12n
0047
(K12n
0047
)
A knot diagram
1
Linearized knot diagam
3 5 6 10 2 10 12 11 5 1 8 7
Solving Sequence
2,5
3
6,10
7 1 11 4 9 8 12
c
2
c
5
c
6
c
1
c
10
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−8u
36
+ 17u
35
+ ··· + 8b + 21, −7u
36
+ 31u
35
+ ··· + 4a + 11, u
37
− 5u
36
+ ··· + u − 1i
I
u
2
= hb
4
− b
3
u − b
3
+ b
2
u − u − 1, a, u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 45 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−8u
36
+ 17u
35
+ · · · + 8b + 21, −7u
36
+ 31u
35
+ · · · + 4a + 11, u
37
−
5u
36
+ · · · + u − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
−u
2
a
6
=
u
u
a
10
=
7
4
u
36
−
31
4
u
35
+ ··· +
7
4
u −
11
4
u
36
−
17
8
u
35
+ ··· +
7
2
u −
21
8
a
7
=
−u
2
− 1
−
1
8
u
36
+
5
8
u
35
+ ··· +
17
8
u −
1
8
a
1
=
u
2
+ 1
−u
4
a
11
=
15
8
u
36
−
67
8
u
35
+ ··· +
3
8
u −
13
8
1
8
u
36
+
5
4
u
35
+ ··· +
33
8
u −
7
2
a
4
=
u
4
+ u
2
+ 1
u
4
a
9
=
7
4
u
36
−
31
4
u
35
+ ··· +
7
4
u −
11
4
11
4
u
36
−
69
8
u
35
+ ··· +
11
4
u −
29
8
a
8
=
5
4
u
36
−
11
2
u
35
+ ··· +
1
2
u −
3
4
−
3
4
u
36
+
61
8
u
35
+ ··· +
19
4
u −
45
8
a
12
=
−
1
8
u
36
+
1
2
u
35
+ ··· −
15
8
u + 1
u
36
−
47
8
u
35
+ ··· −
5
4
u +
15
8
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
13
8
u
36
+ 10u
35
+ ··· +
243
8
u −
13
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
37
+ 23u
36
+ ··· − 25u − 1
c
2
, c
5
u
37
+ 5u
36
+ ··· + u + 1
c
3
u
37
− 5u
36
+ ··· − 7u + 1
c
4
, c
9
u
37
− u
36
+ ··· + 384u + 256
c
6
u
37
+ 3u
36
+ ··· − u + 1
c
7
, c
8
, c
11
c
12
u
37
+ 3u
36
+ ··· + 3u + 1
c
10
u
37
− 13u
36
+ ··· − 2707u − 563
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
37
− 13y
36
+ ··· − 101y −1
c
2
, c
5
y
37
+ 23y
36
+ ··· − 25y −1
c
3
y
37
− 49y
36
+ ··· − 25y −1
c
4
, c
9
y
37
− 45y
36
+ ··· + 507904y −65536
c
6
y
37
− 47y
36
+ ··· − 21y −1
c
7
, c
8
, c
11
c
12
y
37
+ 45y
36
+ ··· − 21y −1
c
10
y
37
− 27y
36
+ ··· + 708095y −316969
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.974418 + 0.080590I
a = 1.70420 + 0.06611I
b = −0.049744 + 0.136325I
−6.68859 − 3.78454I −3.30270 + 3.82626I
u = 0.974418 − 0.080590I
a = 1.70420 − 0.06611I
b = −0.049744 − 0.136325I
−6.68859 + 3.78454I −3.30270 − 3.82626I
u = 0.231315 + 1.006220I
a = −0.542701 + 0.732426I
b = 0.34827 + 2.04966I
−8.03711 + 4.79124I −8.56176 − 2.65048I
u = 0.231315 − 1.006220I
a = −0.542701 − 0.732426I
b = 0.34827 − 2.04966I
−8.03711 − 4.79124I −8.56176 + 2.65048I
u = 1.028220 + 0.124183I
a = −1.74341 − 0.11482I
b = −0.037304 − 0.222366I
−15.2275 − 6.0970I −5.01510 + 2.60803I
u = 1.028220 − 0.124183I
a = −1.74341 + 0.11482I
b = −0.037304 + 0.222366I
−15.2275 + 6.0970I −5.01510 − 2.60803I
u = 0.136558 + 0.938853I
a = 0.637084 − 0.690045I
b = −0.37494 − 1.62263I
−0.94719 + 2.42286I −4.75761 − 3.49030I
u = 0.136558 − 0.938853I
a = 0.637084 + 0.690045I
b = −0.37494 + 1.62263I
−0.94719 − 2.42286I −4.75761 + 3.49030I
u = 0.934595
a = −1.67956
b = 0.109351
−4.18236 0.534920
u = −0.498462 + 0.758486I
a = −0.027415 + 0.606521I
b = 0.231876 + 0.424050I
0.02228 − 1.46962I −3.17240 + 5.64098I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.498462 − 0.758486I
a = −0.027415 − 0.606521I
b = 0.231876 − 0.424050I
0.02228 + 1.46962I −3.17240 − 5.64098I
u = −0.593765 + 0.922350I
a = −0.560730 − 0.309146I
b = −0.562518 − 0.030265I
−0.58389 − 2.98896I −7.13918 + 2.57591I
u = −0.593765 − 0.922350I
a = −0.560730 + 0.309146I
b = −0.562518 + 0.030265I
−0.58389 + 2.98896I −7.13918 − 2.57591I
u = −0.225572 + 1.100310I
a = 0.843421 − 0.554905I
b = 0.474759 − 1.067940I
−3.28963 − 2.82464I −7.40073 + 4.80560I
u = −0.225572 − 1.100310I
a = 0.843421 + 0.554905I
b = 0.474759 + 1.067940I
−3.28963 + 2.82464I −7.40073 − 4.80560I
u = −0.671032 + 0.495259I
a = −0.136744 − 1.209480I
b = −0.372384 − 0.710472I
−6.83822 − 1.47848I −3.18303 + 2.73607I
u = −0.671032 − 0.495259I
a = −0.136744 + 1.209480I
b = −0.372384 + 0.710472I
−6.83822 + 1.47848I −3.18303 − 2.73607I
u = −0.033671 + 0.800781I
a = −0.683908 + 0.699343I
b = 0.295173 + 1.048690I
−0.199573 − 0.983660I −0.88923 + 4.01219I
u = −0.033671 − 0.800781I
a = −0.683908 − 0.699343I
b = 0.295173 − 1.048690I
−0.199573 + 0.983660I −0.88923 − 4.01219I
u = −0.662632 + 1.005330I
a = 0.854489 + 0.362836I
b = 0.841855 + 0.003201I
−8.20792 − 3.66352I −6.31169 + 2.26713I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.662632 − 1.005330I
a = 0.854489 − 0.362836I
b = 0.841855 − 0.003201I
−8.20792 + 3.66352I −6.31169 − 2.26713I
u = −0.227230 + 1.231130I
a = −1.009910 + 0.608801I
b = −0.74907 + 1.20730I
−11.76330 − 3.91769I −8.22395 + 3.01298I
u = −0.227230 − 1.231130I
a = −1.009910 − 0.608801I
b = −0.74907 − 1.20730I
−11.76330 + 3.91769I −8.22395 − 3.01298I
u = 0.486053 + 1.286770I
a = 0.240798 − 1.270660I
b = 0.45076 − 2.68214I
−8.12027 + 5.05520I 0
u = 0.486053 − 1.286770I
a = 0.240798 + 1.270660I
b = 0.45076 + 2.68214I
−8.12027 − 5.05520I 0
u = 0.439816 + 1.321280I
a = −0.347920 + 1.293380I
b = −0.48608 + 2.61912I
−11.09790 + 1.17699I 0
u = 0.439816 − 1.321280I
a = −0.347920 − 1.293380I
b = −0.48608 − 2.61912I
−11.09790 − 1.17699I 0
u = 0.532306 + 1.288570I
a = −0.159576 + 1.311500I
b = −0.46454 + 2.72578I
−10.40170 + 9.18736I 0
u = 0.532306 − 1.288570I
a = −0.159576 − 1.311500I
b = −0.46454 − 2.72578I
−10.40170 − 9.18736I 0
u = 0.56898 + 1.29908I
a = 0.101863 − 1.361970I
b = 0.47767 − 2.74977I
−18.8582 + 11.8123I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.56898 − 1.29908I
a = 0.101863 + 1.361970I
b = 0.47767 + 2.74977I
−18.8582 − 11.8123I 0
u = 0.275325 + 0.494049I
a = 1.091150 − 0.485101I
b = −0.900722 − 0.407991I
−6.61511 − 2.32856I −4.23901 + 4.39570I
u = 0.275325 − 0.494049I
a = 1.091150 + 0.485101I
b = −0.900722 + 0.407991I
−6.61511 + 2.32856I −4.23901 − 4.39570I
u = 0.41598 + 1.37365I
a = 0.43080 − 1.36032I
b = 0.56071 − 2.59083I
19.4284 − 1.0070I 0
u = 0.41598 − 1.37365I
a = 0.43080 + 1.36032I
b = 0.56071 + 2.59083I
19.4284 + 1.0070I 0
u = −0.143905 + 0.264400I
a = −0.85171 + 1.55890I
b = 0.261547 + 0.481132I
−0.001965 − 1.039350I −0.15393 + 6.52218I
u = −0.143905 − 0.264400I
a = −0.85171 − 1.55890I
b = 0.261547 − 0.481132I
−0.001965 + 1.039350I −0.15393 − 6.52218I
8
II. I
u
2
= hb
4
− b
3
u − b
3
+ b
2
u − u − 1, a, u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u + 1
a
6
=
u
u
a
10
=
0
b
a
7
=
u
b
2
u + u
a
1
=
−u
−u
a
11
=
bu + b
bu + 2b
a
4
=
0
u
a
9
=
0
b
a
8
=
−b
3
u
−2b
3
u − b
3
+ b
a
12
=
−b
2
− u
−b
3
u − b
3
+ b
2
u − b
2
− 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3b
2
u − 5b
2
+ 5bu + b + 3u + 2
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
4
, c
9
u
8
c
6
, c
10
(u
4
+ u
3
+ u
2
+ 1)
2
c
7
, c
8
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
11
, c
12
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
4
c
4
, c
9
y
8
c
6
, c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
7
, c
8
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0
b = 0.447930 − 0.664845I
0.21101 − 3.44499I 2.00436 + 8.24669I
u = −0.500000 + 0.866025I
a = 0
b = −0.799738 + 0.055496I
0.211005 − 0.614778I −0.99907 − 2.29114I
u = −0.500000 + 0.866025I
a = 0
b = −0.363298 + 1.193330I
−6.79074 − 5.19385I −1.85285 + 5.62657I
u = −0.500000 + 0.866025I
a = 0
b = 1.215110 + 0.282041I
−6.79074 + 1.13408I −5.65243 + 1.40826I
u = −0.500000 − 0.866025I
a = 0
b = 0.447930 + 0.664845I
0.21101 + 3.44499I 2.00436 − 8.24669I
u = −0.500000 − 0.866025I
a = 0
b = −0.799738 − 0.055496I
0.211005 + 0.614778I −0.99907 + 2.29114I
u = −0.500000 − 0.866025I
a = 0
b = −0.363298 − 1.193330I
−6.79074 + 5.19385I −1.85285 − 5.62657I
u = −0.500000 − 0.866025I
a = 0
b = 1.215110 − 0.282041I
−6.79074 − 1.13408I −5.65243 − 1.40826I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
37
+ 23u
36
+ ··· − 25u − 1)
c
2
((u
2
+ u + 1)
4
)(u
37
+ 5u
36
+ ··· + u + 1)
c
3
((u
2
− u + 1)
4
)(u
37
− 5u
36
+ ··· − 7u + 1)
c
4
, c
9
u
8
(u
37
− u
36
+ ··· + 384u + 256)
c
5
((u
2
− u + 1)
4
)(u
37
+ 5u
36
+ ··· + u + 1)
c
6
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
37
+ 3u
36
+ ··· − u + 1)
c
7
, c
8
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
37
+ 3u
36
+ ··· + 3u + 1)
c
10
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
37
− 13u
36
+ ··· − 2707u − 563)
c
11
, c
12
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
37
+ 3u
36
+ ··· + 3u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
37
− 13y
36
+ ··· − 101y −1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
37
+ 23y
36
+ ··· − 25y −1)
c
3
((y
2
+ y + 1)
4
)(y
37
− 49y
36
+ ··· − 25y −1)
c
4
, c
9
y
8
(y
37
− 45y
36
+ ··· + 507904y −65536)
c
6
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
37
− 47y
36
+ ··· − 21y −1)
c
7
, c
8
, c
11
c
12
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
37
+ 45y
36
+ ··· − 21y −1)
c
10
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
37
− 27y
36
+ ··· + 708095y −316969)
14
