12n
0096
(K12n
0096
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 8 4 11 5 12 7 10 9
Solving Sequence
7,11 4,8
3 6 5 2 1 10 12 9
c
7
c
3
c
6
c
5
c
2
c
1
c
10
c
11
c
9
c
4
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−4915u
16
+ 20254u
15
+ ··· + 7156b + 9167, −123805u
16
+ 530300u
15
+ ··· + 7156a + 154261,
u
17
− 5u
16
+ ··· − 9u + 1i
I
u
2
= hu
2
+ b, a + u + 2, u
3
+ u
2
− 1i
I
u
3
= hb, 3u
4
− u
3
− u
2
+ a + 3u + 4, u
5
− u
4
+ u
2
+ u − 1i
I
u
4
= h−3u
2
a − 2au − 4u
2
+ 5b − a − u + 2, a
2
+ 2u
2
+ a + 2u, u
3
+ u
2
− 1i
* 4 irreducible components of dim
C
= 0, with total 31 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−4915u
16
+ 20254u
15
+ · · · + 7156b + 9167, −1.24 × 10
5
u
16
+ 5.30 ×
10
5
u
15
+ · · · + 7156a + 1.54 × 10
5
, u
17
− 5u
16
+ · · · − 9u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
17.3009u
16
− 74.1056u
15
+ ··· + 179.512u − 21.5569
0.686836u
16
− 2.83035u
15
+ ··· + 5.34754u − 1.28102
a
8
=
1
u
2
a
3
=
17.9877u
16
− 76.9360u
15
+ ··· + 184.860u − 22.8379
0.686836u
16
− 2.83035u
15
+ ··· + 5.34754u − 1.28102
a
6
=
−4.26593u
16
+ 18.3215u
15
+ ··· − 48.9090u + 8.75545
−0.297652u
16
+ 1.56051u
15
+ ··· − 5.66867u + 0.378144
a
5
=
−2.46297u
16
+ 10.6539u
15
+ ··· − 31.7707u + 6.12549
0.563164u
16
− 2.41965u
15
+ ··· + 4.65246u − 0.968977
a
2
=
18.8118u
16
− 80.6283u
15
+ ··· + 197.318u − 25.5869
0.563164u
16
− 2.41965u
15
+ ··· + 4.65246u − 0.968977
a
1
=
u
7
+ 2u
3
u
7
− u
5
+ 2u
3
− u
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
440209
1789
u
16
+
7557829
7156
u
15
+ ··· −
18828293
7156
u +
602132
1789
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
+ 25u
16
+ ··· + 349u + 1
c
2
, c
4
u
17
− 9u
16
+ ··· − 23u − 1
c
3
, c
6
u
17
− 4u
16
+ ··· − 808u
2
+ 32
c
5
, c
8
u
17
− 9u
16
+ ··· + 1536u + 512
c
7
, c
10
u
17
+ 5u
16
+ ··· − 9u − 1
c
9
, c
11
, c
12
u
17
+ 9u
16
+ ··· + 19u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
− 57y
16
+ ··· + 110253y − 1
c
2
, c
4
y
17
− 25y
16
+ ··· + 349y − 1
c
3
, c
6
y
17
− 48y
16
+ ··· + 51712y − 1024
c
5
, c
8
y
17
− 49y
16
+ ··· + 4063232y − 262144
c
7
, c
10
y
17
− 9y
16
+ ··· + 19y − 1
c
9
, c
11
, c
12
y
17
+ 3y
16
+ ··· − 149y − 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.838900 + 0.274006I
a = −0.541331 + 1.242290I
b = 0.022802 + 1.171600I
1.70067 + 3.40197I −9.83492 − 9.12548I
u = −0.838900 − 0.274006I
a = −0.541331 − 1.242290I
b = 0.022802 − 1.171600I
1.70067 − 3.40197I −9.83492 + 9.12548I
u = −0.888050 + 0.699587I
a = 0.963662 + 0.607402I
b = 0.599165 + 0.085043I
2.22871 + 2.69541I −2.36795 − 0.48316I
u = −0.888050 − 0.699587I
a = 0.963662 − 0.607402I
b = 0.599165 − 0.085043I
2.22871 − 2.69541I −2.36795 + 0.48316I
u = 0.702958
a = 8.08327
b = 0.212168
−2.60036 −117.690
u = 0.638788 + 1.195210I
a = −0.615352 + 0.257656I
b = 2.18853 − 1.40547I
−10.83360 + 4.83632I −9.98493 − 0.99160I
u = 0.638788 − 1.195210I
a = −0.615352 − 0.257656I
b = 2.18853 + 1.40547I
−10.83360 − 4.83632I −9.98493 + 0.99160I
u = 0.932524 + 0.992733I
a = 0.024900 + 0.723358I
b = −1.316500 − 0.396988I
8.46454 − 3.58781I −11.43442 + 3.20089I
u = 0.932524 − 0.992733I
a = 0.024900 − 0.723358I
b = −1.316500 + 0.396988I
8.46454 + 3.58781I −11.43442 − 3.20089I
u = 0.596043
a = 0.723994
b = −0.233910
−0.842519 −11.7040
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.18905 + 0.84637I
a = 1.44006 − 1.34507I
b = 1.69244 + 1.67258I
−12.6169 − 12.0697I −10.79554 + 4.97668I
u = 1.18905 − 0.84637I
a = 1.44006 + 1.34507I
b = 1.69244 − 1.67258I
−12.6169 + 12.0697I −10.79554 − 4.97668I
u = 1.45050 + 0.33561I
a = −1.67908 + 1.23512I
b = −2.95952 + 0.58336I
−4.97113 − 2.77667I −12.19520 + 1.72835I
u = 1.45050 − 0.33561I
a = −1.67908 − 1.23512I
b = −2.95952 − 0.58336I
−4.97113 + 2.77667I −12.19520 − 1.72835I
u = 0.248594 + 0.150644I
a = 1.39946 − 0.69681I
b = −0.634067 + 0.017100I
−0.943827 + 0.013133I −9.47910 + 0.58994I
u = 0.248594 − 0.150644I
a = 1.39946 + 0.69681I
b = −0.634067 − 0.017100I
−0.943827 − 0.013133I −9.47910 − 0.58994I
u = −1.76401
a = 2.20808
b = 4.83602
19.2915 −12.4240
6
II. I
u
2
= hu
2
+ b, a + u + 2, u
3
+ u
2
− 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
−u − 2
−u
2
a
8
=
1
u
2
a
3
=
−u
2
− u − 2
−u
2
a
6
=
−u
2
−u
2
− u + 1
a
5
=
−u
2
−u
2
− u + 1
a
2
=
−2u
2
− u − 1
−u
2
− u + 1
a
1
=
−1
0
a
10
=
u
u
a
12
=
u
2
− 1
u
2
+ u − 1
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
2
− 5u − 14
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
9
u
3
− u
2
+ 2u − 1
c
2
, c
7
u
3
+ u
2
− 1
c
4
, c
10
u
3
− u
2
+ 1
c
5
, c
8
u
3
c
6
, c
11
, c
12
u
3
+ u
2
+ 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
9
, c
11
, c
12
y
3
+ 3y
2
+ 2y − 1
c
2
, c
4
, c
7
c
10
y
3
− y
2
+ 2y − 1
c
5
, c
8
y
3
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −1.122560 − 0.744862I
b = −0.215080 + 1.307140I
6.04826 + 5.65624I −9.18265 − 6.33859I
u = −0.877439 − 0.744862I
a = −1.122560 + 0.744862I
b = −0.215080 − 1.307140I
6.04826 − 5.65624I −9.18265 + 6.33859I
u = 0.754878
a = −2.75488
b = −0.569840
−2.22691 −16.6350
10
III. I
u
3
= hb, 3u
4
− u
3
− u
2
+ a + 3u + 4, u
5
− u
4
+ u
2
+ u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
−3u
4
+ u
3
+ u
2
− 3u − 4
0
a
8
=
1
u
2
a
3
=
−3u
4
+ u
3
+ u
2
− 3u − 4
0
a
6
=
1
0
a
5
=
−u
2
+ 1
−u
4
a
2
=
−3u
4
+ u
3
+ 2u
2
− 3u − 5
u
4
a
1
=
u
2
− 1
u
4
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
9
=
u
4
− u
2
+ 1
u
4
− u
3
− u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 10u
4
− 7u
3
+ u
2
+ 10u + 7
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
6
u
5
c
4
(u + 1)
5
c
5
, c
9
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
7
u
5
− u
4
+ u
2
+ u − 1
c
8
, c
11
, c
12
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
10
u
5
+ u
4
− u
2
+ u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
5
c
3
, c
6
y
5
c
5
, c
8
, c
9
c
11
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1
c
7
, c
10
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.758138 + 0.584034I
a = 1.036940 − 0.588205I
b = 0
0.17487 + 2.21397I −10.02401 − 4.83884I
u = −0.758138 − 0.584034I
a = 1.036940 + 0.588205I
b = 0
0.17487 − 2.21397I −10.02401 + 4.83884I
u = 0.935538 + 0.903908I
a = 0.348360 + 0.023996I
b = 0
9.31336 − 3.33174I −1.83654 + 1.25445I
u = 0.935538 − 0.903908I
a = 0.348360 − 0.023996I
b = 0
9.31336 + 3.33174I −1.83654 − 1.25445I
u = 0.645200
a = −5.77061
b = 0
−2.52712 13.7210
14
IV. I
u
4
= h−3u
2
a − 2au − 4u
2
+ 5b − a − u + 2, a
2
+ 2u
2
+ a + 2u, u
3
+ u
2
− 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
a
3
5
u
2
a +
4
5
u
2
+ ··· +
1
5
a −
2
5
a
8
=
1
u
2
a
3
=
3
5
u
2
a +
4
5
u
2
+ ··· +
6
5
a −
2
5
3
5
u
2
a +
4
5
u
2
+ ··· +
1
5
a −
2
5
a
6
=
−
1
5
u
2
a +
2
5
u
2
+ ··· +
3
5
a +
9
5
1
5
u
2
a +
3
5
u
2
+ ··· +
2
5
a +
6
5
a
5
=
−
1
5
u
2
a +
2
5
u
2
+ ··· +
3
5
a +
9
5
1
5
u
2
a +
3
5
u
2
+ ··· +
2
5
a +
6
5
a
2
=
u
2
+ a + 2u + 1
1
5
u
2
a +
3
5
u
2
+ ··· +
2
5
a +
6
5
a
1
=
−1
0
a
10
=
u
u
a
12
=
u
2
− 1
u
2
+ u − 1
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
13
5
u
2
a −
17
5
au +
26
5
u
2
+
29
5
a +
24
5
u −
58
5
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
9
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
7
(u
3
+ u
2
− 1)
2
c
4
, c
10
(u
3
− u
2
+ 1)
2
c
5
, c
8
u
6
c
6
, c
11
, c
12
(u
3
+ u
2
+ 2u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
9
, c
11
, c
12
(y
3
+ 3y
2
+ 2y − 1)
2
c
2
, c
4
, c
7
c
10
(y
3
− y
2
+ 2y − 1)
2
c
5
, c
8
y
6
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.824718 + 0.424452I
b = −0.215080 − 1.307140I
6.04826 −8.27833 + 0.98317I
u = −0.877439 + 0.744862I
a = −1.82472 − 0.42445I
b = −0.569840
1.91067 + 2.82812I −29.3323 − 8.2928I
u = −0.877439 − 0.744862I
a = 0.824718 − 0.424452I
b = −0.215080 + 1.307140I
6.04826 −8.27833 − 0.98317I
u = −0.877439 − 0.744862I
a = −1.82472 + 0.42445I
b = −0.569840
1.91067 − 2.82812I −29.3323 + 8.2928I
u = 0.754878
a = −0.50000 + 1.54901I
b = −0.215080 + 1.307140I
1.91067 + 2.82812I −5.88933 + 2.71361I
u = 0.754878
a = −0.50000 − 1.54901I
b = −0.215080 − 1.307140I
1.91067 − 2.82812I −5.88933 − 2.71361I
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
3
− u
2
+ 2u − 1)
3
(u
17
+ 25u
16
+ ··· + 349u + 1)
c
2
((u − 1)
5
)(u
3
+ u
2
− 1)
3
(u
17
− 9u
16
+ ··· − 23u − 1)
c
3
u
5
(u
3
− u
2
+ 2u − 1)
3
(u
17
− 4u
16
+ ··· − 808u
2
+ 32)
c
4
((u + 1)
5
)(u
3
− u
2
+ 1)
3
(u
17
− 9u
16
+ ··· − 23u − 1)
c
5
u
9
(u
5
− u
4
+ ··· + 3u − 1)(u
17
− 9u
16
+ ··· + 1536u + 512)
c
6
u
5
(u
3
+ u
2
+ 2u + 1)
3
(u
17
− 4u
16
+ ··· − 808u
2
+ 32)
c
7
((u
3
+ u
2
− 1)
3
)(u
5
− u
4
+ u
2
+ u − 1)(u
17
+ 5u
16
+ ··· − 9u − 1)
c
8
u
9
(u
5
+ u
4
+ ··· + 3u + 1)(u
17
− 9u
16
+ ··· + 1536u + 512)
c
9
(u
3
− u
2
+ 2u − 1)
3
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
· (u
17
+ 9u
16
+ ··· + 19u + 1)
c
10
((u
3
− u
2
+ 1)
3
)(u
5
+ u
4
− u
2
+ u + 1)(u
17
+ 5u
16
+ ··· − 9u − 1)
c
11
, c
12
(u
3
+ u
2
+ 2u + 1)
3
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
· (u
17
+ 9u
16
+ ··· + 19u + 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
5
)(y
3
+ 3y
2
+ 2y − 1)
3
(y
17
− 57y
16
+ ··· + 110253y − 1)
c
2
, c
4
((y − 1)
5
)(y
3
− y
2
+ 2y − 1)
3
(y
17
− 25y
16
+ ··· + 349y − 1)
c
3
, c
6
y
5
(y
3
+ 3y
2
+ 2y − 1)
3
(y
17
− 48y
16
+ ··· + 51712y − 1024)
c
5
, c
8
y
9
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)
· (y
17
− 49y
16
+ ··· + 4063232y − 262144)
c
7
, c
10
(y
3
− y
2
+ 2y − 1)
3
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1)
· (y
17
− 9y
16
+ ··· + 19y − 1)
c
9
, c
11
, c
12
(y
3
+ 3y
2
+ 2y − 1)
3
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)
· (y
17
+ 3y
16
+ ··· − 149y − 1)
20
