12n
0571
(K12n
0571
)
A knot diagram
1
Linearized knot diagam
3 7 9 10 8 2 12 3 4 5 1 8
Solving Sequence
3,9
4 10 5 8
6,12
1 7 2 11
c
3
c
9
c
4
c
8
c
5
c
12
c
7
c
2
c
11
c
1
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
9
− u
8
+ 6u
7
+ 4u
6
− 12u
5
− 5u
4
+ 7u
3
+ 2u
2
+ b − u + 1,
3u
9
+ 3u
8
− 17u
7
− 12u
6
+ 30u
5
+ 15u
4
− 11u
3
− 6u
2
+ 2a − u − 4,
u
10
+ 3u
9
− 3u
8
− 14u
7
+ 21u
5
+ 7u
4
− 8u
3
− 3u
2
− 2i
I
u
2
= hb + 1, a
2
+ 3u
2
− 3a − u − 6, u
3
− 3u − 1i
I
u
3
= hb − u + 1, 3a + 4u − 3, u
2
− 3i
I
u
4
= hb + 1, a − 2, u − 1i
I
u
5
= hb + 1, a − 1, u − 1i
I
u
6
= hb, a + 1, u + 1i
I
u
7
= hb + 2, a − 3, u − 1i
I
v
1
= ha, b + 1, v + 1i
* 8 irreducible components of dim
C
= 0, with total 23 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
9
−u
8
+· · · +b + 1, 3u
9
+3u
8
+· · · +2a − 4, u
10
+3u
9
+· · · −3u
2
−2i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
10
=
u
−u
3
+ u
a
5
=
−u
2
+ 1
u
4
− 2u
2
a
8
=
−u
u
a
6
=
u
6
− 3u
4
+ 1
−u
6
+ 4u
4
− 3u
2
a
12
=
−
3
2
u
9
−
3
2
u
8
+ ··· +
1
2
u + 2
u
9
+ u
8
− 6u
7
− 4u
6
+ 12u
5
+ 5u
4
− 7u
3
− 2u
2
+ u − 1
a
1
=
1
2
u
9
+
1
2
u
8
+ ··· − u
2
+
3
2
u
−u
9
− u
8
+ 5u
7
+ 4u
6
− 7u
5
− 5u
4
+ u
3
+ 2u
2
+ 1
a
7
=
−
3
2
u
9
−
3
2
u
8
+ ··· −
3
2
u + 1
u
9
+ u
8
− 5u
7
− 3u
6
+ 8u
5
+ 2u
4
− 3u
3
+ u − 1
a
2
=
3
2
u
9
+
3
2
u
8
+ ··· +
3
2
u − 1
−u
9
− u
8
+ 5u
7
+ 4u
6
− 7u
5
− 5u
4
+ u
3
+ 2u
2
+ 1
a
11
=
−u
3
+ 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
7
− 12u
5
+ 20u
3
− 8u + 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
10
+ u
9
+ 12u
8
+ 5u
7
+ 41u
6
+ 7u
5
+ 39u
4
+ 15u
3
+ 2u
2
+ 4u + 1
c
2
, c
6
, c
7
c
12
u
10
− u
9
+ u
7
+ 5u
6
− 3u
5
+ u
4
+ 3u
3
+ 2u
2
− 1
c
3
, c
4
, c
8
c
9
, c
10
u
10
+ 3u
9
− 3u
8
− 14u
7
+ 21u
5
+ 7u
4
− 8u
3
− 3u
2
− 2
c
5
u
10
+ 15u
9
+ ··· + 340u + 142
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
10
+ 23y
9
+ ··· − 12y + 1
c
2
, c
6
, c
7
c
12
y
10
− y
9
+ 12y
8
− 5y
7
+ 41y
6
− 7y
5
+ 39y
4
− 15y
3
+ 2y
2
− 4y + 1
c
3
, c
4
, c
8
c
9
, c
10
y
10
− 15y
9
+ ··· + 12y + 4
c
5
y
10
− 39y
9
+ ··· − 13076y + 20164
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.785856 + 0.428779I
a = −1.37380 − 0.39872I
b = 1.149750 − 0.803732I
2.82913 − 4.20392I 4.59667 + 6.41727I
u = −0.785856 − 0.428779I
a = −1.37380 + 0.39872I
b = 1.149750 + 0.803732I
2.82913 + 4.20392I 4.59667 − 6.41727I
u = 0.884247
a = 1.30093
b = −0.719291
1.75072 5.11200
u = 1.42487 + 0.24108I
a = −2.18633 − 0.04391I
b = 1.60933 + 0.82750I
10.16070 + 6.73545I 5.08933 − 4.78933I
u = 1.42487 − 0.24108I
a = −2.18633 + 0.04391I
b = 1.60933 − 0.82750I
10.16070 − 6.73545I 5.08933 + 4.78933I
u = 0.129165 + 0.461050I
a = 0.444664 + 0.542292I
b = 0.527577 + 0.512570I
0.055292 + 1.193090I 1.05588 − 5.24459I
u = 0.129165 − 0.461050I
a = 0.444664 − 0.542292I
b = 0.527577 − 0.512570I
0.055292 − 1.193090I 1.05588 + 5.24459I
u = −1.71307
a = 1.96680
b = −1.57906
11.1577 7.60720
u = −1.85377 + 0.06834I
a = −2.51839 + 0.18104I
b = 1.86251 − 0.75572I
−17.0320 − 8.3664I 4.89852 + 3.81014I
u = −1.85377 − 0.06834I
a = −2.51839 − 0.18104I
b = 1.86251 + 0.75572I
−17.0320 + 8.3664I 4.89852 − 3.81014I
5
II. I
u
2
= hb + 1, a
2
+ 3u
2
− 3a − u − 6, u
3
− 3u − 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
10
=
u
−2u − 1
a
5
=
−u
2
+ 1
u
2
+ u
a
8
=
−u
u
a
6
=
3u + 2
−2u − 1
a
12
=
a
−1
a
1
=
−u
2
a + u
2
+ a
u
2
a − u
2
− 1
a
7
=
2au + u
2
− 4u − 3
−au + 2u
a
2
=
−2u
2
a + 2u
2
+ a + 1
u
2
a − u
2
− 1
a
11
=
−u − 1
u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
6
+ 6u
4
+ 10u
3
+ 9u
2
+ 18u + 9
c
2
, c
6
, c
7
c
12
u
6
+ 2u
3
+ 3u
2
− 3
c
3
, c
4
, c
8
c
9
, c
10
(u
3
− 3u − 1)
2
c
5
(u
3
− 6u
2
+ 3u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
6
+ 12y
5
+ 54y
4
+ 26y
3
− 171y
2
− 162y + 81
c
2
, c
6
, c
7
c
12
y
6
+ 6y
4
− 10y
3
+ 9y
2
− 18y + 9
c
3
, c
4
, c
8
c
9
, c
10
(y
3
− 6y
2
+ 9y −1)
2
c
5
(y
3
− 30y
2
+ 21y −1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.53209
a = 1.50000 + 0.56919I
b = −1.00000
10.4179 6.00000
u = −1.53209
a = 1.50000 − 0.56919I
b = −1.00000
10.4179 6.00000
u = −0.347296
a = −1.24606
b = −1.00000
−2.74156 6.00000
u = −0.347296
a = 4.24606
b = −1.00000
−2.74156 6.00000
u = 1.87939
a = 1.50000 + 0.68329I
b = −1.00000
−15.9010 6.00000
u = 1.87939
a = 1.50000 − 0.68329I
b = −1.00000
−15.9010 6.00000
9
III. I
u
3
= hb − u + 1, 3a + 4u − 3, u
2
− 3i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−3
a
10
=
u
−2u
a
5
=
−2
3
a
8
=
−u
u
a
6
=
1
0
a
12
=
−
4
3
u + 1
u − 1
a
1
=
−
1
3
u + 1
−1
a
7
=
1
3
u − 1
1
a
2
=
−
1
3
u + 2
−1
a
11
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
11
(u − 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
2
− 3
c
6
, c
12
(u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y −3)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.73205
a = −1.30940
b = 0.732051
9.86960 0
u = −1.73205
a = 3.30940
b = −2.73205
9.86960 0
13
IV. I
u
4
= hb + 1, a − 2, u − 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
1
a
4
=
1
−1
a
10
=
1
0
a
5
=
0
−1
a
8
=
−1
1
a
6
=
−1
0
a
12
=
2
−1
a
1
=
1
0
a
7
=
1
0
a
2
=
1
0
a
11
=
1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
c
3
, c
4
, c
7
c
8
, c
9
, c
10
c
12
u − 1
c
5
, c
11
u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y
c
3
, c
4
, c
5
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 2.00000
b = −1.00000
1.64493 6.00000
17
V. I
u
5
= hb + 1, a − 1, u − 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
1
a
4
=
1
−1
a
10
=
1
0
a
5
=
0
−1
a
8
=
−1
1
a
6
=
−1
0
a
12
=
1
−1
a
1
=
1
−1
a
7
=
−1
1
a
2
=
2
−1
a
11
=
1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u + 1
c
2
, c
3
, c
4
c
6
, c
8
, c
9
c
10
u − 1
c
7
, c
11
, c
12
u
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
9
, c
10
y −1
c
7
, c
11
, c
12
y
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = −1.00000
1.64493 6.00000
21
VI. I
u
6
= hb, a + 1, u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
−1
a
4
=
1
−1
a
10
=
−1
0
a
5
=
0
−1
a
8
=
1
−1
a
6
=
−1
0
a
12
=
−1
0
a
1
=
0
−1
a
7
=
0
−1
a
2
=
1
−1
a
11
=
−1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
8
, c
9
, c
10
c
11
, c
12
u − 1
c
2
, c
3
, c
4
c
7
u + 1
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
0 0
25
VII. I
u
7
= hb + 2, a − 3, u − 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
1
a
4
=
1
−1
a
10
=
1
0
a
5
=
0
−1
a
8
=
−1
1
a
6
=
−1
0
a
12
=
3
−2
a
1
=
2
−1
a
7
=
2
−1
a
2
=
3
−1
a
11
=
1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
26
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
11
, c
12
u − 1
c
2
, c
5
, c
7
c
8
, c
9
, c
10
u + 1
27
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
28
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 3.00000
b = −2.00000
0 0
29
VIII. I
v
1
= ha, b + 1, v + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
−1
0
a
4
=
1
0
a
10
=
−1
0
a
5
=
1
0
a
8
=
−1
0
a
6
=
1
0
a
12
=
0
−1
a
1
=
1
−1
a
7
=
−1
1
a
2
=
2
−1
a
11
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
30
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
11
u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
c
6
, c
12
u + 1
31
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
y −1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
32
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
33
IX. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
11
u(u − 1)
5
(u + 1)(u
6
+ 6u
4
+ 10u
3
+ 9u
2
+ 18u + 9)
· (u
10
+ u
9
+ 12u
8
+ 5u
7
+ 41u
6
+ 7u
5
+ 39u
4
+ 15u
3
+ 2u
2
+ 4u + 1)
c
2
, c
7
u(u − 1)
4
(u + 1)
2
(u
6
+ 2u
3
+ 3u
2
− 3)
· (u
10
− u
9
+ u
7
+ 5u
6
− 3u
5
+ u
4
+ 3u
3
+ 2u
2
− 1)
c
3
, c
4
, c
8
c
9
, c
10
u(u − 1)
3
(u + 1)(u
2
− 3)(u
3
− 3u − 1)
2
· (u
10
+ 3u
9
− 3u
8
− 14u
7
+ 21u
5
+ 7u
4
− 8u
3
− 3u
2
− 2)
c
5
u(u − 1)(u + 1)
3
(u
2
− 3)(u
3
− 6u
2
+ 3u + 1)
2
· (u
10
+ 15u
9
+ ··· + 340u + 142)
c
6
, c
12
u(u − 1)
3
(u + 1)
3
(u
6
+ 2u
3
+ 3u
2
− 3)
· (u
10
− u
9
+ u
7
+ 5u
6
− 3u
5
+ u
4
+ 3u
3
+ 2u
2
− 1)
34
X. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
11
y(y −1)
6
(y
6
+ 12y
5
+ 54y
4
+ 26y
3
− 171y
2
− 162y + 81)
· (y
10
+ 23y
9
+ ··· − 12y + 1)
c
2
, c
6
, c
7
c
12
y(y −1)
6
(y
6
+ 6y
4
− 10y
3
+ 9y
2
− 18y + 9)
· (y
10
− y
9
+ 12y
8
− 5y
7
+ 41y
6
− 7y
5
+ 39y
4
− 15y
3
+ 2y
2
− 4y + 1)
c
3
, c
4
, c
8
c
9
, c
10
y(y −3)
2
(y −1)
4
(y
3
− 6y
2
+ 9y −1)
2
(y
10
− 15y
9
+ ··· + 12y + 4)
c
5
y(y −3)
2
(y −1)
4
(y
3
− 30y
2
+ 21y −1)
2
· (y
10
− 39y
9
+ ··· − 13076y + 20164)
35
