12n
0268
(K12n
0268
)
A knot diagram
1
Linearized knot diagam
3 5 8 6 2 10 12 4 7 6 8 11
Solving Sequence
6,10
7
2,11
5 3 1 4 9 8 12
c
6
c
10
c
5
c
2
c
1
c
4
c
9
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−3.29204 × 10
20
u
19
+ 9.27352 × 10
20
u
18
+ ··· + 2.43864 × 10
22
b − 1.34033 × 10
22
,
− 4.39429 × 10
21
u
19
+ 1.24794 × 10
22
u
18
+ ··· + 9.75456 × 10
22
a − 4.90353 × 10
23
,
u
20
− 3u
19
+ ··· + 109u + 34i
I
u
2
= h−u
2
a − u
2
+ b − a − 1, 2u
3
a − 4u
2
a − u
3
+ 4a
2
+ 6au − 2u
2
+ 2a − u + 3, u
4
− u
3
+ 3u
2
− 2u + 1i
I
u
3
= h16a
3
u + 7a
3
− 67a
2
u + 5a
2
+ 153au + 61b − 36a − 158u + 91,
a
4
− 3a
3
u − 5a
3
+ 14a
2
u + 9a
2
− 26au − 5a + 18u − 5, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 36 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−3.29 × 10
20
u
19
+ 9.27 × 10
20
u
18
+ · · · + 2.44 × 10
22
b − 1.34 ×
10
22
, −4.39 × 10
21
u
19
+ 1.25 × 10
22
u
18
+ · · · + 9.75 × 10
22
a − 4.90 ×
10
23
, u
20
− 3u
19
+ · · · + 109u + 34i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
u
2
a
2
=
0.0450486u
19
− 0.127934u
18
+ ··· + 22.3785u + 5.02691
0.0134995u
19
− 0.0380274u
18
+ ··· + 4.80738u + 0.549621
a
11
=
−u
u
a
5
=
0.00528585u
19
− 0.0271975u
18
+ ··· + 10.3946u − 0.455911
0.00385593u
19
− 0.00521186u
18
+ ··· + 2.81112u − 0.156325
a
3
=
0.0193868u
19
− 0.0630592u
18
+ ··· + 16.0098u + 1.70343
0.00436855u
19
− 0.00558340u
18
+ ··· + 4.17123u + 0.514711
a
1
=
0.0197609u
19
− 0.0588829u
18
+ ··· + 10.4067u + 2.39866
0.00436280u
19
− 0.0129388u
18
+ ··· + 2.70245u + 0.389051
a
4
=
0.00914178u
19
− 0.0324093u
18
+ ··· + 13.2058u − 0.612236
0.00385593u
19
− 0.00521186u
18
+ ··· + 2.81112u − 0.156325
a
9
=
u
u
3
+ u
a
8
=
0.0119922u
19
− 0.0407718u
18
+ ··· + 4.81606u − 2.27540
0.000306872u
19
− 0.00141344u
18
+ ··· + 0.273172u − 0.657170
a
12
=
0.0193285u
19
− 0.0576787u
18
+ ··· + 9.52664u + 2.37998
0.00479521u
19
− 0.0141430u
18
+ ··· + 3.58255u + 0.407734
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
18686925740007476253183
195091181117356896960512
u
19
+
11735621234030037886189
48772795279339224240128
u
18
+ ··· −
9107616583783871001778045
195091181117356896960512
u −
82148572343533635819015
5737975915216379322368
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
20
+ 15u
19
+ ··· − 353u + 16
c
2
, c
5
u
20
+ 7u
19
+ ··· + 35u + 4
c
3
, c
8
u
20
− u
19
+ ··· + 2560u + 2048
c
6
, c
9
, c
10
u
20
− 3u
19
+ ··· + 109u + 34
c
7
, c
11
u
20
− 3u
19
+ ··· − 33u + 34
c
12
u
20
+ 31u
19
+ ··· + 32843u + 1156
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
20
− 13y
19
+ ··· − 93409y + 256
c
2
, c
5
y
20
+ 15y
19
+ ··· − 353y + 16
c
3
, c
8
y
20
+ 71y
19
+ ··· + 13893632y + 4194304
c
6
, c
9
, c
10
y
20
+ 3y
19
+ ··· + 30619y + 1156
c
7
, c
11
y
20
+ 31y
19
+ ··· + 32843y + 1156
c
12
y
20
− 73y
19
+ ··· − 208085985y + 1336336
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.016951 + 0.789231I
a = −0.147460 − 1.134770I
b = 0.508184 + 0.603985I
1.11325 + 1.38275I 7.09119 − 4.02054I
u = −0.016951 − 0.789231I
a = −0.147460 + 1.134770I
b = 0.508184 − 0.603985I
1.11325 − 1.38275I 7.09119 + 4.02054I
u = −0.409805 + 0.654210I
a = 0.913159 − 0.159905I
b = −0.266779 + 0.021705I
0.11722 + 1.46637I 1.49691 − 4.73539I
u = −0.409805 − 0.654210I
a = 0.913159 + 0.159905I
b = −0.266779 − 0.021705I
0.11722 − 1.46637I 1.49691 + 4.73539I
u = 0.154412 + 0.621344I
a = −1.91384 − 1.58582I
b = 0.530147 − 0.944937I
0.12177 − 2.86051I 4.18275 − 0.60909I
u = 0.154412 − 0.621344I
a = −1.91384 + 1.58582I
b = 0.530147 + 0.944937I
0.12177 + 2.86051I 4.18275 + 0.60909I
u = −0.18250 + 1.41448I
a = 1.66337 + 0.44996I
b = −0.729471 − 0.781970I
7.73611 + 5.17350I 7.90966 − 4.84500I
u = −0.18250 − 1.41448I
a = 1.66337 − 0.44996I
b = −0.729471 + 0.781970I
7.73611 − 5.17350I 7.90966 + 4.84500I
u = −0.120113 + 0.255851I
a = 2.56623 + 3.01463I
b = 0.203906 + 0.753665I
−1.66738 + 1.58596I −6.75642 − 4.62170I
u = −0.120113 − 0.255851I
a = 2.56623 − 3.01463I
b = 0.203906 − 0.753665I
−1.66738 − 1.58596I −6.75642 + 4.62170I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.57886 + 1.65257I
a = 1.15768 − 1.00754I
b = −0.73327 + 1.21762I
6.46005 − 0.37754I 1.69270 + 0.75958I
u = −0.57886 − 1.65257I
a = 1.15768 + 1.00754I
b = −0.73327 − 1.21762I
6.46005 + 0.37754I 1.69270 − 0.75958I
u = 0.83952 + 1.68636I
a = 1.91623 − 0.59074I
b = −0.80508 + 1.30894I
−13.2323 − 12.8519I 1.25558 + 5.23566I
u = 0.83952 − 1.68636I
a = 1.91623 + 0.59074I
b = −0.80508 − 1.30894I
−13.2323 + 12.8519I 1.25558 − 5.23566I
u = 1.22594 + 1.55065I
a = 1.69696 + 0.72298I
b = −1.37246 − 0.39498I
−10.36440 − 5.35439I 2.02942 + 1.66531I
u = 1.22594 − 1.55065I
a = 1.69696 − 0.72298I
b = −1.37246 + 0.39498I
−10.36440 + 5.35439I 2.02942 − 1.66531I
u = −1.85145 + 1.14515I
a = 0.648918 + 0.734717I
b = −0.25500 − 1.69562I
−5.65348 + 3.36046I 0.05819 − 1.65611I
u = −1.85145 − 1.14515I
a = 0.648918 − 0.734717I
b = −0.25500 + 1.69562I
−5.65348 − 3.36046I 0.05819 + 1.65611I
u = 2.43981 + 0.93238I
a = 0.616388 + 1.146420I
b = −0.58018 − 1.86831I
−17.5295 + 2.0815I − 60.10 − 0.732251I
u = 2.43981 − 0.93238I
a = 0.616388 − 1.146420I
b = −0.58018 + 1.86831I
−17.5295 − 2.0815I − 60.10 + 0.732251I
6
II.
I
u
2
= h−u
2
a − u
2
+ b − a − 1, 2u
3
a − u
3
+ · · · + 2a + 3, u
4
− u
3
+ 3u
2
− 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
u
2
a
2
=
a
u
2
a + u
2
+ a + 1
a
11
=
−u
u
a
5
=
−u
2
a +
1
2
u
3
− 2u
2
+
3
2
u −
1
2
u
2
a + u
2
+ a
a
3
=
1
2
u
3
− u
2
+ a +
3
2
u −
1
2
u
2
a + u
2
+ a
a
1
=
−1
0
a
4
=
1
2
u
3
− u
2
+ a +
3
2
u −
1
2
u
2
a + u
2
+ a
a
9
=
u
u
3
+ u
a
8
=
u
u
3
+ u
a
12
=
−u
2
− 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
7
2
u
3
a − u
2
a −
1
2
u
3
−
11
2
au − 3u
2
−
7
2
a +
11
2
u −
11
2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
3
, c
8
u
8
c
6
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
7
(u
4
− u
3
+ u
2
+ 1)
2
c
9
, c
10
, c
12
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
11
(u
4
+ u
3
+ u
2
+ 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
4
c
3
, c
8
y
8
c
6
, c
9
, c
10
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
7
, c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.395123 + 0.506844I
a = −0.893973 − 1.010300I
b = 0.500000 − 0.866025I
−0.21101 − 3.44499I −2.28131 + 9.48913I
u = 0.395123 + 0.506844I
a = −0.178069 + 0.596972I
b = 0.500000 + 0.866025I
−0.211005 + 0.614778I 0.065036 − 0.652246I
u = 0.395123 − 0.506844I
a = −0.893973 + 1.010300I
b = 0.500000 + 0.866025I
−0.21101 + 3.44499I −2.28131 − 9.48913I
u = 0.395123 − 0.506844I
a = −0.178069 − 0.596972I
b = 0.500000 − 0.866025I
−0.211005 − 0.614778I 0.065036 + 0.652246I
u = 0.10488 + 1.55249I
a = −1.202340 − 0.666019I
b = 0.500000 + 0.866025I
6.79074 − 1.13408I 4.18309 + 3.88645I
u = 0.10488 + 1.55249I
a = −1.47562 + 0.50824I
b = 0.500000 − 0.866025I
6.79074 − 5.19385I −0.84181 + 3.92087I
u = 0.10488 − 1.55249I
a = −1.202340 + 0.666019I
b = 0.500000 − 0.866025I
6.79074 + 1.13408I 4.18309 − 3.88645I
u = 0.10488 − 1.55249I
a = −1.47562 − 0.50824I
b = 0.500000 + 0.866025I
6.79074 + 5.19385I −0.84181 − 3.92087I
10
III.
I
u
3
= h16a
3
u − 67a
2
u + · · · − 36a + 91, −3a
3
u + 14a
2
u + · · · − 5a − 5, u
2
+ 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
−1
a
2
=
a
−0.262295a
3
u + 1.09836a
2
u + ··· + 0.590164a − 1.49180
a
11
=
−u
u
a
5
=
−0.557377a
3
u + 1.45902a
2
u + ··· + 4.75410a − 4.29508
0.0983607a
3
u + 0.213115a
2
u + ··· − 0.721311a − 0.0655738
a
3
=
0.196721a
3
u + 0.426230a
2
u + ··· − 1.44262a − 0.131148
−0.475410a
3
u + 0.803279a
2
u + ··· + 3.81967a − 3.01639
a
1
=
0.114754a
3
u + 0.0819672a
2
u + ··· − 2.50820a + 0.590164
−0.114754a
3
u − 0.0819672a
2
u + ··· + 2.50820a − 0.590164
a
4
=
−0.459016a
3
u + 1.67213a
2
u + ··· + 4.03279a − 4.36066
0.0983607a
3
u + 0.213115a
2
u + ··· − 0.721311a − 0.0655738
a
9
=
u
0
a
8
=
0.262295a
3
u − 1.09836a
2
u + ··· − 1.59016a + 3.49180
−0.262295a
3
u + 1.09836a
2
u + ··· + 1.59016a − 2.49180
a
12
=
0.114754a
3
u + 0.0819672a
2
u + ··· − 2.50820a + 0.590164
−0.114754a
3
u − 0.0819672a
2
u + ··· + 2.50820a − 0.590164
(ii) Obstruction class = 1
(iii) Cusp Shapes =
40
61
a
3
u +
48
61
a
3
−
320
61
a
2
u −
140
61
a
2
+
840
61
au +
32
61
a −
944
61
u +
380
61
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
2
(u
4
− u
3
+ u
2
+ 1)
2
c
3
, c
8
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
c
5
(u
4
+ u
3
+ u
2
+ 1)
2
c
6
, c
7
, c
9
c
10
, c
11
(u
2
+ 1)
4
c
12
(u + 1)
8
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
2
, c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
3
, c
8
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
c
6
, c
7
, c
9
c
10
, c
11
(y + 1)
8
c
12
(y −1)
8
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 1.04872 + 1.17889I
b = 0.351808 + 0.720342I
−0.21101 + 1.41510I 0.17326 − 4.90874I
u = 1.000000I
a = 1.52617 − 1.31052I
b = −0.851808 + 0.911292I
6.79074 − 3.16396I 3.82674 + 2.56480I
u = 1.000000I
a = 2.17745 + 0.51206I
b = −0.851808 − 0.911292I
6.79074 + 3.16396I 3.82674 − 2.56480I
u = 1.000000I
a = 0.24766 + 2.61957I
b = 0.351808 − 0.720342I
−0.21101 − 1.41510I 0.17326 + 4.90874I
u = − 1.000000I
a = 1.04872 − 1.17889I
b = 0.351808 − 0.720342I
−0.21101 − 1.41510I 0.17326 + 4.90874I
u = − 1.000000I
a = 1.52617 + 1.31052I
b = −0.851808 − 0.911292I
6.79074 + 3.16396I 3.82674 − 2.56480I
u = − 1.000000I
a = 2.17745 − 0.51206I
b = −0.851808 + 0.911292I
6.79074 − 3.16396I 3.82674 + 2.56480I
u = − 1.000000I
a = 0.24766 − 2.61957I
b = 0.351808 + 0.720342I
−0.21101 + 1.41510I 0.17326 − 4.90874I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
2
− u + 1)
4
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
· (u
20
+ 15u
19
+ ··· − 353u + 16)
c
2
((u
2
+ u + 1)
4
)(u
4
− u
3
+ u
2
+ 1)
2
(u
20
+ 7u
19
+ ··· + 35u + 4)
c
3
, c
8
u
8
(u
8
− 5u
6
+ ··· − 2u
2
+ 1)(u
20
− u
19
+ ··· + 2560u + 2048)
c
5
((u
2
− u + 1)
4
)(u
4
+ u
3
+ u
2
+ 1)
2
(u
20
+ 7u
19
+ ··· + 35u + 4)
c
6
((u
2
+ 1)
4
)(u
4
− u
3
+ 3u
2
− 2u + 1)
2
(u
20
− 3u
19
+ ··· + 109u + 34)
c
7
((u
2
+ 1)
4
)(u
4
− u
3
+ u
2
+ 1)
2
(u
20
− 3u
19
+ ··· − 33u + 34)
c
9
, c
10
((u
2
+ 1)
4
)(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
(u
20
− 3u
19
+ ··· + 109u + 34)
c
11
((u
2
+ 1)
4
)(u
4
+ u
3
+ u
2
+ 1)
2
(u
20
− 3u
19
+ ··· − 33u + 34)
c
12
(u + 1)
8
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
· (u
20
+ 31u
19
+ ··· + 32843u + 1156)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
2
+ y + 1)
4
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
20
− 13y
19
+ ··· − 93409y + 256)
c
2
, c
5
(y
2
+ y + 1)
4
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
20
+ 15y
19
+ ··· − 353y + 16)
c
3
, c
8
y
8
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
· (y
20
+ 71y
19
+ ··· + 13893632y + 4194304)
c
6
, c
9
, c
10
(y + 1)
8
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
20
+ 3y
19
+ ··· + 30619y + 1156)
c
7
, c
11
(y + 1)
8
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
20
+ 31y
19
+ ··· + 32843y + 1156)
c
12
(y −1)
8
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
20
− 73y
19
+ ··· − 208085985y + 1336336)
16
