12a
0422
(K12a
0422
)
A knot diagram
1
Linearized knot diagam
3 6 10 7 2 5 11 12 1 4 8 9
Solving Sequence
8,11
12 9 1
4,7
5 6 10 3 2
c
11
c
8
c
12
c
7
c
4
c
6
c
10
c
3
c
2
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−47u
38
+ 152u
37
+ ··· + 2b + 26, 101u
38
− 320u
37
+ ··· + 4a − 62, u
39
− 5u
38
+ ··· − u + 1i
I
u
2
= hb, a
3
− a
2
u − a
2
+ 2au + 4a − 2u − 3, u
2
+ u − 1i
I
u
3
= hb + 1, a − 2, u + 1i
* 3 irreducible components of dim
C
= 0, with total 46 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−47u
38
+ 152u
37
+ · · · + 2b + 26, 101u
38
− 320u
37
+ · · · + 4a −
62, u
39
− 5u
38
+ · · · − u + 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
4
=
−
101
4
u
38
+ 80u
37
+ ··· −
27
4
u +
31
2
47
2
u
38
− 76u
37
+ ··· +
11
2
u − 13
a
7
=
−u
u
a
5
=
−17.2500u
38
+ 54.7500u
37
+ ··· − 3.75000u + 10.7500
31
2
u
38
−
203
4
u
37
+ ··· +
5
2
u −
33
4
a
6
=
−
1
2
u
38
+
7
4
u
37
+ ··· − 6u +
5
4
3
4
u
38
−
5
2
u
37
+ ··· +
1
4
u −
1
2
a
10
=
−u
3
+ 2u
u
5
− 3u
3
+ u
a
3
=
93
4
u
38
− 77u
37
+ ··· +
19
4
u −
23
2
−
75
2
u
38
+ 122u
37
+ ··· −
19
2
u + 22
a
2
=
−
1
4
u
38
+
3
4
u
37
+ ··· +
19
4
u +
1
4
u
11
− 7u
9
+ 16u
7
− 2u
6
− 13u
5
+ 8u
4
+ 3u
3
− 6u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −64u
38
+ 210u
37
+ ··· − 6u +
81
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
u
39
+ 8u
38
+ ··· + 42u + 1
c
2
, c
5
u
39
+ 2u
38
+ ··· − 6u + 1
c
3
, c
10
u
39
+ 2u
38
+ ··· − 160u − 64
c
7
, c
8
, c
9
c
11
, c
12
u
39
− 5u
38
+ ··· − u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
y
39
+ 48y
38
+ ··· + 978y −1
c
2
, c
5
y
39
− 8y
38
+ ··· + 42y −1
c
3
, c
10
y
39
− 34y
38
+ ··· + 25600y −4096
c
7
, c
8
, c
9
c
11
, c
12
y
39
− 55y
38
+ ··· − 9y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.053960 + 0.224778I
a = 1.61876 + 0.68220I
b = −1.163720 + 0.312418I
4.49453 − 5.61479I 8.25857 + 7.26249I
u = −1.053960 − 0.224778I
a = 1.61876 − 0.68220I
b = −1.163720 − 0.312418I
4.49453 + 5.61479I 8.25857 − 7.26249I
u = 0.862747 + 0.116296I
a = −0.069535 − 0.131159I
b = −0.221452 + 0.784770I
1.42290 + 1.53553I 7.01176 − 4.72842I
u = 0.862747 − 0.116296I
a = −0.069535 + 0.131159I
b = −0.221452 − 0.784770I
1.42290 − 1.53553I 7.01176 + 4.72842I
u = 1.139060 + 0.026381I
a = −0.038812 + 0.168804I
b = −0.025448 + 1.175110I
8.36476 + 3.13639I 0
u = 1.139060 − 0.026381I
a = −0.038812 − 0.168804I
b = −0.025448 − 1.175110I
8.36476 − 3.13639I 0
u = −1.151060 + 0.133458I
a = −1.46113 − 0.36382I
b = 1.226750 − 0.126705I
6.34383 − 1.26782I 0
u = −1.151060 − 0.133458I
a = −1.46113 + 0.36382I
b = 1.226750 + 0.126705I
6.34383 + 1.26782I 0
u = 0.415029 + 0.681141I
a = −0.293962 − 0.902195I
b = −1.43879 − 0.09612I
8.68168 − 1.01341I 8.20452 − 0.45089I
u = 0.415029 − 0.681141I
a = −0.293962 + 0.902195I
b = −1.43879 + 0.09612I
8.68168 + 1.01341I 8.20452 + 0.45089I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.379187 + 0.691000I
a = 0.296338 + 0.946075I
b = 1.43542 + 0.18852I
8.57368 + 5.46017I 7.82438 − 5.30651I
u = 0.379187 − 0.691000I
a = 0.296338 − 0.946075I
b = 1.43542 − 0.18852I
8.57368 − 5.46017I 7.82438 + 5.30651I
u = −1.160140 + 0.396351I
a = 1.23807 + 0.87778I
b = −1.52336 + 0.47798I
13.3910 − 9.1863I 0
u = −1.160140 − 0.396351I
a = 1.23807 − 0.87778I
b = −1.52336 − 0.47798I
13.3910 + 9.1863I 0
u = −1.187780 + 0.377695I
a = −1.21277 − 0.82919I
b = 1.54397 − 0.40480I
13.72490 − 2.63237I 0
u = −1.187780 − 0.377695I
a = −1.21277 + 0.82919I
b = 1.54397 + 0.40480I
13.72490 + 2.63237I 0
u = 0.467926 + 0.375228I
a = 0.068401 − 0.656383I
b = −0.834315 + 0.134239I
1.190300 − 0.320599I 8.01474 − 0.06231I
u = 0.467926 − 0.375228I
a = 0.068401 + 0.656383I
b = −0.834315 − 0.134239I
1.190300 + 0.320599I 8.01474 + 0.06231I
u = 0.239901 + 0.477819I
a = −0.049714 + 1.108090I
b = 0.899573 + 0.293268I
0.46281 + 3.25190I 3.30026 − 8.26387I
u = 0.239901 − 0.477819I
a = −0.049714 − 1.108090I
b = 0.899573 − 0.293268I
0.46281 − 3.25190I 3.30026 + 8.26387I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.487731
a = 0.343685
b = −0.408979
0.740705 13.6330
u = −1.59861
a = −0.240120
b = 0.465924
8.08529 0
u = −0.345833 + 0.036532I
a = 0.12571 + 3.44175I
b = −0.000094 + 0.479895I
3.55637 − 2.89653I −3.84297 + 3.88300I
u = −0.345833 − 0.036532I
a = 0.12571 − 3.44175I
b = −0.000094 − 0.479895I
3.55637 + 2.89653I −3.84297 − 3.88300I
u = −1.67511 + 0.02701I
a = −0.088705 − 0.412032I
b = 0.229021 + 0.890141I
10.40850 − 2.06539I 0
u = −1.67511 − 0.02701I
a = −0.088705 + 0.412032I
b = 0.229021 − 0.890141I
10.40850 + 2.06539I 0
u = 1.73748
a = −2.04429
b = 1.35510
11.5719 0
u = 1.74621 + 0.05370I
a = −1.91054 + 0.23326I
b = 1.42871 + 0.37526I
14.5888 + 6.7421I 0
u = 1.74621 − 0.05370I
a = −1.91054 − 0.23326I
b = 1.42871 − 0.37526I
14.5888 − 6.7421I 0
u = −1.76716 + 0.00715I
a = −0.013757 − 0.629655I
b = 0.04950 + 1.56385I
18.9593 − 3.2843I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.76716 − 0.00715I
a = −0.013757 + 0.629655I
b = 0.04950 − 1.56385I
18.9593 + 3.2843I 0
u = 1.76755 + 0.03169I
a = 1.88452 − 0.11380I
b = −1.56905 − 0.21871I
16.9501 + 1.9651I 0
u = 1.76755 − 0.03169I
a = 1.88452 + 0.11380I
b = −1.56905 + 0.21871I
16.9501 − 1.9651I 0
u = 1.77210 + 0.10705I
a = −1.71021 + 0.26961I
b = 1.61742 + 0.71877I
−15.5908 + 11.3747I 0
u = 1.77210 − 0.10705I
a = −1.71021 − 0.26961I
b = 1.61742 − 0.71877I
−15.5908 − 11.3747I 0
u = −0.042751 + 0.219509I
a = −0.63189 + 2.24770I
b = 0.311409 + 0.398108I
−1.261040 − 0.319837I −6.08687 + 0.82925I
u = −0.042751 − 0.219509I
a = −0.63189 − 2.24770I
b = 0.311409 − 0.398108I
−1.261040 + 0.319837I −6.08687 − 0.82925I
u = 1.78078 + 0.09884I
a = 1.71959 − 0.23944I
b = −1.67156 − 0.66408I
−15.0724 + 4.7212I 0
u = 1.78078 − 0.09884I
a = 1.71959 + 0.23944I
b = −1.67156 + 0.66408I
−15.0724 − 4.7212I 0
8
II. I
u
2
= hb, a
3
− a
2
u − a
2
+ 2au + 4a − 2u − 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
4
=
a
0
a
7
=
−u
u
a
5
=
au
−au + a
a
6
=
−a
2
u + a
2
− u
2a
2
u − a
2
+ u
a
10
=
1
0
a
3
=
a
0
a
2
=
a
2
u + u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 10a
2
u − 9a
2
+ 6au − a + 3u − 1
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
3
, c
10
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
, c
9
(u
2
− u − 1)
3
c
11
, c
12
(u
2
+ u − 1)
3
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
3
, c
10
y
6
c
7
, c
8
, c
9
c
11
, c
12
(y
2
− 3y + 1)
3
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0.922021
b = 0
−0.126494 0.954070
u = 0.618034
a = 0.34801 + 2.11500I
b = 0
4.01109 − 2.82812I 14.0681 + 1.5771I
u = 0.618034
a = 0.34801 − 2.11500I
b = 0
4.01109 + 2.82812I 14.0681 − 1.5771I
u = −1.61803
a = −0.132927 + 0.807858I
b = 0
11.90680 + 2.82812I 11.55793 − 3.24268I
u = −1.61803
a = −0.132927 − 0.807858I
b = 0
11.90680 − 2.82812I 11.55793 + 3.24268I
u = −1.61803
a = −0.352181
b = 0
7.76919 −5.20600
12
III. I
u
3
= hb + 1, a − 2, u + 1i
(i) Arc colorings
a
8
=
0
−1
a
11
=
1
0
a
12
=
1
−1
a
9
=
−1
0
a
1
=
0
−1
a
4
=
2
−1
a
7
=
1
−1
a
5
=
1
0
a
6
=
0
−1
a
10
=
−1
1
a
3
=
1
0
a
2
=
1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
8
, c
9
, c
11
c
12
u + 1
c
3
, c
10
u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 2.00000
b = −1.00000
1.64493 6.00000
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
(u + 1)(u
3
− u
2
+ 2u − 1)
2
(u
39
+ 8u
38
+ ··· + 42u + 1)
c
2
(u + 1)(u
3
+ u
2
− 1)
2
(u
39
+ 2u
38
+ ··· − 6u + 1)
c
3
, c
10
u
6
(u − 1)(u
39
+ 2u
38
+ ··· − 160u − 64)
c
5
(u + 1)(u
3
− u
2
+ 1)
2
(u
39
+ 2u
38
+ ··· − 6u + 1)
c
6
(u + 1)(u
3
+ u
2
+ 2u + 1)
2
(u
39
+ 8u
38
+ ··· + 42u + 1)
c
7
, c
8
, c
9
(u + 1)(u
2
− u − 1)
3
(u
39
− 5u
38
+ ··· − u + 1)
c
11
, c
12
(u + 1)(u
2
+ u − 1)
3
(u
39
− 5u
38
+ ··· − u + 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
(y −1)(y
3
+ 3y
2
+ 2y −1)
2
(y
39
+ 48y
38
+ ··· + 978y −1)
c
2
, c
5
(y −1)(y
3
− y
2
+ 2y −1)
2
(y
39
− 8y
38
+ ··· + 42y −1)
c
3
, c
10
y
6
(y −1)(y
39
− 34y
38
+ ··· + 25600y −4096)
c
7
, c
8
, c
9
c
11
, c
12
(y −1)(y
2
− 3y + 1)
3
(y
39
− 55y
38
+ ··· − 9y −1)
18
