12n
0233
(K12n
0233
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 10 11 12 3 5 6 7 9
Solving Sequence
6,10
11 7 12
3,5
2 1 4 9 8
c
10
c
6
c
11
c
5
c
2
c
1
c
4
c
9
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
16
− 22u
14
+ ··· + b − 2, −u
16
+ u
15
+ ··· + a − 3u, u
17
+ 2u
16
+ ··· − u − 1i
I
u
2
= hb − u, u
2
+ a − 2, u
3
− u
2
− 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 20 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
= h2u
16
−22u
14
+· · ·+b−2, −u
16
+u
15
+· · ·+a−3u, u
17
+2u
16
+· · ·−u−1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
u
16
− u
15
+ ··· − 9u
2
+ 3u
−2u
16
+ 22u
14
+ ··· + u + 2
a
5
=
−u
u
a
2
=
−u
15
+ 10u
13
+ ··· + 3u + 1
−u
16
+ 11u
14
+ ··· + u + 1
a
1
=
u
8
− 5u
6
+ 7u
4
− 4u
2
+ 1
−u
8
+ 4u
6
− 2u
4
− 2u
2
a
4
=
u
16
+ u
15
+ ··· − 4u − 1
u
6
− 4u
4
+ 3u
2
a
9
=
−u
2
+ 1
u
2
a
8
=
−u
3
+ 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shap es = −u
16
+ 11u
14
− 2u
13
− 49u
12
+ 17u
11
+ 116u
10
− 50u
9
− 163u
8
+
56u
7
+ 146u
6
− 2u
5
− 88u
4
− 47u
3
+ 32u
2
+ 26u + 7
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
+ 22u
16
+ ··· + 120u + 1
c
2
, c
4
u
17
− 4u
16
+ ··· + 12u − 1
c
3
, c
8
u
17
+ u
16
+ ··· + 20u − 8
c
5
, c
6
, c
7
c
9
, c
10
, c
11
u
17
+ 2u
16
+ ··· − u − 1
c
12
u
17
+ 18u
15
+ ··· + u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
− 50y
16
+ ··· + 11308y −1
c
2
, c
4
y
17
− 22y
16
+ ··· + 120y −1
c
3
, c
8
y
17
+ 21y
16
+ ··· + 656y −64
c
5
, c
6
, c
7
c
9
, c
10
, c
11
y
17
− 24y
16
+ ··· + 3y −1
c
12
y
17
+ 36y
16
+ ··· + 3y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.982279 + 0.151832I
a = 0.168994 − 0.909972I
b = −0.73475 + 1.67481I
1.92118 + 2.31259I 9.54984 − 4.08399I
u = 0.982279 − 0.151832I
a = 0.168994 + 0.909972I
b = −0.73475 − 1.67481I
1.92118 − 2.31259I 9.54984 + 4.08399I
u = −0.865450
a = 1.36121
b = 0.158508
0.152271 10.7240
u = 1.134090 + 0.377025I
a = −0.290069 + 0.747188I
b = 1.58149 − 1.34476I
−6.34608 + 5.60143I 7.40158 − 3.76696I
u = 1.134090 − 0.377025I
a = −0.290069 − 0.747188I
b = 1.58149 + 1.34476I
−6.34608 − 5.60143I 7.40158 + 3.76696I
u = −1.19563
a = −0.435884
b = 0.101957
5.72559 17.0640
u = −0.374547 + 0.647974I
a = −0.529092 − 1.284900I
b = −0.554996 − 0.754476I
−11.06150 − 2.09782I 3.78406 + 2.85716I
u = −0.374547 − 0.647974I
a = −0.529092 + 1.284900I
b = −0.554996 + 0.754476I
−11.06150 + 2.09782I 3.78406 − 2.85716I
u = 0.373542
a = 0.533717
b = −0.273837
0.571638 17.3900
u = −0.161903 + 0.300607I
a = 0.05405 + 2.11440I
b = 0.442012 + 0.466984I
−1.59083 − 0.74897I −0.41320 + 3.78790I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.161903 − 0.300607I
a = 0.05405 − 2.11440I
b = 0.442012 − 0.466984I
−1.59083 + 0.74897I −0.41320 − 3.78790I
u = 1.70139
a = 0.253107
b = −1.16331
9.39467 9.61340
u = −1.72231 + 0.03669I
a = −0.59162 − 2.45052I
b = 0.95501 + 2.99216I
11.62350 − 3.05566I 10.30225 + 2.57182I
u = −1.72231 − 0.03669I
a = −0.59162 + 2.45052I
b = 0.95501 − 2.99216I
11.62350 + 3.05566I 10.30225 − 2.57182I
u = −1.75947 + 0.10503I
a = 1.37142 + 1.82000I
b = −2.29897 − 2.04908I
3.97600 − 7.68149I 8.50094 + 3.18214I
u = −1.75947 − 0.10503I
a = 1.37142 − 1.82000I
b = −2.29897 + 2.04908I
3.97600 + 7.68149I 8.50094 − 3.18214I
u = 1.78986
a = −0.0795251
b = 0.397105
16.7200 17.9570
6
II. I
u
2
= hb − u, u
2
+ a − 2, u
3
− u
2
− 2u + 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
2
− u + 1
a
12
=
−u
2
+ 1
u
2
+ u − 1
a
3
=
−u
2
+ 2
u
a
5
=
−u
u
a
2
=
−u
2
+ u + 2
0
a
1
=
u
−u
a
4
=
−u
2
+ 2
u
a
9
=
−u
2
+ 1
u
2
a
8
=
−u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
− u + 5
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
7
u
3
+ u
2
− 2u − 1
c
9
, c
10
, 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
2
, c
4
(y −1)
3
c
3
, c
8
y
3
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = 0.445042
b = −1.24698
4.69981 7.80190
u = 0.445042
a = 1.80194
b = 0.445042
−0.939962 4.75300
u = 1.80194
a = −1.24698
b = 1.80194
15.9794 6.44500
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
17
+ 22u
16
+ ··· + 120u + 1)
c
2
((u − 1)
3
)(u
17
− 4u
16
+ ··· + 12u − 1)
c
3
, c
8
u
3
(u
17
+ u
16
+ ··· + 20u − 8)
c
4
((u + 1)
3
)(u
17
− 4u
16
+ ··· + 12u − 1)
c
5
, c
6
, c
7
(u
3
+ u
2
− 2u − 1)(u
17
+ 2u
16
+ ··· − u − 1)
c
9
, c
10
, c
11
(u
3
− u
2
− 2u + 1)(u
17
+ 2u
16
+ ··· − u − 1)
c
12
(u
3
− u
2
− 2u + 1)(u
17
+ 18u
15
+ ··· + u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
3
)(y
17
− 50y
16
+ ··· + 11308y −1)
c
2
, c
4
((y −1)
3
)(y
17
− 22y
16
+ ··· + 120y −1)
c
3
, c
8
y
3
(y
17
+ 21y
16
+ ··· + 656y −64)
c
5
, c
6
, c
7
c
9
, c
10
, c
11
(y
3
− 5y
2
+ 6y −1)(y
17
− 24y
16
+ ··· + 3y −1)
c
12
(y
3
− 5y
2
+ 6y −1)(y
17
+ 36y
16
+ ··· + 3y −1)
12
