 − 0.94398I
b = 0.34749 − 1.38106I
−8.06992 + 2.05324I −6.14708 − 0.44806I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.432575 + 0.376085I
a = −1.94487 + 0.34224I
b = 0.40285 − 1.84963I
−2.72128 − 1.34903I −0.70979 + 4.63030I
u = −0.432575 − 0.376085I
a = −1.94487 − 0.34224I
b = 0.40285 + 1.84963I
−2.72128 + 1.34903I −0.70979 − 4.63030I
u = −0.523312 + 0.099839I
a = 0.392896 + 0.049847I
b = −0.497458 + 0.355216I
0.918277 − 0.180115I 10.59554 + 1.05153I
u = −0.523312 − 0.099839I
a = 0.392896 − 0.049847I
b = −0.497458 − 0.355216I
0.918277 + 0.180115I 10.59554 − 1.05153I
u = −1.48986 + 0.04819I
a = −0.213305 − 0.758850I
b = −0.383824 + 1.018250I
4.60096 − 0.59150I 0
u = −1.48986 − 0.04819I
a = −0.213305 + 0.758850I
b = −0.383824 − 1.018250I
4.60096 + 0.59150I 0
u = 1.47972 + 0.21886I
a = −2.61247 − 0.77557I
b = 1.52280 + 0.76083I
−9.13237 − 1.69237I 0
u = 1.47972 − 0.21886I
a = −2.61247 + 0.77557I
b = 1.52280 − 0.76083I
−9.13237 + 1.69237I 0
u = 1.50421 + 0.08278I
a = 2.26526 + 2.33731I
b = −1.43825 − 2.50580I
3.72115 + 2.85609I 0
u = 1.50421 − 0.08278I
a = 2.26526 − 2.33731I
b = −1.43825 + 2.50580I
3.72115 − 2.85609I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.50288 + 0.20473I
a = 0.247045 + 0.583522I
b = 0.259330 − 0.087519I
−1.58696 + 2.41508I 0
u = 1.50288 − 0.20473I
a = 0.247045 − 0.583522I
b = 0.259330 + 0.087519I
−1.58696 − 2.41508I 0
u = −1.51678 + 0.17371I
a = 1.27783 − 0.63401I
b = −0.991957 + 0.734002I
3.01331 − 4.79464I 0
u = −1.51678 − 0.17371I
a = 1.27783 + 0.63401I
b = −0.991957 − 0.734002I
3.01331 + 4.79464I 0
u = −1.51343 + 0.20796I
a = −3.33763 + 1.93254I
b = 2.35728 − 2.09370I
−3.64700 − 5.36770I 0
u = −1.51343 − 0.20796I
a = −3.33763 − 1.93254I
b = 2.35728 + 2.09370I
−3.64700 + 5.36770I 0
u = 1.53588 + 0.02926I
a = −1.104940 − 0.849103I
b = 1.04875 + 1.10169I
7.88396 + 0.66037I 0
u = 1.53588 − 0.02926I
a = −1.104940 + 0.849103I
b = 1.04875 − 1.10169I
7.88396 − 0.66037I 0
u = −1.53405 + 0.08867I
a = −0.256516 + 1.388440I
b = 0.10053 − 1.82714I
6.40849 − 4.54154I 0
u = −1.53405 − 0.08867I
a = −0.256516 − 1.388440I
b = 0.10053 + 1.82714I
6.40849 + 4.54154I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.52299 + 0.20532I
a = 1.60846 + 0.80229I
b = −1.32719 − 1.42793I
−1.27126 + 8.23471I 0
u = 1.52299 − 0.20532I
a = 1.60846 − 0.80229I
b = −1.32719 + 1.42793I
−1.27126 − 8.23471I 0
u = 0.270980 + 0.357042I
a = 1.336840 − 0.453135I
b = 0.0735636 − 0.0211419I
−1.313390 − 0.394009I −4.59693 − 0.11426I
u = 0.270980 − 0.357042I
a = 1.336840 + 0.453135I
b = 0.0735636 + 0.0211419I
−1.313390 + 0.394009I −4.59693 + 0.11426I
u = 1.53896 + 0.21280I
a = −2.44076 − 2.71196I
b = 1.46316 + 3.05025I
−8.2472 + 12.6896I 0
u = 1.53896 − 0.21280I
a = −2.44076 + 2.71196I
b = 1.46316 − 3.05025I
−8.2472 − 12.6896I 0
u = −1.57081 + 0.11496I
a = 0.54586 − 2.32961I
b = 0.35877 + 2.51364I
0.85171 − 7.47143I 0
u = −1.57081 − 0.11496I
a = 0.54586 + 2.32961I
b = 0.35877 − 2.51364I
0.85171 + 7.47143I 0
u = 1.59062
a = 0.786115
b = −1.65951
3.85485 0
u = 0.284196
a = 2.66910
b = 0.479450
−1.25005 −12.8860
9
II.
I
u
2
= hu
4
− 2u
2
+ b, −u
5
+ 3u
3
+ a − 2u − 1, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
u
5
− 3u
3
+ 2u + 1
−u
4
+ 2u
2
a
5
=
−u
u
a
3
=
u
5
− 3u
3
+ u + 1
−u
4
+ 2u
2
+ u
a
8
=
u
−u
3
+ u
a
4
=
u
5
− 3u
3
+ u + 1
−u
4
+ 2u
2
+ u
a
1
=
u
−u
a
9
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
5
− u
4
− 14u
3
+ u
2
+ 14u + 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
7
u
6
c
4
(u + 1)
6
c
5
, c
6
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
8
, c
9
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
10
, c
11
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
12
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
7
y
6
c
5
, c
6
, c
10
c
11
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
8
, c
9
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.493180 + 0.575288I
a = −0.997760 + 0.232521I
b = 0.138835 − 1.234450I
−4.60518 − 1.97241I −5.56070 + 3.48596I
u = −0.493180 − 0.575288I
a = −0.997760 − 0.232521I
b = 0.138835 + 1.234450I
−4.60518 + 1.97241I −5.56070 − 3.48596I
u = 0.483672
a = 1.65437
b = 0.413150
−0.906083 11.4460
u = 1.52087 + 0.16310I
a = 1.05885 + 1.20667I
b = −0.408802 − 1.276380I
2.05064 + 4.59213I −1.33400 − 2.48468I
u = 1.52087 − 0.16310I
a = 1.05885 − 1.20667I
b = −0.408802 + 1.276380I
2.05064 − 4.59213I −1.33400 + 2.48468I
u = −1.53904
a = 0.223460
b = −0.873214
6.01515 6.34350
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
6
)(u
50
− 7u
49
+ ··· + 2u − 1)
c
3
, c
7
u
6
(u
50
− u
49
+ ··· + 224u
2
− 64)
c
4
((u + 1)
6
)(u
50
− 7u
49
+ ··· + 2u − 1)
c
5
, c
6
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
50
+ 2u
49
+ ··· + 3u + 1)
c
8
, c
9
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
50
+ 6u
49
+ ··· + 45u − 9)
c
10
, c
11
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
50
+ 2u
49
+ ··· + 3u + 1)
c
12
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
50
+ 6u
49
+ ··· + 45u − 9)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
6
)(y
50
− 53y
49
+ ··· + 24y + 1)
c
3
, c
7
y
6
(y
50
− 39y
49
+ ··· − 28672y + 4096)
c
5
, c
6
, c
10
c
11
(y
6
− 7y
5
+ ··· − 5y + 1)(y
50
− 54y
49
+ ··· − 23y + 1)
c
8
, c
9
, c
12
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
50
+ 54y
49
+ ··· − 2871y + 81)
15
