12n
0478
(K12n
0478
)
A knot diagram
1
Linearized knot diagam
3 6 11 10 2 11 12 1 3 4 8 7
Solving Sequence
8,11 4,12
3 7 1 6 2 5 10 9
c
11
c
3
c
7
c
12
c
6
c
2
c
5
c
10
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−271083u
22
+ 222451u
21
+ ··· + 3670918b − 2822596,
4443679u
22
− 7680194u
21
+ ··· + 11012754a − 1652360, u
23
− 2u
22
+ ··· + u + 3i
I
u
2
= hb, −u
2
+ a − 1, u
3
− u
2
+ 2u − 1i
I
u
3
= h−u
2
a − 2au − 3u
2
+ 5b + a − u − 2, −2u
2
a + a
2
+ 9u
2
− 2a + 7u + 18, u
3
+ u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 32 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−2.71 × 10
5
u
22
+ 2.22 × 10
5
u
21
+ · · · + 3.67 × 10
6
b − 2.82 × 10
6
, 4.44 ×
10
6
u
22
− 7.68 × 10
6
u
21
+ · · · + 1.10 × 10
7
a − 1.65 × 10
6
, u
23
− 2u
22
+ · · · + u + 3i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
4
=
−0.403503u
22
+ 0.697391u
21
+ ··· − 2.67020u + 0.150041
0.0738461u
22
− 0.0605982u
21
+ ··· − 0.383893u + 0.768907
a
12
=
1
−u
2
a
3
=
−0.329657u
22
+ 0.636793u
21
+ ··· − 3.05409u + 0.918948
0.0738461u
22
− 0.0605982u
21
+ ··· − 0.383893u + 0.768907
a
7
=
−u
u
3
+ u
a
1
=
u
2
+ 1
−u
4
− 2u
2
a
6
=
−u
3
− 2u
u
3
+ u
a
2
=
−0.366882u
22
+ 0.480327u
21
+ ··· − 3.32904u + 0.381697
0.00183360u
22
+ 0.212168u
21
+ ··· − 0.461364u + 0.884270
a
5
=
−0.280544u
22
+ 0.210668u
21
+ ··· + 0.417855u − 1.55795
0.0945336u
22
+ 0.274635u
21
+ ··· + 0.828867u + 0.895667
a
10
=
−0.302116u
22
+ 1.00913u
21
+ ··· − 1.05470u + 1.52594
0.382872u
22
− 0.834070u
21
+ ··· + 1.92264u + 0.119951
a
9
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
2296634
1835459
u
22
+
2776211
1835459
u
21
+ ··· +
26517006
1835459
u −
9454146
1835459
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
23
+ 28u
21
+ ··· − 166u + 289
c
2
, c
5
u
23
+ 4u
22
+ ··· − 36u + 17
c
3
, c
4
, c
10
u
23
+ u
22
+ ··· + 16u + 8
c
6
, c
8
u
23
+ 2u
22
+ ··· − 11u + 3
c
7
, c
11
, c
12
u
23
− 2u
22
+ ··· + u + 3
c
9
u
23
− u
22
+ ··· + 41840u + 16424
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
23
+ 56y
22
+ ··· + 17730y −83521
c
2
, c
5
y
23
+ 28y
21
+ ··· − 166y −289
c
3
, c
4
, c
10
y
23
+ 37y
22
+ ··· − 384y −64
c
6
, c
8
y
23
− 38y
22
+ ··· + 235y −9
c
7
, c
11
, c
12
y
23
+ 18y
22
+ ··· + 91y −9
c
9
y
23
+ 121y
22
+ ··· − 5779227136y −269747776
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.264719 + 0.995849I
a = 0.027236 + 0.839660I
b = 0.655248 − 0.393424I
−0.87049 + 1.94619I −1.46427 − 4.12692I
u = 0.264719 − 0.995849I
a = 0.027236 − 0.839660I
b = 0.655248 + 0.393424I
−0.87049 − 1.94619I −1.46427 + 4.12692I
u = 1.044080 + 0.096245I
a = −0.18594 − 4.05927I
b = −0.15025 + 1.91664I
−19.4801 + 4.8275I 2.71629 − 2.13422I
u = 1.044080 − 0.096245I
a = −0.18594 + 4.05927I
b = −0.15025 − 1.91664I
−19.4801 − 4.8275I 2.71629 + 2.13422I
u = −0.120294 + 0.936784I
a = −1.30367 + 2.24060I
b = −0.17471 − 1.45617I
1.85828 − 0.56096I −1.43838 − 0.02221I
u = −0.120294 − 0.936784I
a = −1.30367 − 2.24060I
b = −0.17471 + 1.45617I
1.85828 + 0.56096I −1.43838 + 0.02221I
u = −0.869378 + 0.140608I
a = 0.73050 − 3.00673I
b = −0.40991 + 1.42355I
7.57385 + 1.52871I 4.02521 − 0.99137I
u = −0.869378 − 0.140608I
a = 0.73050 + 3.00673I
b = −0.40991 − 1.42355I
7.57385 − 1.52871I 4.02521 + 0.99137I
u = −0.149742 + 1.181310I
a = −0.396431 − 1.221600I
b = −0.393766 + 0.448599I
−4.34222 − 1.67723I −4.81280 − 1.55068I
u = −0.149742 − 1.181310I
a = −0.396431 + 1.221600I
b = −0.393766 − 0.448599I
−4.34222 + 1.67723I −4.81280 + 1.55068I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.473129 + 1.217800I
a = 0.56482 − 2.13187I
b = 0.582006 + 1.271790I
4.27129 − 6.38038I 0.09560 + 5.13604I
u = −0.473129 − 1.217800I
a = 0.56482 + 2.13187I
b = 0.582006 − 1.271790I
4.27129 + 6.38038I 0.09560 − 5.13604I
u = 0.178779 + 1.353110I
a = −0.400421 − 0.046602I
b = −0.093563 + 0.525227I
−4.07440 + 3.45935I −1.04248 − 6.69109I
u = 0.178779 − 1.353110I
a = −0.400421 + 0.046602I
b = −0.093563 − 0.525227I
−4.07440 − 3.45935I −1.04248 + 6.69109I
u = 0.584239 + 1.282930I
a = −0.96792 − 2.83129I
b = 0.04217 + 1.93464I
16.3629 + 0.9230I 0.444071 − 0.918165I
u = 0.584239 − 1.282930I
a = −0.96792 + 2.83129I
b = 0.04217 − 1.93464I
16.3629 − 0.9230I 0.444071 + 0.918165I
u = −0.302901 + 1.377090I
a = −0.93784 + 1.52418I
b = 0.20153 − 1.48391I
2.69974 − 2.64776I 0.74921 + 1.92747I
u = −0.302901 − 1.377090I
a = −0.93784 − 1.52418I
b = 0.20153 + 1.48391I
2.69974 + 2.64776I 0.74921 − 1.92747I
u = 0.505836 + 0.270726I
a = 0.610789 + 0.807599I
b = −0.138145 − 0.624160I
0.99412 + 1.06305I 3.28669 − 4.14999I
u = 0.505836 − 0.270726I
a = 0.610789 − 0.807599I
b = −0.138145 + 0.624160I
0.99412 − 1.06305I 3.28669 + 4.14999I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.48292 + 1.39630I
a = 1.22794 + 2.66452I
b = 0.21259 − 1.84694I
15.3060 + 10.2657I −0.43766 − 4.56567I
u = 0.48292 − 1.39630I
a = 1.22794 − 2.66452I
b = 0.21259 + 1.84694I
15.3060 − 10.2657I −0.43766 + 4.56567I
u = −0.290242
a = 1.72852
b = 0.333618
−1.11943 −12.2430
7
II. I
u
2
= hb, −u
2
+ a − 1, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
4
=
u
2
+ 1
0
a
12
=
1
−u
2
a
3
=
u
2
+ 1
0
a
7
=
−u
u
2
− u + 1
a
1
=
u
2
+ 1
−u
2
+ u − 1
a
6
=
−u
2
− 1
u
2
− u + 1
a
2
=
2u
2
+ 2
−u
2
+ u − 1
a
5
=
u
2
+ 1
0
a
10
=
1
0
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
2
− 4u + 4
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
9
c
10
u
3
c
5
(u + 1)
3
c
6
, c
8
u
3
− u
2
+ 1
c
7
u
3
+ u
2
+ 2u + 1
c
11
, c
12
u
3
− u
2
+ 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
3
c
3
, c
4
, c
9
c
10
y
3
c
6
, c
8
y
3
− y
2
+ 2y −1
c
7
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −0.662359 + 0.562280I
b = 0
−4.66906 + 2.82812I −6.83447 − 1.85489I
u = 0.215080 − 1.307140I
a = −0.662359 − 0.562280I
b = 0
−4.66906 − 2.82812I −6.83447 + 1.85489I
u = 0.569840
a = 1.32472
b = 0
−0.531480 3.66890
11
III. I
u
3
= h−u
2
a − 2au − 3u
2
+ 5b + a − u − 2, −2u
2
a + a
2
+ 9u
2
− 2a + 7u +
18, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
4
=
a
1
5
u
2
a +
3
5
u
2
+ ··· −
1
5
a +
2
5
a
12
=
1
−u
2
a
3
=
1
5
u
2
a +
3
5
u
2
+ ··· +
4
5
a +
2
5
1
5
u
2
a +
3
5
u
2
+ ··· −
1
5
a +
2
5
a
7
=
−u
−u
2
− u − 1
a
1
=
u
2
+ 1
−u
2
− u − 1
a
6
=
u
2
+ 1
−u
2
− u − 1
a
2
=
1
5
u
2
a +
8
5
u
2
+ ··· +
4
5
a +
7
5
1
5
u
2
a −
2
5
u
2
+ ··· −
1
5
a −
3
5
a
5
=
−
1
5
u
2
a −
3
5
u
2
+ ··· −
4
5
a −
2
5
−
1
5
u
2
a −
3
5
u
2
+ ··· +
1
5
a −
2
5
a
10
=
3
5
u
2
a −
11
5
u
2
+ ··· +
2
5
a −
29
5
2
a
9
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
+ 4u + 4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
6
c
2
(u + 1)
6
c
3
, c
4
, c
9
c
10
(u
2
+ 2)
3
c
6
, c
8
(u
3
+ u
2
− 1)
2
c
7
(u
3
− u
2
+ 2u − 1)
2
c
11
, c
12
(u
3
+ u
2
+ 2u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
9
c
10
(y + 2)
6
c
6
, c
8
(y
3
− y
2
+ 2y −1)
2
c
7
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −1.71575 + 1.02526I
b = − 1.414210I
0.26574 − 2.82812I −3.50976 + 2.97945I
u = −0.215080 + 1.307140I
a = 0.39103 − 2.14982I
b = 1.414210I
0.26574 − 2.82812I −3.50976 + 2.97945I
u = −0.215080 − 1.307140I
a = −1.71575 − 1.02526I
b = 1.414210I
0.26574 + 2.82812I −3.50976 − 2.97945I
u = −0.215080 − 1.307140I
a = 0.39103 + 2.14982I
b = − 1.414210I
0.26574 + 2.82812I −3.50976 − 2.97945I
u = −0.569840
a = 1.32472 + 3.89599I
b = − 1.414210I
4.40332 3.01950
u = −0.569840
a = 1.32472 − 3.89599I
b = 1.414210I
4.40332 3.01950
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
23
+ 28u
21
+ ··· − 166u + 289)
c
2
((u − 1)
3
)(u + 1)
6
(u
23
+ 4u
22
+ ··· − 36u + 17)
c
3
, c
4
, c
10
u
3
(u
2
+ 2)
3
(u
23
+ u
22
+ ··· + 16u + 8)
c
5
((u − 1)
6
)(u + 1)
3
(u
23
+ 4u
22
+ ··· − 36u + 17)
c
6
, c
8
(u
3
− u
2
+ 1)(u
3
+ u
2
− 1)
2
(u
23
+ 2u
22
+ ··· − 11u + 3)
c
7
((u
3
− u
2
+ 2u − 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
23
− 2u
22
+ ··· + u + 3)
c
9
u
3
(u
2
+ 2)
3
(u
23
− u
22
+ ··· + 41840u + 16424)
c
11
, c
12
(u
3
− u
2
+ 2u − 1)(u
3
+ u
2
+ 2u + 1)
2
(u
23
− 2u
22
+ ··· + u + 3)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
23
+ 56y
22
+ ··· + 17730y −83521)
c
2
, c
5
((y −1)
9
)(y
23
+ 28y
21
+ ··· − 166y −289)
c
3
, c
4
, c
10
y
3
(y + 2)
6
(y
23
+ 37y
22
+ ··· − 384y −64)
c
6
, c
8
((y
3
− y
2
+ 2y −1)
3
)(y
23
− 38y
22
+ ··· + 235y −9)
c
7
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
23
+ 18y
22
+ ··· + 91y −9)
c
9
y
3
(y + 2)
6
(y
23
+ 121y
22
+ ··· − 5.77923 × 10
9
y −2.69748 × 10
8
)
17
