12a
0792
(K12a
0792
)
A knot diagram
1
Linearized knot diagam
3 8 11 12 10 9 2 7 6 1 5 4
Solving Sequence
2,7
8 3 9 1 6 10 11 4 5 12
c
7
c
2
c
8
c
1
c
6
c
9
c
10
c
3
c
5
c
12
c
4
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
42
− u
41
+ ··· − u + 1i
* 1 irreducible components of dim
C
= 0, with total 42 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
42
− u
41
+ · · · − u + 1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
9
=
−u
2
+ 1
u
2
a
1
=
u
3
u
5
− u
3
+ u
a
6
=
u
4
− u
2
+ 1
−u
4
a
10
=
−u
6
+ u
4
− 2u
2
+ 1
u
6
+ u
2
a
11
=
u
14
− u
12
+ 4u
10
− 3u
8
+ 2u
6
− 2u
2
+ 1
u
16
− 2u
14
+ 6u
12
− 8u
10
+ 10u
8
− 6u
6
+ 4u
4
a
4
=
u
27
− 2u
25
+ ··· + 12u
5
− 5u
3
u
29
− 3u
27
+ ··· − u
3
+ u
a
5
=
u
8
− u
6
+ 3u
4
− 2u
2
+ 1
−u
8
− 2u
4
a
12
=
u
32
− 3u
30
+ ··· − 2u
2
+ 1
−u
32
+ 2u
30
+ ··· − 6u
6
+ 4u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
41
− 16u
39
+ ··· − 4u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
8
, c
9
u
42
+ 7u
41
+ ··· + 3u + 1
c
2
, c
7
u
42
+ u
41
+ ··· + u + 1
c
3
u
42
− u
41
+ ··· + 7u + 1
c
4
, c
11
, c
12
u
42
+ u
41
+ ··· + 3u + 1
c
10
u
42
+ 11u
41
+ ··· − 5u + 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
8
, c
9
y
42
+ 57y
41
+ ··· + 21y + 1
c
2
, c
7
y
42
− 7y
41
+ ··· − 3y + 1
c
3
y
42
− 7y
41
+ ··· − 3y + 1
c
4
, c
11
, c
12
y
42
+ 37y
41
+ ··· − 3y + 1
c
10
y
42
− 3y
41
+ ··· − 535y + 9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.681333 + 0.754455I
0.36250 − 4.45884I 0.58642 + 2.49782I
u = −0.681333 − 0.754455I
0.36250 + 4.45884I 0.58642 − 2.49782I
u = 0.815313 + 0.544997I
−3.06628 − 1.97224I −3.93875 + 4.40296I
u = 0.815313 − 0.544997I
−3.06628 + 1.97224I −3.93875 − 4.40296I
u = 0.718210 + 0.740792I
5.29261 + 0.88504I 5.49396 − 1.23926I
u = 0.718210 − 0.740792I
5.29261 − 0.88504I 5.49396 + 1.23926I
u = −0.809919 + 0.678495I
2.91299 + 2.57210I 0.86185 − 2.86212I
u = −0.809919 − 0.678495I
2.91299 − 2.57210I 0.86185 + 2.86212I
u = −0.785663 + 0.713246I
3.00195 + 2.59627I 1.81535 − 3.78463I
u = −0.785663 − 0.713246I
3.00195 − 2.59627I 1.81535 + 3.78463I
u = 0.862775 + 0.279474I
−5.58236 − 6.06191I −7.57750 + 8.26085I
u = 0.862775 − 0.279474I
−5.58236 + 6.06191I −7.57750 − 8.26085I
u = 0.874849 + 0.672104I
4.77268 − 6.11758I 3.66668 + 7.81265I
u = 0.874849 − 0.672104I
4.77268 + 6.11758I 3.66668 − 7.81265I
u = −0.900846 + 0.657474I
−0.36475 + 9.68044I −1.43034 − 8.54464I
u = −0.900846 − 0.657474I
−0.36475 − 9.68044I −1.43034 + 8.54464I
u = −0.844548 + 0.120880I
−6.39911 − 1.65910I −10.56317 − 0.31309I
u = −0.844548 − 0.120880I
−6.39911 + 1.65910I −10.56317 + 0.31309I
u = −0.797938 + 0.292374I
−0.41890 + 3.10279I −2.43445 − 9.36586I
u = −0.797938 − 0.292374I
−0.41890 − 3.10279I −2.43445 + 9.36586I
u = 0.718845 + 0.151909I
−1.185460 − 0.386187I −7.35672 + 0.55843I
u = 0.718845 − 0.151909I
−1.185460 + 0.386187I −7.35672 − 0.55843I
u = 0.486298 + 0.530301I
−2.30134 − 1.96887I 0.29013 + 3.56121I
u = 0.486298 − 0.530301I
−2.30134 + 1.96887I 0.29013 − 3.56121I
u = −0.937864 + 0.907045I
6.05226 + 3.34288I −2.00000 − 2.33342I
u = −0.937864 − 0.907045I
6.05226 − 3.34288I −2.00000 + 2.33342I
u = 0.925718 + 0.938959I
10.35550 + 4.97697I 0. − 2.32708I
u = 0.925718 − 0.938959I
10.35550 − 4.97697I 0. + 2.32708I
u = −0.932788 + 0.936544I
15.4788 − 1.0421I 5.04588 + 0.I
u = −0.932788 − 0.936544I
15.4788 + 1.0421I 5.04588 + 0.I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.941607 + 0.930102I
13.34440 − 3.10724I 2.23436 + 3.37048I
u = 0.941607 − 0.930102I
13.34440 + 3.10724I 2.23436 − 3.37048I
u = 0.953922 + 0.922825I
13.30320 − 3.70457I 2.13928 + 0.I
u = 0.953922 − 0.922825I
13.30320 + 3.70457I 2.13928 + 0.I
u = −0.963591 + 0.918930I
15.3766 + 7.8617I 4.81414 − 5.73365I
u = −0.963591 − 0.918930I
15.3766 − 7.8617I 4.81414 + 5.73365I
u = 0.968815 + 0.914465I
10.2127 − 11.7890I 0. + 6.81392I
u = 0.968815 − 0.914465I
10.2127 + 11.7890I 0. − 6.81392I
u = 0.155278 + 0.536125I
−3.38219 + 3.26579I 0.36679 − 2.81006I
u = 0.155278 − 0.536125I
−3.38219 − 3.26579I 0.36679 + 2.81006I
u = −0.267141 + 0.437516I
1.191110 − 0.437816I 6.95012 + 1.07134I
u = −0.267141 − 0.437516I
1.191110 + 0.437816I 6.95012 − 1.07134I
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
8
, c
9
u
42
+ 7u
41
+ ··· + 3u + 1
c
2
, c
7
u
42
+ u
41
+ ··· + u + 1
c
3
u
42
− u
41
+ ··· + 7u + 1
c
4
, c
11
, c
12
u
42
+ u
41
+ ··· + 3u + 1
c
10
u
42
+ 11u
41
+ ··· − 5u + 3
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
8
, c
9
y
42
+ 57y
41
+ ··· + 21y + 1
c
2
, c
7
y
42
− 7y
41
+ ··· − 3y + 1
c
3
y
42
− 7y
41
+ ··· − 3y + 1
c
4
, c
11
, c
12
y
42
+ 37y
41
+ ··· − 3y + 1
c
10
y
42
− 3y
41
+ ··· − 535y + 9
8
