12a
0736
(K12a
0736
)
A knot diagram
1
Linearized knot diagam
3 8 9 11 12 10 2 7 1 6 5 4
Solving Sequence
2,7
8 3 9 4 1 10 6 11 12 5
c
7
c
2
c
8
c
3
c
1
c
9
c
6
c
10
c
12
c
5
c
4
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
69
− 11u
67
+ ··· + 2u
2
+ 1i
I
u
2
= hu + 1i
* 2 irreducible components of dim
C
= 0, with total 70 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
69
− 11u
67
+ · · · + 2u
2
+ 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
4
=
u
7
− 2u
5
+ 2u
3
− 2u
−u
7
+ u
5
− 2u
3
+ u
a
1
=
u
3
u
5
− u
3
+ u
a
10
=
−u
10
+ u
8
− 2u
6
+ u
4
− u
2
+ 1
−u
12
+ 2u
10
− 4u
8
+ 4u
6
− 3u
4
+ 2u
2
a
6
=
u
22
− 3u
20
+ ··· + 2u
2
+ 1
u
24
− 4u
22
+ ··· − 12u
6
+ 4u
4
a
11
=
−u
34
+ 5u
32
+ ··· + u
2
+ 1
−u
36
+ 6u
34
+ ··· − 3u
4
+ 2u
2
a
12
=
−u
19
+ 4u
17
− 10u
15
+ 18u
13
− 23u
11
+ 24u
9
− 18u
7
+ 10u
5
− 3u
3
u
19
− 3u
17
+ 8u
15
− 13u
13
+ 17u
11
− 17u
9
+ 12u
7
− 6u
5
+ u
3
+ u
a
5
=
−u
62
+ 11u
60
+ ··· + 2u
2
+ 1
u
62
− 10u
60
+ ··· + 8u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
67
− 40u
65
+ ··· − 12u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
69
+ 22u
68
+ ··· − 4u + 1
c
2
, c
7
u
69
− 11u
67
+ ··· + 2u
2
+ 1
c
3
u
69
− 2u
68
+ ··· − 466u + 61
c
4
, c
5
, c
11
u
69
− 2u
68
+ ··· + 2u
2
+ 1
c
6
, c
10
, c
12
u
69
+ 3u
68
+ ··· + 96u + 15
c
9
u
69
+ 8u
68
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
69
+ 50y
68
+ ··· + 12y −1
c
2
, c
7
y
69
− 22y
68
+ ··· − 4y −1
c
3
y
69
− 10y
68
+ ··· + 70756y −3721
c
4
, c
5
, c
11
y
69
− 54y
68
+ ··· − 4y −1
c
6
, c
10
, c
12
y
69
+ 69y
68
+ ··· − 2124y −225
c
9
y
69
+ 2y
68
+ ··· − 308y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.991972 + 0.143862I
1.34578 + 4.89493I −0.86940 − 7.00797I
u = −0.991972 − 0.143862I
1.34578 − 4.89493I −0.86940 + 7.00797I
u = 0.991475 + 0.091911I
−3.34547 − 2.48635I −7.92585 + 6.49145I
u = 0.991475 − 0.091911I
−3.34547 + 2.48635I −7.92585 − 6.49145I
u = −0.658926 + 0.796334I
−0.433628 − 0.203581I 0
u = −0.658926 − 0.796334I
−0.433628 + 0.203581I 0
u = 0.736871 + 0.614606I
2.92904 + 0.21710I 1.212247 − 0.558100I
u = 0.736871 − 0.614606I
2.92904 − 0.21710I 1.212247 + 0.558100I
u = 0.664003 + 0.805623I
−4.00701 + 4.76117I 0
u = 0.664003 − 0.805623I
−4.00701 − 4.76117I 0
u = −0.708316 + 0.768193I
2.43578 − 2.14828I 0. + 4.59076I
u = −0.708316 − 0.768193I
2.43578 + 2.14828I 0. − 4.59076I
u = 0.745411 + 0.741861I
3.11017 − 0.75564I 0
u = 0.745411 − 0.741861I
3.11017 + 0.75564I 0
u = −0.669215 + 0.811820I
0.28757 − 9.26473I 0
u = −0.669215 − 0.811820I
0.28757 + 9.26473I 0
u = −0.935368
−1.88030 −3.00500
u = 0.715362 + 0.794901I
7.54934 + 4.39001I 0
u = 0.715362 − 0.794901I
7.54934 − 4.39001I 0
u = 1.068480 + 0.101467I
−6.62522 + 0.10424I 0
u = 1.068480 − 0.101467I
−6.62522 − 0.10424I 0
u = −1.069720 + 0.112307I
−10.30900 + 4.49993I −9.11338 + 0.I
u = −1.069720 − 0.112307I
−10.30900 − 4.49993I −9.11338 + 0.I
u = 1.069200 + 0.121692I
−6.09738 − 9.05674I 0
u = 1.069200 − 0.121692I
−6.09738 + 9.05674I 0
u = −0.768907 + 0.776307I
8.45985 + 1.96096I 0
u = −0.768907 − 0.776307I
8.45985 − 1.96096I 0
u = −0.892627 + 0.630156I
−0.52815 + 2.44731I 0
u = −0.892627 − 0.630156I
−0.52815 − 2.44731I 0
u = −0.972878 + 0.520213I
−3.78853 − 2.83456I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.972878 − 0.520213I
−3.78853 + 2.83456I 0
u = 0.977801 + 0.533864I
−7.85518 − 1.72176I 0
u = 0.977801 − 0.533864I
−7.85518 + 1.72176I 0
u = −0.981067 + 0.547830I
−4.01496 + 6.30286I 0
u = −0.981067 − 0.547830I
−4.01496 − 6.30286I 0
u = 0.841536 + 0.762661I
3.17898 − 6.92216I 0
u = 0.841536 − 0.762661I
3.17898 + 6.92216I 0
u = −0.858320 + 0.746163I
−0.89607 + 2.81825I 0
u = −0.858320 − 0.746163I
−0.89607 − 2.81825I 0
u = 0.887210 + 0.744059I
3.03339 + 1.23094I 0
u = 0.887210 − 0.744059I
3.03339 − 1.23094I 0
u = 0.960144 + 0.658096I
2.22365 − 5.28707I 0
u = 0.960144 − 0.658096I
2.22365 + 5.28707I 0
u = 0.964185 + 0.699578I
2.44076 − 4.74437I 0
u = 0.964185 − 0.699578I
2.44076 + 4.74437I 0
u = −0.955842 + 0.728092I
7.88682 + 3.72650I 0
u = −0.955842 − 0.728092I
7.88682 − 3.72650I 0
u = −0.989530 + 0.707234I
1.58470 + 7.74715I 0
u = −0.989530 − 0.707234I
1.58470 − 7.74715I 0
u = 0.993720 + 0.722492I
6.70266 − 10.11200I 0
u = 0.993720 − 0.722492I
6.70266 + 10.11200I 0
u = −1.019650 + 0.704891I
−1.51932 + 5.86547I 0
u = −1.019650 − 0.704891I
−1.51932 − 5.86547I 0
u = 1.021010 + 0.710171I
−5.08551 − 10.46680I 0
u = 1.021010 − 0.710171I
−5.08551 + 10.46680I 0
u = −1.021170 + 0.714495I
−0.7784 + 15.0026I 0
u = −1.021170 − 0.714495I
−0.7784 − 15.0026I 0
u = 0.709030 + 0.232698I
2.70697 + 0.12234I 1.57891 + 1.07217I
u = 0.709030 − 0.232698I
2.70697 − 0.12234I 1.57891 − 1.07217I
u = −0.295637 + 0.611584I
−2.29380 − 2.06576I 0.948201 + 0.462919I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.295637 − 0.611584I
−2.29380 + 2.06576I 0.948201 − 0.462919I
u = 0.267368 + 0.620746I
−6.03688 − 2.42235I −2.36442 + 2.92314I
u = 0.267368 − 0.620746I
−6.03688 + 2.42235I −2.36442 − 2.92314I
u = −0.243750 + 0.627445I
−1.88595 + 6.88787I 1.87116 − 5.76329I
u = −0.243750 − 0.627445I
−1.88595 − 6.88787I 1.87116 + 5.76329I
u = 0.097783 + 0.542313I
4.72615 − 2.75194I 7.92617 + 4.59750I
u = 0.097783 − 0.542313I
4.72615 + 2.75194I 7.92617 − 4.59750I
u = −0.145390 + 0.409475I
0.081720 + 0.949994I 1.58080 − 7.18327I
u = −0.145390 − 0.409475I
0.081720 − 0.949994I 1.58080 + 7.18327I
7
II. I
u
2
= hu + 1i
(i) Arc colorings
a
2
=
0
−1
a
7
=
1
0
a
8
=
1
1
a
3
=
1
0
a
9
=
0
1
a
4
=
1
1
a
1
=
−1
−1
a
10
=
−1
0
a
6
=
1
0
a
11
=
−1
0
a
12
=
−1
−1
a
5
=
2
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
8
, c
11
u + 1
c
6
, c
10
, c
12
u
c
9
u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
8
, c
9
, c
11
y −1
c
6
, c
10
, c
12
y
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
−1.64493 −6.00000
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
8
(u + 1)(u
69
+ 22u
68
+ ··· − 4u + 1)
c
2
, c
7
(u + 1)(u
69
− 11u
67
+ ··· + 2u
2
+ 1)
c
3
(u + 1)(u
69
− 2u
68
+ ··· − 466u + 61)
c
4
, c
5
, c
11
(u + 1)(u
69
− 2u
68
+ ··· + 2u
2
+ 1)
c
6
, c
10
, c
12
u(u
69
+ 3u
68
+ ··· + 96u + 15)
c
9
(u − 1)(u
69
+ 8u
68
+ ··· − 2u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y −1)(y
69
+ 50y
68
+ ··· + 12y −1)
c
2
, c
7
(y −1)(y
69
− 22y
68
+ ··· − 4y −1)
c
3
(y −1)(y
69
− 10y
68
+ ··· + 70756y −3721)
c
4
, c
5
, c
11
(y −1)(y
69
− 54y
68
+ ··· − 4y −1)
c
6
, c
10
, c
12
y(y
69
+ 69y
68
+ ··· − 2124y −225)
c
9
(y −1)(y
69
+ 2y
68
+ ··· − 308y −1)
13
