12a
1215
(K12a
1215
)
A knot diagram
1
Linearized knot diagam
5 6 7 10 2 11 4 12 1 3 8 9
Solving Sequence
8,12
9 1
4,10
5 7 3 11 6 2
c
8
c
12
c
9
c
4
c
7
c
3
c
11
c
6
c
2
c
1
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h8.53148 × 10
61
u
60
− 2.63560 × 10
62
u
59
+ ··· + 2.09325 × 10
62
b − 3.25181 × 10
62
,
− 2.51402 × 10
63
u
60
+ 8.62823 × 10
63
u
59
+ ··· + 2.09325 × 10
62
a + 4.65225 × 10
63
, u
61
− 4u
60
+ ··· − 2u + 1i
I
u
2
= hb + 1, a
2
+ 2a − 1, u − 1i
I
u
3
= hb − 1, a − 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 64 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
= h8.53×10
61
u
60
−2.64×10
62
u
59
+· · ·+2.09×10
62
b−3.25×10
62
, −2.51×
10
63
u
60
+8.63×10
63
u
59
+· · ·+2.09×10
62
a+4.65×10
63
, u
61
−4u
60
+· · ·−2u+1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
u
2
a
1
=
−u
−u
3
+ u
a
4
=
12.0101u
60
− 41.2193u
59
+ ··· + 70.6756u − 22.2250
−0.407570u
60
+ 1.25909u
59
+ ··· + 1.20398u + 1.55347
a
10
=
−u
2
+ 1
−u
4
+ 2u
2
a
5
=
7.23475u
60
− 25.7491u
59
+ ··· + 71.7086u − 14.2434
−5.32045u
60
+ 17.2364u
59
+ ··· − 2.23417u + 8.89499
a
7
=
12.4952u
60
− 42.3313u
59
+ ··· + 65.7255u − 22.9269
−0.0236838u
60
+ 0.0968098u
59
+ ··· + 2.88901u + 0.876791
a
3
=
−1.23183u
60
+ 2.22698u
59
+ ··· + 10.3657u + 2.23332
−1.82728u
60
+ 4.13641u
59
+ ··· − 3.93849u + 1.46849
a
11
=
u
u
a
6
=
7.14430u
60
− 25.0978u
59
+ ··· + 62.9492u − 15.2793
−5.37461u
60
+ 17.3303u
59
+ ··· + 0.112788u + 8.52436
a
2
=
−8.78259u
60
+ 27.8143u
59
+ ··· + 85.7450u + 9.78144
5.73116u
60
− 20.6175u
59
+ ··· + 6.59694u − 9.06352
(ii) Obstruction class = −1
(iii) Cusp Shapes = −112.270u
60
+ 381.730u
59
+ ··· − 78.0643u + 180.153
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
61
+ 5u
60
+ ··· − 6u + 2
c
3
, c
7
u
61
− 2u
60
+ ··· − 18u − 1
c
4
u
61
+ 14u
60
+ ··· − 20506u + 253751
c
6
u
61
− 16u
60
+ ··· − 298u − 71
c
8
, c
9
, c
11
c
12
u
61
+ 4u
60
+ ··· − 2u − 1
c
10
u
61
+ 2u
60
+ ··· + 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
61
− 67y
60
+ ··· + 68y −4
c
3
, c
7
y
61
− 34y
60
+ ··· + 206y −1
c
4
y
61
− 654y
60
+ ··· − 751092018078y −64389570001
c
6
y
61
− 674y
60
+ ··· + 204818y −5041
c
8
, c
9
, c
11
c
12
y
61
− 74y
60
+ ··· − 98y −1
c
10
y
61
+ 2y
60
+ ··· + 102y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.746185 + 0.718329I
a = 0.782846 + 0.678292I
b = −0.677631 + 0.453749I
−7.75122 − 2.10321I 0
u = 0.746185 − 0.718329I
a = 0.782846 − 0.678292I
b = −0.677631 − 0.453749I
−7.75122 + 2.10321I 0
u = −0.967076 + 0.398497I
a = −0.312053 + 0.858947I
b = −0.128262 + 0.982641I
−10.05700 + 6.18751I 0
u = −0.967076 − 0.398497I
a = −0.312053 − 0.858947I
b = −0.128262 − 0.982641I
−10.05700 − 6.18751I 0
u = −0.814186 + 0.491497I
a = −0.27925 + 1.52542I
b = 1.238410 + 0.560509I
0.54223 + 8.36059I 0
u = −0.814186 − 0.491497I
a = −0.27925 − 1.52542I
b = 1.238410 − 0.560509I
0.54223 − 8.36059I 0
u = 1.07507
a = 1.21685
b = 0.0908214
−6.55025 0
u = −0.919328 + 0.600137I
a = 0.37668 − 1.39219I
b = −1.252060 − 0.562320I
−6.64410 + 11.69040I 0
u = −0.919328 − 0.600137I
a = 0.37668 + 1.39219I
b = −1.252060 + 0.562320I
−6.64410 − 11.69040I 0
u = 1.035030 + 0.370907I
a = 0.389492 − 0.314417I
b = 0.923451 + 0.227131I
−0.586497 + 0.638281I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.035030 − 0.370907I
a = 0.389492 + 0.314417I
b = 0.923451 − 0.227131I
−0.586497 − 0.638281I 0
u = −0.022074 + 0.884978I
a = 0.760719 + 0.081071I
b = −1.116060 + 0.470898I
−3.89787 − 6.79256I 0
u = −0.022074 − 0.884978I
a = 0.760719 − 0.081071I
b = −1.116060 − 0.470898I
−3.89787 + 6.79256I 0
u = −0.784824 + 0.190256I
a = 0.337219 − 1.157560I
b = 0.146848 − 1.059540I
−2.79892 + 2.75782I −12.5719 − 8.9610I
u = −0.784824 − 0.190256I
a = 0.337219 + 1.157560I
b = 0.146848 + 1.059540I
−2.79892 − 2.75782I −12.5719 + 8.9610I
u = 0.609495 + 0.503426I
a = −0.901420 − 0.967851I
b = 0.770857 − 0.254010I
−0.88592 − 1.76991I −10.70286 + 7.10949I
u = 0.609495 − 0.503426I
a = −0.901420 + 0.967851I
b = 0.770857 + 0.254010I
−0.88592 + 1.76991I −10.70286 − 7.10949I
u = −0.701647 + 0.311220I
a = −0.02710 − 1.68276I
b = −1.217220 − 0.602617I
1.19274 + 3.49963I −4.99930 − 7.32299I
u = −0.701647 − 0.311220I
a = −0.02710 + 1.68276I
b = −1.217220 + 0.602617I
1.19274 − 3.49963I −4.99930 + 7.32299I
u = −0.760212 + 0.073205I
a = 0.457673 − 0.810260I
b = 1.46334 − 0.51937I
−5.23021 + 0.94975I −19.2405 − 6.6283I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.760212 − 0.073205I
a = 0.457673 + 0.810260I
b = 1.46334 + 0.51937I
−5.23021 − 0.94975I −19.2405 + 6.6283I
u = 1.086370 + 0.613746I
a = −0.145908 + 0.115784I
b = −0.881548 − 0.426034I
−7.18543 + 1.66168I 0
u = 1.086370 − 0.613746I
a = −0.145908 − 0.115784I
b = −0.881548 + 0.426034I
−7.18543 − 1.66168I 0
u = 0.740049
a = −0.465551
b = 0.111226
−1.28803 −7.85780
u = 0.217246 + 0.690863I
a = 0.585924 + 0.057283I
b = −0.219547 − 0.599860I
−6.43015 − 2.58923I −10.03195 + 2.41589I
u = 0.217246 − 0.690863I
a = 0.585924 − 0.057283I
b = −0.219547 + 0.599860I
−6.43015 + 2.58923I −10.03195 − 2.41589I
u = −0.077144 + 0.695385I
a = −0.802000 + 0.155188I
b = 1.138390 − 0.392051I
2.78020 − 4.38394I −1.53700 + 6.52121I
u = −0.077144 − 0.695385I
a = −0.802000 − 0.155188I
b = 1.138390 + 0.392051I
2.78020 + 4.38394I −1.53700 − 6.52121I
u = 0.623061 + 0.152291I
a = −0.39384 + 3.45519I
b = −0.923865 + 0.057323I
0.488805 − 0.376151I 9.8390 − 11.6732I
u = 0.623061 − 0.152291I
a = −0.39384 − 3.45519I
b = −0.923865 − 0.057323I
0.488805 + 0.376151I 9.8390 + 11.6732I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.624285
a = −36.9272
b = 1.01546
−4.28785 381.270
u = 1.38099
a = −0.737085
b = −1.32091
−1.75297 0
u = −0.146537 + 0.414207I
a = 0.599805 − 0.923128I
b = −1.184330 + 0.269776I
2.80305 − 0.88629I 0.82315 − 2.43589I
u = −0.146537 − 0.414207I
a = 0.599805 + 0.923128I
b = −1.184330 − 0.269776I
2.80305 + 0.88629I 0.82315 + 2.43589I
u = −1.60938 + 0.03982I
a = −0.35124 − 1.78283I
b = −0.934436 − 0.284454I
−7.29629 + 1.06963I 0
u = −1.60938 − 0.03982I
a = −0.35124 + 1.78283I
b = −0.934436 + 0.284454I
−7.29629 − 1.06963I 0
u = −1.61455 + 0.13274I
a = 0.040334 + 1.272930I
b = 0.970713 + 0.461820I
−8.57121 + 4.07733I 0
u = −1.61455 − 0.13274I
a = 0.040334 − 1.272930I
b = 0.970713 − 0.461820I
−8.57121 − 4.07733I 0
u = −1.62277
a = −5.90882
b = 1.08441
−12.2377 0
u = 1.62759 + 0.06997I
a = −0.77048 + 1.60057I
b = −1.33629 + 0.86263I
−6.89593 − 4.83384I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.62759 − 0.06997I
a = −0.77048 − 1.60057I
b = −1.33629 − 0.86263I
−6.89593 + 4.83384I 0
u = 1.64847 + 0.04117I
a = 0.30494 + 1.78899I
b = 0.22966 + 1.43807I
−11.30720 − 3.57722I 0
u = 1.64847 − 0.04117I
a = 0.30494 − 1.78899I
b = 0.22966 − 1.43807I
−11.30720 + 3.57722I 0
u = 1.64947 + 0.01439I
a = 1.12320 + 0.98345I
b = 1.74629 + 0.70290I
−13.72070 − 1.24651I 0
u = 1.64947 − 0.01439I
a = 1.12320 − 0.98345I
b = 1.74629 − 0.70290I
−13.72070 + 1.24651I 0
u = 1.65001 + 0.13394I
a = 0.50978 − 1.60334I
b = 1.31764 − 0.70879I
−7.91826 − 10.72950I 0
u = 1.65001 − 0.13394I
a = 0.50978 + 1.60334I
b = 1.31764 + 0.70879I
−7.91826 + 10.72950I 0
u = 0.123537 + 0.319863I
a = −1.021060 − 0.481385I
b = 0.014084 + 0.425125I
−0.233943 − 0.995944I −4.40045 + 6.40501I
u = 0.123537 − 0.319863I
a = −1.021060 + 0.481385I
b = 0.014084 − 0.425125I
−0.233943 + 0.995944I −4.40045 − 6.40501I
u = −1.65674 + 0.03555I
a = 0.362387 − 0.621120I
b = 0.449534 − 0.418699I
−9.99487 + 0.25967I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.65674 − 0.03555I
a = 0.362387 + 0.621120I
b = 0.449534 + 0.418699I
−9.99487 − 0.25967I 0
u = −1.67432 + 0.22629I
a = 0.020974 − 1.105390I
b = −0.981377 − 0.576312I
−16.0070 + 5.8071I 0
u = −1.67432 − 0.22629I
a = 0.020974 + 1.105390I
b = −0.981377 + 0.576312I
−16.0070 − 5.8071I 0
u = 1.68740 + 0.17394I
a = −0.34603 + 1.54557I
b = −1.34464 + 0.64636I
−15.5798 − 14.7360I 0
u = 1.68740 − 0.17394I
a = −0.34603 − 1.54557I
b = −1.34464 − 0.64636I
−15.5798 + 14.7360I 0
u = 1.69400 + 0.10712I
a = −0.32490 − 1.47775I
b = −0.139744 − 1.234560I
−19.3348 − 8.1899I 0
u = 1.69400 − 0.10712I
a = −0.32490 + 1.47775I
b = −0.139744 + 1.234560I
−19.3348 + 8.1899I 0
u = −1.73658 + 0.10391I
a = −0.204374 + 0.738826I
b = −0.512503 + 0.660988I
−17.3526 + 1.0261I 0
u = −1.73658 − 0.10391I
a = −0.204374 − 0.738826I
b = −0.512503 − 0.660988I
−17.3526 − 1.0261I 0
u = −0.0120702 + 0.1398260I
a = −1.36141 + 8.95063I
b = 0.949803 + 0.197306I
−3.17091 − 0.11856I −2.06650 − 1.49742I
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.0120702 − 0.1398260I
a = −1.36141 − 8.95063I
b = 0.949803 − 0.197306I
−3.17091 + 0.11856I −2.06650 + 1.49742I
11
II. I
u
2
= hb + 1, a
2
+ 2a − 1, u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
1
a
9
=
1
1
a
1
=
−1
0
a
4
=
a
−1
a
10
=
0
1
a
5
=
a
−a − 1
a
7
=
−a + 1
1
a
3
=
1
0
a
11
=
1
1
a
6
=
1
a + 1
a
2
=
−a
−2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
2
− 2
c
3
, c
8
, c
9
(u − 1)
2
c
4
u
2
+ 2u − 1
c
6
u
2
− 2u − 1
c
7
, c
10
, c
11
c
12
(u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −2)
2
c
3
, c
7
, c
8
c
9
, c
10
, c
11
c
12
(y −1)
2
c
4
, c
6
y
2
− 6y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.414214
b = −1.00000
−4.93480 −8.00000
u = 1.00000
a = −2.41421
b = −1.00000
−4.93480 −8.00000
15
III. I
u
3
= hb − 1, a − 1, u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
1
a
9
=
1
1
a
1
=
−1
0
a
4
=
1
1
a
10
=
0
1
a
5
=
1
0
a
7
=
2
1
a
3
=
−1
0
a
11
=
1
1
a
6
=
1
0
a
2
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
c
3
, c
11
, c
12
u + 1
c
4
, c
6
, c
7
c
8
, c
9
, c
10
u − 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
c
3
, c
4
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.00000
0 0
19
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u(u
2
− 2)(u
61
+ 5u
60
+ ··· − 6u + 2)
c
3
((u − 1)
2
)(u + 1)(u
61
− 2u
60
+ ··· − 18u − 1)
c
4
(u − 1)(u
2
+ 2u − 1)(u
61
+ 14u
60
+ ··· − 20506u + 253751)
c
6
(u − 1)(u
2
− 2u − 1)(u
61
− 16u
60
+ ··· − 298u − 71)
c
7
(u − 1)(u + 1)
2
(u
61
− 2u
60
+ ··· − 18u − 1)
c
8
, c
9
((u − 1)
3
)(u
61
+ 4u
60
+ ··· − 2u − 1)
c
10
(u − 1)(u + 1)
2
(u
61
+ 2u
60
+ ··· + 2u − 1)
c
11
, c
12
((u + 1)
3
)(u
61
+ 4u
60
+ ··· − 2u − 1)
20
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y(y −2)
2
(y
61
− 67y
60
+ ··· + 68y −4)
c
3
, c
7
((y −1)
3
)(y
61
− 34y
60
+ ··· + 206y −1)
c
4
(y −1)(y
2
− 6y + 1)
· (y
61
− 654y
60
+ ··· − 751092018078y −64389570001)
c
6
(y −1)(y
2
− 6y + 1)(y
61
− 674y
60
+ ··· + 204818y −5041)
c
8
, c
9
, c
11
c
12
((y −1)
3
)(y
61
− 74y
60
+ ··· − 98y −1)
c
10
((y −1)
3
)(y
61
+ 2y
60
+ ··· + 102y −1)
21
