12a
0815
(K12a
0815
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 10 3 11 12 1 6 8 9
Solving Sequence
8,11
12 9 1
3,7
4 2 6 10 5
c
11
c
8
c
12
c
7
c
3
c
1
c
6
c
10
c
5
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.39811 × 10
17
u
53
+ 3.48531 × 10
17
u
52
+ ··· + 3.74793 × 10
16
b + 1.64653 × 10
17
,
6.79652 × 10
16
u
53
− 2.14430 × 10
17
u
52
+ ··· + 3.74793 × 10
16
a − 3.66519 × 10
17
, u
54
− 4u
53
+ ··· − 12u + 1i
I
u
2
= hu
2
+ b − u − 2, −u
2
+ a + u + 2, u
3
− u
2
− 2u + 1i
I
u
3
= hb + u − 1, a + 3, u
2
+ u − 1i
I
u
4
= hb + 1, a − 2, u
2
+ u − 1i
* 4 irreducible components of dim
C
= 0, with total 61 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
= h−1.40 × 10
17
u
53
+ 3.49 × 10
17
u
52
+ · · · + 3.75 × 10
16
b + 1.65 ×
10
17
, 6.80 × 10
16
u
53
− 2.14 × 10
17
u
52
+ · · · + 3.75 × 10
16
a − 3.67 ×
10
17
, u
54
− 4u
53
+ · · · − 12u + 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
9
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
−1.81341u
53
+ 5.72128u
52
+ ··· − 57.8240u + 9.77923
3.73034u
53
− 9.29930u
52
+ ··· + 36.1231u − 4.39318
a
7
=
−u
u
a
4
=
−9.13774u
53
+ 22.8480u
52
+ ··· − 104.984u + 13.8690
11.0547u
53
− 26.4260u
52
+ ··· + 83.2829u − 8.48291
a
2
=
11.7271u
53
− 28.5773u
52
+ ··· + 105.648u − 12.1490
−11.8102u
53
+ 27.9993u
52
+ ··· − 86.3491u + 8.53503
a
6
=
−4.40884u
53
+ 10.2375u
52
+ ··· − 46.3267u + 6.19547
6.42322u
53
− 15.2092u
52
+ ··· + 47.7853u − 4.63561
a
10
=
−u
3
+ 2u
u
5
− 3u
3
+ u
a
5
=
6.54314u
53
− 14.0720u
52
+ ··· + 18.5879u + 0.684875
−6.46008u
53
+ 14.6500u
52
+ ··· − 37.8870u + 2.92907
(ii) Obstruction class = −1
(iii) Cusp Shapes =
363185510972410645
37479301199371286
u
53
−
688675840018976189
18739650599685643
u
52
+···+
4805743652784223102
18739650599685643
u−
1430274773919045159
37479301199371286
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
54
− 6u
53
+ ··· − 37u − 1
c
3
, c
6
u
54
+ 3u
53
+ ··· − 4u + 8
c
5
, c
10
u
54
+ 2u
53
+ ··· − 64u − 16
c
7
, c
8
, c
9
c
11
, c
12
u
54
− 4u
53
+ ··· − 12u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
54
− 50y
53
+ ··· − 985y + 1
c
3
, c
6
y
54
− 27y
53
+ ··· − 3088y + 64
c
5
, c
10
y
54
− 30y
53
+ ··· − 7808y + 256
c
7
, c
8
, c
9
c
11
, c
12
y
54
− 72y
53
+ ··· − 24y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.955893 + 0.280895I
a = −0.657144 + 0.964305I
b = −0.167798 + 0.004865I
0.99214 − 4.46696I 0
u = −0.955893 − 0.280895I
a = −0.657144 − 0.964305I
b = −0.167798 − 0.004865I
0.99214 + 4.46696I 0
u = −0.954075 + 0.196551I
a = −2.44568 + 0.32192I
b = 1.39296 − 0.94545I
1.98617 − 1.55786I 0
u = −0.954075 − 0.196551I
a = −2.44568 − 0.32192I
b = 1.39296 + 0.94545I
1.98617 + 1.55786I 0
u = −1.003860 + 0.333824I
a = 1.98014 − 0.51608I
b = −1.15335 + 1.40699I
4.09810 − 6.73935I 0
u = −1.003860 − 0.333824I
a = 1.98014 + 0.51608I
b = −1.15335 − 1.40699I
4.09810 + 6.73935I 0
u = 1.037550 + 0.306742I
a = −0.949401 − 0.673804I
b = 0.037724 + 1.330030I
−3.90243 + 4.30298I 0
u = 1.037550 − 0.306742I
a = −0.949401 + 0.673804I
b = 0.037724 − 1.330030I
−3.90243 − 4.30298I 0
u = −1.004380 + 0.437767I
a = −1.67226 + 0.47123I
b = 0.92436 − 1.53705I
−1.30619 − 11.23270I 0
u = −1.004380 − 0.437767I
a = −1.67226 − 0.47123I
b = 0.92436 + 1.53705I
−1.30619 + 11.23270I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.889050 + 0.150577I
a = 0.895885 + 0.803841I
b = −0.126696 − 1.360540I
1.36794 + 1.65747I 6.44317 − 4.64193I
u = 0.889050 − 0.150577I
a = 0.895885 − 0.803841I
b = −0.126696 + 1.360540I
1.36794 − 1.65747I 6.44317 + 4.64193I
u = −1.113440 + 0.153181I
a = 0.382959 − 0.729976I
b = 0.0606512 + 0.0742692I
6.28050 − 1.40683I 0
u = −1.113440 − 0.153181I
a = 0.382959 + 0.729976I
b = 0.0606512 − 0.0742692I
6.28050 + 1.40683I 0
u = 0.647324 + 0.583844I
a = 1.140030 + 0.621418I
b = 0.132989 − 1.059230I
−3.55059 − 3.17785I 2.00000 + 1.62776I
u = 0.647324 − 0.583844I
a = 1.140030 − 0.621418I
b = 0.132989 + 1.059230I
−3.55059 + 3.17785I 2.00000 − 1.62776I
u = 0.708312 + 0.161436I
a = −0.110891 + 1.210160I
b = −0.21103 − 1.94212I
−0.861342 + 0.425422I 8.90403 + 2.84052I
u = 0.708312 − 0.161436I
a = −0.110891 − 1.210160I
b = −0.21103 + 1.94212I
−0.861342 − 0.425422I 8.90403 − 2.84052I
u = −0.724781
a = 3.04269
b = −0.702104
−7.67173 19.8130
u = 0.180101 + 0.700187I
a = 0.088171 − 0.373846I
b = −0.49531 − 1.40016I
−4.94514 + 7.40203I −1.00662 − 6.28181I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.180101 − 0.700187I
a = 0.088171 + 0.373846I
b = −0.49531 + 1.40016I
−4.94514 − 7.40203I −1.00662 + 6.28181I
u = 0.484782 + 0.418903I
a = −1.25208 − 0.74775I
b = −0.023304 + 0.876278I
1.232030 − 0.400002I 7.11175 + 0.11005I
u = 0.484782 − 0.418903I
a = −1.25208 + 0.74775I
b = −0.023304 − 0.876278I
1.232030 + 0.400002I 7.11175 − 0.11005I
u = 0.212049 + 0.561353I
a = −0.355668 + 0.444206I
b = 0.58234 + 1.33649I
0.34502 + 3.69748I 2.76292 − 6.81307I
u = 0.212049 − 0.561353I
a = −0.355668 − 0.444206I
b = 0.58234 − 1.33649I
0.34502 − 3.69748I 2.76292 + 6.81307I
u = −0.281238 + 0.495314I
a = 0.769983 + 1.023740I
b = 0.106414 + 1.105760I
−7.98083 − 1.54477I −5.36049 + 2.77593I
u = −0.281238 − 0.495314I
a = 0.769983 − 1.023740I
b = 0.106414 − 1.105760I
−7.98083 + 1.54477I −5.36049 − 2.77593I
u = −1.45229 + 0.18721I
a = −0.812390 + 0.483425I
b = 0.156984 − 0.422947I
3.36449 + 0.31719I 0
u = −1.45229 − 0.18721I
a = −0.812390 − 0.483425I
b = 0.156984 + 0.422947I
3.36449 − 0.31719I 0
u = 0.139977 + 0.472620I
a = 1.76198 + 0.54098I
b = 0.201388 − 0.650535I
−2.37654 + 1.88570I −0.67343 − 3.61371I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.139977 − 0.472620I
a = 1.76198 − 0.54098I
b = 0.201388 + 0.650535I
−2.37654 − 1.88570I −0.67343 + 3.61371I
u = 0.487736
a = −0.807237
b = 0.441336
0.740706 13.6380
u = −1.60169
a = 2.38830
b = −2.02710
8.10579 0
u = −1.65983 + 0.02504I
a = 0.87862 + 1.94748I
b = −0.58285 − 2.35560I
7.60543 − 1.00097I 0
u = −1.65983 − 0.02504I
a = 0.87862 − 1.94748I
b = −0.58285 + 2.35560I
7.60543 + 1.00097I 0
u = 1.66740
a = −2.44614
b = 1.33681
0.921274 0
u = −1.69751 + 0.03635I
a = −0.96750 + 1.32403I
b = 0.48614 − 1.68807I
10.60180 − 2.36821I 0
u = −1.69751 − 0.03635I
a = −0.96750 − 1.32403I
b = 0.48614 + 1.68807I
10.60180 + 2.36821I 0
u = 1.70775 + 0.07197I
a = 0.258719 + 0.302731I
b = 0.129943 + 0.458522I
10.43440 + 5.86156I 0
u = 1.70775 − 0.07197I
a = 0.258719 − 0.302731I
b = 0.129943 − 0.458522I
10.43440 − 5.86156I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.70917 + 0.05180I
a = 2.59054 + 0.66443I
b = −1.81746 − 0.96175I
11.47430 + 2.55306I 0
u = 1.70917 − 0.05180I
a = 2.59054 − 0.66443I
b = −1.81746 + 0.96175I
11.47430 − 2.55306I 0
u = 0.043095 + 0.280243I
a = 0.89056 − 1.82691I
b = −0.611450 − 0.877287I
−1.250130 − 0.131994I −5.38771 − 0.73638I
u = 0.043095 − 0.280243I
a = 0.89056 + 1.82691I
b = −0.611450 + 0.877287I
−1.250130 + 0.131994I −5.38771 + 0.73638I
u = 1.71938 + 0.08814I
a = −2.28411 − 1.00360I
b = 1.57801 + 1.44618I
13.7389 + 8.4458I 0
u = 1.71938 − 0.08814I
a = −2.28411 + 1.00360I
b = 1.57801 − 1.44618I
13.7389 − 8.4458I 0
u = 1.71805 + 0.12084I
a = 1.98176 + 1.07195I
b = −1.28426 − 1.61703I
8.2449 + 13.5020I 0
u = 1.71805 − 0.12084I
a = 1.98176 − 1.07195I
b = −1.28426 + 1.61703I
8.2449 − 13.5020I 0
u = −1.72791 + 0.08472I
a = 0.849535 − 1.106390I
b = −0.25652 + 1.47457I
5.92114 − 5.92817I 0
u = −1.72791 − 0.08472I
a = 0.849535 + 1.106390I
b = −0.25652 − 1.47457I
5.92114 + 5.92817I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.74587 + 0.03327I
a = −0.107778 − 0.244366I
b = −0.087691 − 0.343352I
16.5619 + 2.1570I 0
u = 1.74587 − 0.03327I
a = −0.107778 + 0.244366I
b = −0.087691 + 0.343352I
16.5619 − 2.1570I 0
u = 1.82048
a = 0.109703
b = 0.193921
15.6372 0
u = 0.166814
a = 4.00471
b = −1.18722
−1.16691 −12.8120
10
II. I
u
2
= hu
2
+ b − u − 2, −u
2
+ a + u + 2, u
3
− u
2
− 2u + 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
9
=
u
−u
2
− u + 1
a
1
=
−u
2
+ 1
u
2
+ u − 1
a
3
=
u
2
− u − 2
−u
2
+ u + 2
a
7
=
−u
u
a
4
=
u
2
− u − 2
−u
2
+ u + 2
a
2
=
−u − 1
2u + 1
a
6
=
−u
u
a
10
=
−u
2
+ 1
u
2
a
5
=
u
2
− 1
−u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u
2
+ 7u + 30
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
6
u
3
c
4
(u + 1)
3
c
5
, c
7
, c
8
c
9
u
3
+ u
2
− 2u − 1
c
10
, c
11
, c
12
u
3
− u
2
− 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
6
y
3
c
5
, c
7
, c
8
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = 0.801938
b = −0.801938
4.69981 8.83150
u = 0.445042
a = −2.24698
b = 2.24698
−0.939962 31.5310
u = 1.80194
a = −0.554958
b = 0.554958
15.9794 16.6380
14
III. I
u
3
= hb + u − 1, a + 3, u
2
+ u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u − 1
a
9
=
u
−u + 1
a
1
=
u
−u
a
3
=
−3
−u + 1
a
7
=
−u
u
a
4
=
−2
−u
a
2
=
3u + 2
−u + 1
a
6
=
2u + 3
0
a
10
=
1
0
a
5
=
2u + 3
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −16
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
11
, c
12
u
2
+ u − 1
c
4
, c
6
, c
7
c
8
, c
9
u
2
− u − 1
c
5
, c
10
u
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
9
, c
11
c
12
y
2
− 3y + 1
c
5
, c
10
y
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −3.00000
b = 0.381966
−7.89568 −16.0000
u = −1.61803
a = −3.00000
b = 2.61803
7.89568 −16.0000
18
IV. I
u
4
= hb + 1, a − 2, u
2
+ u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u − 1
a
9
=
u
−u + 1
a
1
=
u
−u
a
3
=
2
−1
a
7
=
−u
u
a
4
=
u + 1
−u
a
2
=
u + 1
−1
a
6
=
u
0
a
10
=
1
0
a
5
=
u
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −1
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
11
, c
12
u
2
+ u − 1
c
4
, c
6
, c
7
c
8
, c
9
u
2
− u − 1
c
5
, c
10
u
2
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
9
, c
11
c
12
y
2
− 3y + 1
c
5
, c
10
y
2
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 2.00000
b = −1.00000
0 −1.00000
u = −1.61803
a = 2.00000
b = −1.00000
0 −1.00000
22
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
3
)(u
2
+ u − 1)
2
(u
54
− 6u
53
+ ··· − 37u − 1)
c
3
u
3
(u
2
+ u − 1)
2
(u
54
+ 3u
53
+ ··· − 4u + 8)
c
4
((u + 1)
3
)(u
2
− u − 1)
2
(u
54
− 6u
53
+ ··· − 37u − 1)
c
5
u
4
(u
3
+ u
2
− 2u − 1)(u
54
+ 2u
53
+ ··· − 64u − 16)
c
6
u
3
(u
2
− u − 1)
2
(u
54
+ 3u
53
+ ··· − 4u + 8)
c
7
, c
8
, c
9
((u
2
− u − 1)
2
)(u
3
+ u
2
− 2u − 1)(u
54
− 4u
53
+ ··· − 12u + 1)
c
10
u
4
(u
3
− u
2
− 2u + 1)(u
54
+ 2u
53
+ ··· − 64u − 16)
c
11
, c
12
((u
2
+ u − 1)
2
)(u
3
− u
2
− 2u + 1)(u
54
− 4u
53
+ ··· − 12u + 1)
23
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
3
)(y
2
− 3y + 1)
2
(y
54
− 50y
53
+ ··· − 985y + 1)
c
3
, c
6
y
3
(y
2
− 3y + 1)
2
(y
54
− 27y
53
+ ··· − 3088y + 64)
c
5
, c
10
y
4
(y
3
− 5y
2
+ 6y −1)(y
54
− 30y
53
+ ··· − 7808y + 256)
c
7
, c
8
, c
9
c
11
, c
12
((y
2
− 3y + 1)
2
)(y
3
− 5y
2
+ 6y −1)(y
54
− 72y
53
+ ··· − 24y + 1)
24
