12a
1218
(K12a
1218
)
A knot diagram
1
Linearized knot diagam
5 6 7 10 11 3 4 12 1 2 8 9
Solving Sequence
3,6
7 4
8,11
12 2 5 1 10 9
c
6
c
3
c
7
c
11
c
2
c
5
c
1
c
10
c
9
c
4
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−2.88764 × 10
38
u
57
+ 1.08762 × 10
39
u
56
+ ··· + 1.25794 × 10
38
b + 4.48859 × 10
38
,
− 2.38160 × 10
37
u
57
− 3.45850 × 10
38
u
56
+ ··· + 1.25794 × 10
38
a − 1.21766 × 10
39
,
u
58
− 4u
57
+ ··· + 10u + 1i
I
u
2
= hb − u, a − 1, u
2
+ u − 1i
I
u
3
= hb + a, a
2
+ a − 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 62 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−2.89 × 10
38
u
57
+ 1.09 × 10
39
u
56
+ · · · + 1.26 × 10
38
b + 4.49 ×
10
38
, −2.38 × 10
37
u
57
− 3.46 × 10
38
u
56
+ · · · + 1.26 × 10
38
a − 1.22 ×
10
39
, u
58
− 4u
57
+ · · · + 10u + 1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
0
a
7
=
1
−u
2
a
4
=
u
−u
3
+ u
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
11
=
0.189325u
57
+ 2.74933u
56
+ ··· + 51.6861u + 9.67976
2.29553u
57
− 8.64604u
56
+ ··· − 52.0009u − 3.56820
a
12
=
−3.31034u
57
+ 11.1720u
56
+ ··· + 74.9248u + 11.0418
5.11921u
57
− 15.5570u
56
+ ··· − 75.3805u − 5.77324
a
2
=
−u
u
a
5
=
1.35642u
57
− 2.73919u
56
+ ··· − 31.4858u − 5.53784
−3.83315u
57
+ 10.5013u
56
+ ··· + 40.7130u + 3.05633
a
1
=
−1.73986u
57
+ 2.87590u
56
+ ··· − 2.41112u − 0.766687
1.73986u
57
− 2.87590u
56
+ ··· + 2.41112u − 0.233313
a
10
=
−5.92742u
57
+ 17.2187u
56
+ ··· + 94.5982u + 13.7225
8.41228u
57
− 23.1154u
56
+ ··· − 94.9130u − 7.61093
a
9
=
0.212480u
57
+ 0.0136304u
56
+ ··· + 18.3659u + 6.33053
1.59639u
57
− 4.39858u
56
+ ··· − 18.8215u − 1.06192
(ii) Obstruction class = −1
(iii) Cusp Shapes = 21.4511u
57
− 79.0212u
56
+ ··· − 524.981u − 52.4582
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
58
+ 3u
57
+ ··· + 4u + 4
c
2
, c
3
, c
6
c
7
u
58
− 4u
57
+ ··· + 10u + 1
c
4
u
58
− u
57
+ ··· + 519u + 83
c
5
u
58
+ u
57
+ ··· − 519u + 83
c
8
, c
9
, c
11
c
12
u
58
+ 4u
57
+ ··· − 10u + 1
c
10
u
58
− 3u
57
+ ··· − 4u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
58
− 17y
57
+ ··· − 120y + 16
c
2
, c
3
, c
6
c
7
, c
8
, c
9
c
11
, c
12
y
58
− 68y
57
+ ··· − 90y + 1
c
4
, c
5
y
58
+ 51y
57
+ ··· − 36463y + 6889
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.803979 + 0.621852I
a = −0.271991 + 1.151380I
b = −1.08545 − 1.13142I
−8.28668 − 10.95660I 0
u = −0.803979 − 0.621852I
a = −0.271991 − 1.151380I
b = −1.08545 + 1.13142I
−8.28668 + 10.95660I 0
u = −0.793217 + 0.521649I
a = 0.498912 − 1.122630I
b = 0.946234 + 1.016640I
− 8.39682I 0
u = −0.793217 − 0.521649I
a = 0.498912 + 1.122630I
b = 0.946234 − 1.016640I
8.39682I 0
u = 0.577533 + 0.730806I
a = −0.634988 − 0.459765I
b = −0.098643 + 1.074010I
−5.27688 + 2.48624I 0
u = 0.577533 − 0.730806I
a = −0.634988 + 0.459765I
b = −0.098643 − 1.074010I
−5.27688 − 2.48624I 0
u = 1.008620 + 0.380806I
a = −0.594450 − 0.045398I
b = 0.149754 + 0.464822I
1.242260 − 0.522127I 0
u = 1.008620 − 0.380806I
a = −0.594450 + 0.045398I
b = 0.149754 − 0.464822I
1.242260 + 0.522127I 0
u = −0.749389 + 0.385619I
a = −0.849917 + 1.075030I
b = −0.781148 − 0.827305I
2.11157 − 4.30929I 3.71388 + 6.70404I
u = −0.749389 − 0.385619I
a = −0.849917 − 1.075030I
b = −0.781148 + 0.827305I
2.11157 + 4.30929I 3.71388 − 6.70404I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.154793 + 0.824875I
a = −0.217281 − 0.442528I
b = 0.869009 − 0.759634I
−10.25510 + 6.17839I −4.76560 − 4.41616I
u = −0.154793 − 0.824875I
a = −0.217281 + 0.442528I
b = 0.869009 + 0.759634I
−10.25510 − 6.17839I −4.76560 + 4.41616I
u = 0.668037 + 0.496903I
a = 0.637455 + 0.425308I
b = 0.087808 − 0.841653I
1.48873 + 1.56696I 6.38155 − 7.61387I
u = 0.668037 − 0.496903I
a = 0.637455 − 0.425308I
b = 0.087808 + 0.841653I
1.48873 − 1.56696I 6.38155 + 7.61387I
u = 0.805056
a = 0.0586228
b = −0.469605
1.37980 7.84380
u = −0.614408 + 0.433410I
a = −0.65974 − 1.92027I
b = 0.667344 + 0.163534I
−9.33906 − 4.00146I −4.44972 + 5.66506I
u = −0.614408 − 0.433410I
a = −0.65974 + 1.92027I
b = 0.667344 − 0.163534I
−9.33906 + 4.00146I −4.44972 − 5.66506I
u = 1.176150 + 0.494013I
a = 0.885445 − 0.099461I
b = −0.449248 − 0.435686I
−6.21754 − 1.60405I 0
u = 1.176150 − 0.494013I
a = 0.885445 + 0.099461I
b = −0.449248 + 0.435686I
−6.21754 + 1.60405I 0
u = −0.091748 + 0.703159I
a = −0.125386 + 0.534649I
b = −0.680727 + 0.714968I
−2.11157 + 4.30929I −3.71388 − 6.70404I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.091748 − 0.703159I
a = −0.125386 − 0.534649I
b = −0.680727 − 0.714968I
−2.11157 − 4.30929I −3.71388 + 6.70404I
u = 0.635581 + 0.189573I
a = −0.89658 + 3.23104I
b = 0.93416 − 2.56648I
−7.67427 + 0.41875I −15.8309 + 7.9264I
u = 0.635581 − 0.189573I
a = −0.89658 − 3.23104I
b = 0.93416 + 2.56648I
−7.67427 − 0.41875I −15.8309 − 7.9264I
u = −0.550445 + 0.243751I
a = 1.52059 − 0.52439I
b = 0.749569 + 0.393249I
−1.242260 − 0.522127I −4.85746 + 7.66329I
u = −0.550445 − 0.243751I
a = 1.52059 + 0.52439I
b = 0.749569 − 0.393249I
−1.242260 + 0.522127I −4.85746 − 7.66329I
u = 0.584934 + 0.061813I
a = 0.15128 − 2.97701I
b = 0.05666 + 2.44383I
0.213446I 0. + 27.5576I
u = 0.584934 − 0.061813I
a = 0.15128 + 2.97701I
b = 0.05666 − 2.44383I
− 0.213446I 0. − 27.5576I
u = −0.430814 + 0.285650I
a = 0.09371 + 2.24652I
b = −0.371702 + 0.056513I
−1.48873 − 1.56696I −6.38155 + 7.61387I
u = −0.430814 − 0.285650I
a = 0.09371 − 2.24652I
b = −0.371702 − 0.056513I
−1.48873 + 1.56696I −6.38155 − 7.61387I
u = −0.250271 + 0.449093I
a = −0.567142 − 0.727486I
b = −1.268190 − 0.096887I
−10.37260 + 0.86444I −7.08102 + 2.87846I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.250271 − 0.449093I
a = −0.567142 + 0.727486I
b = −1.268190 + 0.096887I
−10.37260 − 0.86444I −7.08102 − 2.87846I
u = 1.50011
a = −0.813827
b = 1.76423
−4.86772 0
u = 1.54778 + 0.03907I
a = −0.22281 + 1.48990I
b = 0.058169 − 0.419643I
5.27688 + 2.48624I 0
u = 1.54778 − 0.03907I
a = −0.22281 − 1.48990I
b = 0.058169 + 0.419643I
5.27688 − 2.48624I 0
u = 0.001775 + 0.445857I
a = 1.000040 − 0.710446I
b = 0.364686 − 0.633641I
1.41091I 0. − 3.50261I
u = 0.001775 − 0.445857I
a = 1.000040 + 0.710446I
b = 0.364686 + 0.633641I
− 1.41091I 0. + 3.50261I
u = 1.57398 + 0.11293I
a = 0.56491 − 1.41977I
b = −0.204122 + 0.485422I
−1.93678 + 5.95845I 0
u = 1.57398 − 0.11293I
a = 0.56491 + 1.41977I
b = −0.204122 − 0.485422I
−1.93678 − 5.95845I 0
u = 1.58805 + 0.06429I
a = 0.055690 − 1.002130I
b = −1.144500 + 0.638936I
6.21754 + 1.60405I 0
u = 1.58805 − 0.06429I
a = 0.055690 + 1.002130I
b = −1.144500 − 0.638936I
6.21754 − 1.60405I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.59261 + 0.05649I
a = −0.10696 + 3.02763I
b = 0.34217 − 2.56487I
− 1.36403I 0
u = −1.59261 − 0.05649I
a = −0.10696 − 3.02763I
b = 0.34217 + 2.56487I
1.36403I 0
u = −1.58253 + 0.21992I
a = 0.43189 − 1.53582I
b = 0.242359 + 1.309890I
1.93678 − 5.95845I 0
u = −1.58253 − 0.21992I
a = 0.43189 + 1.53582I
b = 0.242359 − 1.309890I
1.93678 + 5.95845I 0
u = −1.60277 + 0.00806I
a = 0.94531 − 2.97432I
b = −1.17552 + 2.67763I
7.67427 − 0.41875I 0
u = −1.60277 − 0.00806I
a = 0.94531 + 2.97432I
b = −1.17552 − 2.67763I
7.67427 + 0.41875I 0
u = −1.61745 + 0.14658I
a = −0.199604 + 1.369900I
b = −0.401208 − 1.169100I
9.33906 − 4.00146I 0
u = −1.61745 − 0.14658I
a = −0.199604 − 1.369900I
b = −0.401208 + 1.169100I
9.33906 + 4.00146I 0
u = 1.62381 + 0.10972I
a = −0.13698 + 1.52608I
b = 1.07956 − 1.05237I
10.25510 + 6.17839I 0
u = 1.62381 − 0.10972I
a = −0.13698 − 1.52608I
b = 1.07956 + 1.05237I
10.25510 − 6.17839I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.63751 + 0.15124I
a = 0.28012 − 1.77911I
b = −1.14258 + 1.30370I
8.28668 + 10.95660I 0
u = 1.63751 − 0.15124I
a = 0.28012 + 1.77911I
b = −1.14258 − 1.30370I
8.28668 − 10.95660I 0
u = 1.64399 + 0.18789I
a = −0.39793 + 1.95590I
b = 1.21392 − 1.48592I
14.0571I 0
u = 1.64399 − 0.18789I
a = −0.39793 − 1.95590I
b = 1.21392 + 1.48592I
− 14.0571I 0
u = −1.65526 + 0.07461I
a = −0.184098 − 0.969973I
b = 0.710486 + 0.840147I
10.37260 − 0.86444I 0
u = −1.65526 − 0.07461I
a = −0.184098 + 0.969973I
b = 0.710486 − 0.840147I
10.37260 + 0.86444I 0
u = −1.74741
a = −0.264052
b = −0.316899
4.86772 0
u = −0.113883
a = 5.02027
b = 0.684561
−1.37980 −7.84380
10
II. I
u
2
= hb − u, a − 1, u
2
+ u − 1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
0
a
7
=
1
u − 1
a
4
=
u
−u + 1
a
8
=
u
−u
a
11
=
1
u
a
12
=
−u + 1
2u
a
2
=
−u
u
a
5
=
−u + 1
u − 1
a
1
=
−u
u
a
10
=
−u + 1
2u
a
9
=
1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
2
c
2
, c
3
u
2
− u − 1
c
4
, c
5
, c
6
c
7
u
2
+ u − 1
c
8
, c
9
, c
10
(u − 1)
2
c
11
, c
12
(u + 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
2
c
2
, c
3
, c
4
c
5
, c
6
, c
7
y
2
− 3y + 1
c
8
, c
9
, c
10
c
11
, c
12
(y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.00000
b = 0.618034
−0.657974 5.00000
u = −1.61803
a = 1.00000
b = −1.61803
7.23771 5.00000
14
III. I
u
3
= hb + a, a
2
+ a − 1, u − 1i
(i) Arc colorings
a
3
=
0
1
a
6
=
1
0
a
7
=
1
−1
a
4
=
1
0
a
8
=
0
−1
a
11
=
a
−a
a
12
=
a
0
a
2
=
−1
1
a
5
=
−a + 2
a − 1
a
1
=
−a + 1
a
a
10
=
a
−a
a
9
=
−a + 1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
(u + 1)
2
c
4
, c
5
, c
11
c
12
u
2
− u − 1
c
6
, c
7
(u − 1)
2
c
8
, c
9
u
2
+ u − 1
c
10
u
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
6
, c
7
(y −1)
2
c
4
, c
5
, c
8
c
9
, c
11
, c
12
y
2
− 3y + 1
c
10
y
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.618034
b = −0.618034
0.657974 −5.00000
u = 1.00000
a = −1.61803
b = 1.61803
−7.23771 −5.00000
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
2
(u + 1)
2
(u
58
+ 3u
57
+ ··· + 4u + 4)
c
2
, c
3
((u + 1)
2
)(u
2
− u − 1)(u
58
− 4u
57
+ ··· + 10u + 1)
c
4
(u
2
− u − 1)(u
2
+ u − 1)(u
58
− u
57
+ ··· + 519u + 83)
c
5
(u
2
− u − 1)(u
2
+ u − 1)(u
58
+ u
57
+ ··· − 519u + 83)
c
6
, c
7
((u − 1)
2
)(u
2
+ u − 1)(u
58
− 4u
57
+ ··· + 10u + 1)
c
8
, c
9
((u − 1)
2
)(u
2
+ u − 1)(u
58
+ 4u
57
+ ··· − 10u + 1)
c
10
u
2
(u − 1)
2
(u
58
− 3u
57
+ ··· − 4u + 4)
c
11
, c
12
((u + 1)
2
)(u
2
− u − 1)(u
58
+ 4u
57
+ ··· − 10u + 1)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
2
(y −1)
2
(y
58
− 17y
57
+ ··· − 120y + 16)
c
2
, c
3
, c
6
c
7
, c
8
, c
9
c
11
, c
12
((y −1)
2
)(y
2
− 3y + 1)(y
58
− 68y
57
+ ··· − 90y + 1)
c
4
, c
5
((y
2
− 3y + 1)
2
)(y
58
+ 51y
57
+ ··· − 36463y + 6889)
20
