12a
0827
(K12a
0827
)
A knot diagram
1
Linearized knot diagam
4 5 8 2 10 11 3 1 12 7 6 9
Solving Sequence
6,12
11
3,7
8 10 5 2 4 9 1
c
11
c
6
c
7
c
10
c
5
c
2
c
4
c
9
c
12
c
1
, c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
53
+ u
52
+ ··· + b − 3u, −u
53
+ u
52
+ ··· + a + 1, u
56
− 2u
55
+ ··· − u − 1i
I
u
2
= h−u
2
+ b − 1, a − 1, u
3
+ 2u − 1i
I
u
3
= hu
3
+ u
2
+ b + u + 1, a + u + 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 63 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
53
+u
52
+· · · + b− 3u, −u
53
+u
52
+· · · + a+ 1, u
56
−2u
55
+· · · − u− 1i
(i) Arc colorings
a
6
=
0
u
a
12
=
1
0
a
11
=
1
u
2
a
3
=
u
53
− u
52
+ ··· + 5u − 1
−u
53
− u
52
+ ··· − 2u
2
+ 3u
a
7
=
u
u
3
+ u
a
8
=
−u
12
− 5u
10
− 7u
8
+ 2u
4
− 3u
2
+ 1
u
12
+ 6u
10
+ 12u
8
+ 8u
6
+ u
4
+ 2u
2
a
10
=
u
2
+ 1
u
4
+ 2u
2
a
5
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
2
=
−u
50
+ u
49
+ ··· + 3u − 2
−u
52
+ u
51
+ ··· − 3u
2
+ 2u
a
4
=
−u
53
+ u
52
+ ··· + u − 2
u
53
− u
52
+ ··· − 5u
2
+ 2u
a
9
=
−u
4
− u
2
+ 1
u
4
+ 2u
2
a
1
=
u
8
+ 3u
6
+ u
4
− 2u
2
+ 1
−u
8
− 4u
6
− 4u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
55
− 8u
54
+ ··· + 41u + 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
56
− 8u
55
+ ··· + 4u − 1
c
3
, c
7
u
56
− u
55
+ ··· + 64u + 128
c
5
u
56
+ 2u
55
+ ··· − 6840u − 1480
c
6
, c
10
, c
11
u
56
− 2u
55
+ ··· − u − 1
c
8
, c
9
, c
12
u
56
+ 6u
55
+ ··· + 15u + 19
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
56
− 60y
55
+ ··· + 20y + 1
c
3
, c
7
y
56
− 45y
55
+ ··· − 77824y + 16384
c
5
y
56
+ 30y
55
+ ··· + 5106160y + 2190400
c
6
, c
10
, c
11
y
56
+ 54y
55
+ ··· − 21y + 1
c
8
, c
9
, c
12
y
56
+ 66y
55
+ ··· − 9725y + 361
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.660832 + 0.532382I
a = −2.01116 − 0.10615I
b = −0.35051 − 2.14768I
−15.9022 − 5.0365I −5.83411 + 0.48302I
u = 0.660832 − 0.532382I
a = −2.01116 + 0.10615I
b = −0.35051 + 2.14768I
−15.9022 + 5.0365I −5.83411 − 0.48302I
u = 0.708316 + 0.463836I
a = 1.259210 + 0.179633I
b = 0.37444 + 2.69166I
−15.6591 + 9.6036I −5.25563 − 6.20278I
u = 0.708316 − 0.463836I
a = 1.259210 − 0.179633I
b = 0.37444 − 2.69166I
−15.6591 − 9.6036I −5.25563 + 6.20278I
u = −0.676559 + 0.489617I
a = −0.386353 + 0.785496I
b = 1.073820 − 0.154726I
−10.73890 − 2.24941I −4.62574 + 2.95001I
u = −0.676559 − 0.489617I
a = −0.386353 − 0.785496I
b = 1.073820 + 0.154726I
−10.73890 + 2.24941I −4.62574 − 2.95001I
u = 0.682129 + 0.475711I
a = −1.55512 − 0.39032I
b = −0.49146 − 2.66606I
−8.46348 + 5.18228I −3.67010 − 5.70178I
u = 0.682129 − 0.475711I
a = −1.55512 + 0.39032I
b = −0.49146 + 2.66606I
−8.46348 − 5.18228I −3.67010 + 5.70178I
u = 0.664199 + 0.499108I
a = 1.88326 + 0.30058I
b = 0.59126 + 2.40754I
−8.55005 − 0.70692I −4.00146 − 0.24980I
u = 0.664199 − 0.499108I
a = 1.88326 − 0.30058I
b = 0.59126 − 2.40754I
−8.55005 + 0.70692I −4.00146 + 0.24980I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.185528 + 1.194610I
a = 1.15155 + 1.06757I
b = −0.869359 − 0.937264I
−7.53088 + 3.11306I 0
u = 0.185528 − 1.194610I
a = 1.15155 − 1.06757I
b = −0.869359 + 0.937264I
−7.53088 − 3.11306I 0
u = −0.634915 + 0.454764I
a = 0.148473 − 0.390436I
b = −0.506435 + 0.067605I
−3.97803 − 2.08824I 3.15083 + 3.45018I
u = −0.634915 − 0.454764I
a = 0.148473 + 0.390436I
b = −0.506435 − 0.067605I
−3.97803 + 2.08824I 3.15083 − 3.45018I
u = 0.066056 + 1.253010I
a = −1.231350 − 0.008674I
b = 1.006180 + 0.304650I
−2.42783 + 1.72439I 0
u = 0.066056 − 1.253010I
a = −1.231350 + 0.008674I
b = 1.006180 − 0.304650I
−2.42783 − 1.72439I 0
u = −0.658902 + 0.263840I
a = −0.849914 + 0.073788I
b = 0.44012 + 1.55766I
−6.76817 − 5.79426I −2.57901 + 6.54647I
u = −0.658902 − 0.263840I
a = −0.849914 − 0.073788I
b = 0.44012 − 1.55766I
−6.76817 + 5.79426I −2.57901 − 6.54647I
u = −0.355604 + 0.604584I
a = 1.27737 + 1.18185I
b = 0.176477 − 1.192730I
−8.09448 + 2.23237I −6.24032 − 0.17719I
u = −0.355604 − 0.604584I
a = 1.27737 − 1.18185I
b = 0.176477 + 1.192730I
−8.09448 − 2.23237I −6.24032 + 0.17719I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.032180 + 1.315600I
a = 1.65049 − 1.49020I
b = −1.72250 + 0.53274I
−5.16815 − 0.97608I 0
u = −0.032180 − 1.315600I
a = 1.65049 + 1.49020I
b = −1.72250 − 0.53274I
−5.16815 + 0.97608I 0
u = 0.140776 + 1.329860I
a = 0.112471 − 0.618255I
b = −0.251370 + 0.625724I
−3.44679 + 2.43795I 0
u = 0.140776 − 1.329860I
a = 0.112471 + 0.618255I
b = −0.251370 − 0.625724I
−3.44679 − 2.43795I 0
u = 0.650158
a = 0.687504
b = −0.940476
−3.94742 0.365180
u = −0.550030 + 0.257011I
a = 1.005750 − 0.053759I
b = 0.08417 − 1.41533I
−0.60546 − 3.13990I 0.90903 + 8.93719I
u = −0.550030 − 0.257011I
a = 1.005750 + 0.053759I
b = 0.08417 + 1.41533I
−0.60546 + 3.13990I 0.90903 − 8.93719I
u = −0.194330 + 1.384330I
a = 1.52389 + 1.96311I
b = −0.42665 − 1.78739I
−5.81737 − 5.85679I 0
u = −0.194330 − 1.384330I
a = 1.52389 − 1.96311I
b = −0.42665 + 1.78739I
−5.81737 + 5.85679I 0
u = −0.244927 + 1.385240I
a = −2.01842 − 1.29418I
b = 0.99243 + 1.48272I
−11.9991 − 9.0718I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.244927 − 1.385240I
a = −2.01842 + 1.29418I
b = 0.99243 − 1.48272I
−11.9991 + 9.0718I 0
u = −0.135589 + 1.401420I
a = −0.55425 − 2.40672I
b = −0.40949 + 1.76759I
−6.82917 − 1.43395I 0
u = −0.135589 − 1.401420I
a = −0.55425 + 2.40672I
b = −0.40949 − 1.76759I
−6.82917 + 1.43395I 0
u = 0.169908 + 1.402300I
a = −0.588451 + 0.710634I
b = 0.950712 − 1.025380I
−8.25453 + 3.79866I 0
u = 0.169908 − 1.402300I
a = −0.588451 − 0.710634I
b = 0.950712 + 1.025380I
−8.25453 − 3.79866I 0
u = 0.477090 + 0.308383I
a = 1.096420 + 0.178210I
b = 0.155533 − 0.573459I
−2.81186 + 1.40524I −1.32654 − 4.47405I
u = 0.477090 − 0.308383I
a = 1.096420 − 0.178210I
b = 0.155533 + 0.573459I
−2.81186 − 1.40524I −1.32654 + 4.47405I
u = −0.09346 + 1.46417I
a = 0.20672 + 1.99781I
b = 0.533413 − 1.285270I
−14.6540 + 0.7423I 0
u = −0.09346 − 1.46417I
a = 0.20672 − 1.99781I
b = 0.533413 + 1.285270I
−14.6540 − 0.7423I 0
u = −0.22823 + 1.47473I
a = 0.447474 − 0.510871I
b = −0.437565 + 0.216954I
−10.20770 − 5.24287I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.22823 − 1.47473I
a = 0.447474 + 0.510871I
b = −0.437565 − 0.216954I
−10.20770 + 5.24287I 0
u = 0.482677 + 0.077215I
a = −0.481205 − 0.248126I
b = 0.246960 + 0.272043I
0.970388 + 0.223799I 9.84475 − 1.11489I
u = 0.482677 − 0.077215I
a = −0.481205 + 0.248126I
b = 0.246960 − 0.272043I
0.970388 − 0.223799I 9.84475 + 1.11489I
u = 0.24142 + 1.49209I
a = −0.96174 + 3.18550I
b = −1.14574 − 3.47762I
−14.8427 + 8.5522I 0
u = 0.24142 − 1.49209I
a = −0.96174 − 3.18550I
b = −1.14574 + 3.47762I
−14.8427 − 8.5522I 0
u = 0.25422 + 1.49240I
a = 1.26227 − 3.10390I
b = 0.62907 + 3.47790I
17.4812 + 13.1175I 0
u = 0.25422 − 1.49240I
a = 1.26227 + 3.10390I
b = 0.62907 − 3.47790I
17.4812 − 13.1175I 0
u = 0.22938 + 1.49684I
a = 0.54325 − 2.87749I
b = 1.58029 + 2.89554I
−15.0303 + 2.5470I 0
u = 0.22938 − 1.49684I
a = 0.54325 + 2.87749I
b = 1.58029 − 2.89554I
−15.0303 − 2.5470I 0
u = −0.23625 + 1.49620I
a = −0.956299 + 1.041640I
b = 0.911847 − 0.439648I
−17.1840 − 5.5773I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.23625 − 1.49620I
a = −0.956299 − 1.041640I
b = 0.911847 + 0.439648I
−17.1840 + 5.5773I 0
u = −0.328861 + 0.345682I
a = −1.49250 − 0.54755I
b = −0.337925 + 0.940882I
−1.35678 + 0.40897I −4.58141 + 0.13803I
u = −0.328861 − 0.345682I
a = −1.49250 + 0.54755I
b = −0.337925 − 0.940882I
−1.35678 − 0.40897I −4.58141 − 0.13803I
u = 0.21901 + 1.50844I
a = −0.60237 + 2.27231I
b = −1.30465 − 2.19638I
16.9245 − 1.8446I 0
u = 0.21901 − 1.50844I
a = −0.60237 − 2.27231I
b = −1.30465 + 2.19638I
16.9245 + 1.8446I 0
u = −0.273566
a = −2.44641
b = −1.04562
−1.24400 −12.1870
10
II. I
u
2
= h−u
2
+ b − 1, a − 1, u
3
+ 2u − 1i
(i) Arc colorings
a
6
=
0
u
a
12
=
1
0
a
11
=
1
u
2
a
3
=
1
u
2
+ 1
a
7
=
u
−u + 1
a
8
=
u
−u + 1
a
10
=
u
2
+ 1
u
a
5
=
−u
2
− u
u
2
a
2
=
u
2
+ u + 1
1
a
4
=
1
u
2
+ 1
a
9
=
u
2
− u + 1
u
a
1
=
u
2
+ u
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7u
2
+ 5u + 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
7
u
3
c
4
(u + 1)
3
c
5
u
3
+ 3u
2
+ 5u + 2
c
6
, c
8
, c
9
u
3
+ 2u + 1
c
10
, c
11
, c
12
u
3
+ 2u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
7
y
3
c
5
y
3
+ y
2
+ 13y −4
c
6
, c
8
, c
9
c
10
, c
11
, c
12
y
3
+ 4y
2
+ 4y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.22670 + 1.46771I
a = 1.00000
b = −1.102790 − 0.665457I
−11.08570 − 5.13794I −9.85299 + 2.68036I
u = −0.22670 − 1.46771I
a = 1.00000
b = −1.102790 + 0.665457I
−11.08570 + 5.13794I −9.85299 − 2.68036I
u = 0.453398
a = 1.00000
b = 1.20557
−0.857735 9.70600
14
III. I
u
3
= hu
3
+ u
2
+ b + u + 1, a + u + 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
6
=
0
u
a
12
=
1
0
a
11
=
1
u
2
a
3
=
−u − 1
−u
3
− u
2
− u − 1
a
7
=
u
u
3
+ u
a
8
=
u
u
3
+ u
a
10
=
u
2
+ 1
−u
3
− 2u − 1
a
5
=
−u
3
− 2u − 1
−u
3
− u
2
− u − 2
a
2
=
u
3
+ u
1
a
4
=
−u − 1
−u
3
− u
2
− u − 1
a
9
=
u
3
+ u
2
+ 2u + 2
−u
3
− 2u − 1
a
1
=
u
3
+ 2u + 1
u
3
+ u
2
+ u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
3
− 2u
2
+ 2u − 5
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
7
u
4
c
4
(u + 1)
4
c
5
(u
2
− u + 1)
2
c
6
, c
8
, c
9
u
4
− u
3
+ 2u
2
− 2u + 1
c
10
, c
11
, c
12
u
4
+ u
3
+ 2u
2
+ 2u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
7
y
4
c
5
(y
2
+ y + 1)
2
c
6
, c
8
, c
9
c
10
, c
11
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.621744 + 0.440597I
a = −0.378256 − 0.440597I
b = −0.692440 − 0.318148I
−4.93480 − 2.02988I −6.26314 + 3.25323I
u = −0.621744 − 0.440597I
a = −0.378256 + 0.440597I
b = −0.692440 + 0.318148I
−4.93480 + 2.02988I −6.26314 − 3.25323I
u = 0.121744 + 1.306620I
a = −1.12174 − 1.30662I
b = 1.192440 + 0.547877I
−4.93480 + 2.02988I −3.23686 − 4.54099I
u = 0.121744 − 1.306620I
a = −1.12174 + 1.30662I
b = 1.192440 − 0.547877I
−4.93480 − 2.02988I −3.23686 + 4.54099I
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
7
)(u
56
− 8u
55
+ ··· + 4u − 1)
c
3
, c
7
u
7
(u
56
− u
55
+ ··· + 64u + 128)
c
4
((u + 1)
7
)(u
56
− 8u
55
+ ··· + 4u − 1)
c
5
((u
2
− u + 1)
2
)(u
3
+ 3u
2
+ 5u + 2)(u
56
+ 2u
55
+ ··· − 6840u − 1480)
c
6
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
56
− 2u
55
+ ··· − u − 1)
c
8
, c
9
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
56
+ 6u
55
+ ··· + 15u + 19)
c
10
, c
11
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
56
− 2u
55
+ ··· − u − 1)
c
12
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
56
+ 6u
55
+ ··· + 15u + 19)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
7
)(y
56
− 60y
55
+ ··· + 20y + 1)
c
3
, c
7
y
7
(y
56
− 45y
55
+ ··· − 77824y + 16384)
c
5
(y
2
+ y + 1)
2
(y
3
+ y
2
+ 13y −4)
· (y
56
+ 30y
55
+ ··· + 5106160y + 2190400)
c
6
, c
10
, c
11
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
56
+ 54y
55
+ ··· − 21y + 1)
c
8
, c
9
, c
12
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
56
+ 66y
55
+ ··· − 9725y + 361)
20
