12n
0721
(K12n
0721
)
A knot diagram
1
Linearized knot diagam
4 6 7 9 2 3 11 12 4 7 8 10
Solving Sequence
7,11
8
4,12
3 6 2 5 10 1 9
c
7
c
11
c
3
c
6
c
2
c
5
c
10
c
12
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
15
− 2u
14
+ 7u
13
+ 12u
12
− 22u
11
− 21u
10
+ 47u
9
− 66u
7
+ 33u
6
+ 38u
5
− 32u
4
+ 4u
3
+ 11u
2
+ 2b − u,
5u
15
+ 10u
14
+ ··· + 2a − 4, u
16
+ 3u
15
+ ··· − 6u
2
− 1i
I
u
2
= hb + u − 1, a − u + 1, u
2
− u − 1i
I
u
3
= hb − u, a + u, u
2
− u − 1i
* 3 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
=
h−u
15
−2u
14
+· · ·+2b−u, 5u
15
+10u
14
+· · ·+2a−4, u
16
+3u
15
+· · ·−6u
2
−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
4
=
−
5
2
u
15
− 5u
14
+ ··· +
7
2
u + 2
1
2
u
15
+ u
14
+ ··· −
11
2
u
2
+
1
2
u
a
12
=
u
−u
3
+ u
a
3
=
−2u
15
− 4u
14
+ ··· + 4u + 2
1
2
u
15
+ u
14
+ ··· −
11
2
u
2
+
1
2
u
a
6
=
u
3
− 2u
−
1
2
u
15
− u
14
+ ··· +
7
2
u
2
+
1
2
u
a
2
=
−
1
2
u
15
− u
14
+ ··· +
7
2
u + 1
1
2
u
15
+ u
14
+ ··· −
7
2
u
2
+
1
2
u
a
5
=
1
2
u
15
+ u
14
+ ··· −
7
2
u − 1
−
5
2
u
15
− 4u
14
+ ··· −
1
2
u + 1
a
10
=
−u
u
a
1
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
a
9
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1
2
u
15
−
7
2
u
13
+ 3u
12
+ 10u
11
−
39
2
u
10
−
19
2
u
9
+ 45u
8
− 21u
7
−
77
2
u
6
+ 59u
5
− 8u
4
− 31u
3
+
57
2
u
2
−
27
2
u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
− 15u
15
+ ··· − 1082u − 31
c
2
, c
3
, c
5
c
6
u
16
+ 3u
15
+ ··· + 6u − 1
c
4
, c
9
u
16
+ u
15
+ ··· + 16u + 16
c
7
, c
8
, c
10
c
11
u
16
− 3u
15
+ ··· − 6u
2
− 1
c
12
u
16
− u
15
+ ··· + 10u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 85y
15
+ ··· − 746024y + 961
c
2
, c
3
, c
5
c
6
y
16
− 25y
15
+ ··· − 68y + 1
c
4
, c
9
y
16
+ 25y
15
+ ··· − 2176y + 256
c
7
, c
8
, c
10
c
11
y
16
− 17y
15
+ ··· + 12y + 1
c
12
y
16
+ 43y
15
+ ··· − 72y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.588394 + 0.904691I
a = −0.036403 + 0.941230I
b = 1.85781 − 0.03762I
18.9500 + 2.9686I −2.68910 − 2.21532I
u = 0.588394 − 0.904691I
a = −0.036403 − 0.941230I
b = 1.85781 + 0.03762I
18.9500 − 2.9686I −2.68910 + 2.21532I
u = 1.15773
a = 0.686514
b = −1.71495
−6.77980 2.42280
u = 0.383322 + 0.651485I
a = −0.022997 − 1.230160I
b = −1.41341 + 0.15665I
−8.11407 + 1.96040I −3.80773 − 2.97128I
u = 0.383322 − 0.651485I
a = −0.022997 + 1.230160I
b = −1.41341 − 0.15665I
−8.11407 − 1.96040I −3.80773 + 2.97128I
u = −1.329780 + 0.108886I
a = −0.33314 − 1.59285I
b = 0.429972 + 0.619424I
3.19833 − 2.02641I 3.60242 + 3.44848I
u = −1.329780 − 0.108886I
a = −0.33314 + 1.59285I
b = 0.429972 − 0.619424I
3.19833 + 2.02641I 3.60242 − 3.44848I
u = 1.44035
a = −0.317757
b = 1.11690
3.33662 2.15710
u = −0.527680
a = −0.542315
b = −0.135163
0.784966 13.2500
u = −1.45613 + 0.27551I
a = 0.88578 + 1.40121I
b = −1.259310 − 0.357655I
−2.20746 − 5.42559I 0.36298 + 3.83626I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.45613 − 0.27551I
a = 0.88578 − 1.40121I
b = −1.259310 + 0.357655I
−2.20746 + 5.42559I 0.36298 − 3.83626I
u = 1.59819
a = 0.0934299
b = −0.387958
8.26971 17.0160
u = −1.59142 + 0.33831I
a = −1.16055 − 1.16099I
b = 1.81685 + 0.10280I
−13.4441 − 7.6018I −0.05971 + 3.02517I
u = −1.59142 − 0.33831I
a = −1.16055 + 1.16099I
b = 1.81685 − 0.10280I
−13.4441 + 7.6018I −0.05971 − 3.02517I
u = 0.071327 + 0.313314I
a = 0.70738 + 1.71306I
b = 0.628677 − 0.215988I
−1.188400 + 0.433304I −5.83178 − 2.04218I
u = 0.071327 − 0.313314I
a = 0.70738 − 1.71306I
b = 0.628677 + 0.215988I
−1.188400 − 0.433304I −5.83178 + 2.04218I
6
II. I
u
2
= hb + u − 1, a − u + 1, u
2
− u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u − 1
a
4
=
u − 1
−u + 1
a
12
=
u
−u − 1
a
3
=
0
−u + 1
a
6
=
1
u − 2
a
2
=
−u + 1
u − 2
a
5
=
u − 1
−u + 1
a
10
=
−u
u
a
1
=
−1
0
a
9
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
10
, c
11
, c
12
u
2
+ u − 1
c
4
, c
9
u
2
c
5
, c
6
, c
7
c
8
u
2
− u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
4
, c
9
y
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −1.61803
b = 1.61803
−7.89568 −5.00000
u = 1.61803
a = 0.618034
b = −0.618034
7.89568 −5.00000
10
III. I
u
3
= hb − u, a + u, u
2
− u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u − 1
a
4
=
−u
u
a
12
=
u
−u − 1
a
3
=
0
u
a
6
=
1
−u − 1
a
2
=
u
−u − 1
a
5
=
−u
u
a
10
=
−u
u
a
1
=
−1
0
a
9
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
10
, c
11
, c
12
u
2
+ u − 1
c
4
, c
9
u
2
c
5
, c
6
, c
7
c
8
u
2
− u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
4
, c
9
y
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.618034
b = −0.618034
0 0
u = 1.61803
a = −1.61803
b = 1.61803
0 0
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u − 1)
2
)(u
16
− 15u
15
+ ··· − 1082u − 31)
c
2
, c
3
((u
2
+ u − 1)
2
)(u
16
+ 3u
15
+ ··· + 6u − 1)
c
4
, c
9
u
4
(u
16
+ u
15
+ ··· + 16u + 16)
c
5
, c
6
((u
2
− u − 1)
2
)(u
16
+ 3u
15
+ ··· + 6u − 1)
c
7
, c
8
((u
2
− u − 1)
2
)(u
16
− 3u
15
+ ··· − 6u
2
− 1)
c
10
, c
11
((u
2
+ u − 1)
2
)(u
16
− 3u
15
+ ··· − 6u
2
− 1)
c
12
((u
2
+ u − 1)
2
)(u
16
− u
15
+ ··· + 10u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
− 3y + 1)
2
)(y
16
− 85y
15
+ ··· − 746024y + 961)
c
2
, c
3
, c
5
c
6
((y
2
− 3y + 1)
2
)(y
16
− 25y
15
+ ··· − 68y + 1)
c
4
, c
9
y
4
(y
16
+ 25y
15
+ ··· − 2176y + 256)
c
7
, c
8
, c
10
c
11
((y
2
− 3y + 1)
2
)(y
16
− 17y
15
+ ··· + 12y + 1)
c
12
((y
2
− 3y + 1)
2
)(y
16
+ 43y
15
+ ··· − 72y + 1)
16
