12n
0292
(K12n
0292
)
A knot diagram
1
Linearized knot diagam
3 6 7 9 2 5 11 4 12 7 9 10
Solving Sequence
4,9 5,11
12 10 1 8 7 3 6 2
c
4
c
11
c
9
c
12
c
8
c
7
c
3
c
6
c
2
c
1
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−3.88782 × 10
23
u
13
− 5.12763 × 10
24
u
12
+ ··· + 2.64685 × 10
25
b + 6.25813 × 10
25
,
7.50387 × 10
22
u
13
+ 1.45176 × 10
23
u
12
+ ··· + 5.29369 × 10
25
a + 2.07299 × 10
26
,
u
14
+ 13u
13
+ ··· − 32u + 64i
I
u
2
= hu
7
+ u
6
− 3u
5
− 2u
4
+ 3u
3
+ b + 2, u
7
+ u
6
− 3u
5
− 2u
4
+ 3u
3
+ a + 2, u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1i
I
v
1
= ha, 251v
5
+ 1517v
4
+ 2839v
3
+ 6155v
2
+ 413b + 3834v + 768, v
6
+ 6v
5
+ 11v
4
+ 24v
3
+ 15v
2
+ 3v + 1i
* 3 irreducible components of dim
C
= 0, with total 28 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−3.89 × 10
23
u
13
− 5.13 × 10
24
u
12
+ · · · + 2.65 × 10
25
b + 6.26 ×
10
25
, 7.50 × 10
22
u
13
+ 1.45 × 10
23
u
12
+ · · · + 5.29 × 10
25
a + 2.07 ×
10
26
, u
14
+ 13u
13
+ · · · − 32u + 64i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
5
=
1
u
2
a
11
=
−0.00141751u
13
− 0.00274243u
12
+ ··· − 2.48245u − 3.91596
0.0146885u
13
+ 0.193726u
12
+ ··· − 5.56422u − 2.36437
a
12
=
−0.00141751u
13
− 0.00274243u
12
+ ··· − 2.48245u − 3.91596
0.0101820u
13
+ 0.140392u
12
+ ··· − 4.97157u − 3.36822
a
10
=
−0.00113062u
13
− 0.0125080u
12
+ ··· − 2.90912u − 1.42506
0.0142858u
13
+ 0.186357u
12
+ ··· − 5.61762u − 1.88242
a
1
=
−0.0210436u
13
− 0.266521u
12
+ ··· + 3.80828u − 0.914411
−0.0113459u
13
− 0.139334u
12
+ ··· + 2.52961u − 1.68257
a
8
=
u
u
a
7
=
0.0181604u
13
+ 0.234062u
12
+ ··· − 2.85095u − 0.317190
−0.00288324u
13
− 0.0324588u
12
+ ··· + 0.957338u − 1.23160
a
3
=
−0.00413905u
13
− 0.0558999u
12
+ ··· + 0.679785u + 1.52829
−0.00466437u
13
− 0.0566994u
12
+ ··· + 0.968533u − 0.525572
a
6
=
0.00922038u
13
+ 0.124297u
12
+ ··· − 0.666608u − 1.67826
−0.00752145u
13
− 0.0902797u
12
+ ··· + 1.73607u − 1.64474
a
2
=
−0.0194805u
13
− 0.252321u
12
+ ··· + 2.53552u + 0.574034
−0.0151718u
13
− 0.187049u
12
+ ··· + 2.63035u − 1.84661
(ii) Obstruction class = −1
(iii) Cusp Shapes =
6397031736972176627304075
26468471689247551065891424
u
13
+
85735105186510577411506669
26468471689247551065891424
u
12
+
··· −
114004214719555496517015677
827139740288985970809107
u −
55506637637800758601432107
827139740288985970809107
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
14
+ 9u
13
+ ··· + 16u + 1
c
2
, c
5
u
14
+ 3u
13
+ ··· + 8u + 1
c
3
u
14
− 61u
13
+ ··· + 58652u + 7489
c
4
, c
8
u
14
+ 13u
13
+ ··· − 32u + 64
c
7
, c
10
u
14
+ 41u
13
+ ··· − 640u + 256
c
9
, c
11
, c
12
u
14
− 21u
13
+ ··· + 17u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
14
− 13y
13
+ ··· − 32y + 1
c
2
, c
5
y
14
− 9y
13
+ ··· − 16y + 1
c
3
y
14
− 3381y
13
+ ··· − 1276544916y + 56085121
c
4
, c
8
y
14
− 339y
13
+ ··· − 62464y + 4096
c
7
, c
10
y
14
− 1029y
13
+ ··· − 2539520y + 65536
c
9
, c
11
, c
12
y
14
− 121y
13
+ ··· − 57y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.615322 + 0.027933I
a = 0.270955 + 0.334290I
b = −0.491780 − 0.018263I
−0.755939 − 0.001558I −11.59104 + 0.05980I
u = −0.615322 − 0.027933I
a = 0.270955 − 0.334290I
b = −0.491780 + 0.018263I
−0.755939 + 0.001558I −11.59104 − 0.05980I
u = 0.009870 + 0.545956I
a = −0.167508 + 0.354571I
b = −0.343156 + 1.170610I
2.39955 + 2.72347I −2.27719 − 1.42875I
u = 0.009870 − 0.545956I
a = −0.167508 − 0.354571I
b = −0.343156 − 1.170610I
2.39955 − 2.72347I −2.27719 + 1.42875I
u = 0.538862
a = 2.91010
b = 0.103022
−10.2297 −34.5740
u = −0.527756
a = 0.348493
b = −0.450251
−0.765092 −12.2670
u = 0.328130
a = −2.96230
b = −3.66807
−2.58775 −102.750
u = 2.36859 + 0.76389I
a = 0.286166 − 0.684183I
b = 0.464252 + 0.349923I
−3.81801 − 4.65772I −14.3487 + 2.2929I
u = 2.36859 − 0.76389I
a = 0.286166 + 0.684183I
b = 0.464252 − 0.349923I
−3.81801 + 4.65772I −14.3487 − 2.2929I
u = −1.88270 + 1.66535I
a = −0.845008 − 0.271270I
b = −2.08577 − 0.29493I
15.9880 + 11.7884I −14.3165 − 4.7631I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.88270 − 1.66535I
a = −0.845008 + 0.271270I
b = −2.08577 + 0.29493I
15.9880 − 11.7884I −14.3165 + 4.7631I
u = 2.40458 + 1.69377I
a = 0.839800 − 0.202162I
b = 2.14114 − 0.18650I
17.5291 − 4.7891I −13.20379 + 0.61326I
u = 2.40458 − 1.69377I
a = 0.839800 + 0.202162I
b = 2.14114 + 0.18650I
17.5291 + 4.7891I −13.20379 − 0.61326I
u = −17.9093
a = −0.565105
b = −2.35407
6.82503 0
6
II. I
u
2
= hu
7
+ u
6
− 3u
5
− 2u
4
+ 3u
3
+ b + 2, u
7
+ u
6
− 3u
5
− 2u
4
+ 3u
3
+ a +
2, u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
5
=
1
u
2
a
11
=
−u
7
− u
6
+ 3u
5
+ 2u
4
− 3u
3
− 2
−u
7
− u
6
+ 3u
5
+ 2u
4
− 3u
3
− 2
a
12
=
−u
7
− u
6
+ 3u
5
+ 2u
4
− 3u
3
− 2
−u
7
− u
6
+ 3u
5
+ 2u
4
− 3u
3
− u − 2
a
10
=
−u
7
− u
6
+ 3u
5
+ 2u
4
− 3u
3
− 2
−u
7
− u
6
+ 3u
5
+ 2u
4
− 3u
3
− 2
a
1
=
0
−u
a
8
=
u
u
a
7
=
u
u
a
3
=
−u
2
+ 1
−u
2
a
6
=
−u
3
+ 2u
−u
5
+ u
3
+ u
a
2
=
u
5
− 2u
3
+ u
u
5
− u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u
7
+ 8u
6
− 18u
5
− 12u
4
+ 7u
3
− 3u
2
+ 12u − 3
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
2
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
3
, c
4
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
5
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
6
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
7
, c
10
u
8
c
8
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
9
(u − 1)
8
c
11
, c
12
(u + 1)
8
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
2
, c
5
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
3
, c
4
, c
8
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
7
, c
10
y
8
c
9
, c
11
, c
12
(y −1)
8
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.180120 + 0.268597I
a = −0.805639 + 0.183365I
b = −0.805639 + 0.183365I
−2.68559 − 1.13123I −13.78185 + 1.82144I
u = 1.180120 − 0.268597I
a = −0.805639 − 0.183365I
b = −0.805639 − 0.183365I
−2.68559 + 1.13123I −13.78185 − 1.82144I
u = 0.108090 + 0.747508I
a = −0.189481 + 1.310380I
b = −0.189481 + 1.310380I
0.51448 − 2.57849I −9.42408 + 5.06085I
u = 0.108090 − 0.747508I
a = −0.189481 − 1.310380I
b = −0.189481 − 1.310380I
0.51448 + 2.57849I −9.42408 − 5.06085I
u = −1.37100
a = 0.729394
b = 0.729394
−8.14766 −18.0480
u = −1.334530 + 0.318930I
a = 0.708845 + 0.169402I
b = 0.708845 + 0.169402I
−4.02461 + 6.44354I −15.1664 − 7.9255I
u = −1.334530 − 0.318930I
a = 0.708845 − 0.169402I
b = 0.708845 − 0.169402I
−4.02461 − 6.44354I −15.1664 + 7.9255I
u = 0.463640
a = −2.15684
b = −2.15684
−2.48997 1.79260
10
III.
I
v
1
= ha, 251v
5
+1517v
4
+· · ·+413b+768, v
6
+6v
5
+11v
4
+24v
3
+15v
2
+3v+1i
(i) Arc colorings
a
4
=
1
0
a
9
=
v
0
a
5
=
1
0
a
11
=
0
−0.607748v
5
− 3.67312v
4
+ ··· − 9.28329v −1.85956
a
12
=
0.0290557v
5
+ 0.0242131v
4
+ ··· − 0.687651v −0.0266344
−0.607748v
5
− 3.67312v
4
+ ··· − 9.28329v −1.85956
a
10
=
−0.0290557v
5
− 0.0242131v
4
+ ··· + 0.687651v + 0.0266344
−0.392252v
5
− 2.32688v
4
+ ··· − 5.71671v −1.14044
a
1
=
−v
−0.392252v
5
− 2.32688v
4
+ ··· − 5.71671v −1.14044
a
8
=
v
0
a
7
=
v
0.392252v
5
+ 2.32688v
4
+ ··· + 5.71671v + 1.14044
a
3
=
0.0266344v
5
+ 0.188862v
4
+ ··· + 0.0363196v + 1.39225
−0.421308v
5
− 2.35109v
4
+ ··· − 3.02906v + 0.886199
a
6
=
0.392252v
5
+ 2.32688v
4
+ ··· + 6.71671v + 1.14044
0.392252v
5
+ 2.32688v
4
+ ··· + 5.71671v + 1.14044
a
2
=
0.569007v
5
+ 3.30751v
4
+ ··· + 6.86683v + 1.56174
−0.0290557v
5
− 0.0242131v
4
+ ··· + 2.68765v + 1.02663
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1191
413
v
5
+
6981
413
v
4
+
11931
413
v
3
+
26206
413
v
2
+
13113
413
v −
8216
413
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
8
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
9
(u
2
+ u − 1)
3
c
10
, c
11
, c
12
(u
2
− u − 1)
3
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
4
, c
8
y
6
c
7
, c
9
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.670304
a = 0
b = −0.922021
−2.10041 −18.3450
v = −0.046814 + 0.284512I
a = 0
b = −0.34801 − 2.11500I
2.03717 − 2.82812I −25.9630 + 6.8067I
v = −0.046814 − 0.284512I
a = 0
b = −0.34801 + 2.11500I
2.03717 + 2.82812I −25.9630 − 6.8067I
v = −0.32087 + 1.95007I
a = 0
b = 0.132927 + 0.807858I
−5.85852 − 2.82812I −18.4326 + 1.8100I
v = −0.32087 − 1.95007I
a = 0
b = 0.132927 − 0.807858I
−5.85852 + 2.82812I −18.4326 − 1.8100I
v = −4.59433
a = 0
b = 0.352181
−9.99610 0.135730
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 2u − 1)
2
· (u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
14
+ 9u
13
+ ··· + 16u + 1)
c
2
(u
3
+ u
2
− 1)
2
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
· (u
14
+ 3u
13
+ ··· + 8u + 1)
c
3
(u
3
− u
2
+ 2u − 1)
2
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
· (u
14
− 61u
13
+ ··· + 58652u + 7489)
c
4
u
6
(u
8
+ u
7
+ ··· + 2u − 1)(u
14
+ 13u
13
+ ··· − 32u + 64)
c
5
(u
3
− u
2
+ 1)
2
(u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
· (u
14
+ 3u
13
+ ··· + 8u + 1)
c
6
(u
3
+ u
2
+ 2u + 1)
2
· (u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
14
+ 9u
13
+ ··· + 16u + 1)
c
7
u
8
(u
2
+ u − 1)
3
(u
14
+ 41u
13
+ ··· − 640u + 256)
c
8
u
6
(u
8
− u
7
+ ··· − 2u − 1)(u
14
+ 13u
13
+ ··· − 32u + 64)
c
9
((u − 1)
8
)(u
2
+ u − 1)
3
(u
14
− 21u
13
+ ··· + 17u + 1)
c
10
u
8
(u
2
− u − 1)
3
(u
14
+ 41u
13
+ ··· − 640u + 256)
c
11
, c
12
((u + 1)
8
)(u
2
− u − 1)
3
(u
14
− 21u
13
+ ··· + 17u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
· (y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
14
− 13y
13
+ ··· − 32y + 1)
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
· (y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
14
− 9y
13
+ ··· − 16y + 1)
c
3
(y
3
+ 3y
2
+ 2y −1)
2
· (y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
14
− 3381y
13
+ ··· − 1276544916y + 56085121)
c
4
, c
8
y
6
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
14
− 339y
13
+ ··· − 62464y + 4096)
c
7
, c
10
y
8
(y
2
− 3y + 1)
3
(y
14
− 1029y
13
+ ··· − 2539520y + 65536)
c
9
, c
11
, c
12
((y −1)
8
)(y
2
− 3y + 1)
3
(y
14
− 121y
13
+ ··· − 57y + 1)
16
