12n
0386
(K12n
0386
)
A knot diagram
1
Linearized knot diagam
3 6 8 10 2 12 3 11 4 9 6 7
Solving Sequence
6,11
12
3,7
8 9 2 1 5 10 4
c
11
c
6
c
7
c
8
c
2
c
1
c
5
c
10
c
4
c
3
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
− 3u
10
− 13u
9
+ 39u
8
+ 46u
7
− 146u
6
+ 10u
5
− 54u
4
− 15u
3
− 91u
2
+ 32b + 3u − 1, a − 1,
u
13
− 3u
12
− 14u
11
+ 42u
10
+ 59u
9
− 185u
8
− 36u
7
+ 92u
6
− 25u
5
− 5u
4
+ 18u
3
− 6u
2
− 3u + 1i
I
u
2
= hb
4
− b
2
+ 2, a + 1, u − 1i
I
u
3
= hb − 1, a + 1, u − 1i
I
u
4
= hb + 1, a + 1, u − 1i
I
u
5
= hb − 1, a, u + 1i
I
u
6
= hb, a + 1, u + 1i
I
u
7
= hb
4
+ 1, a + 1, u + 1i
I
v
1
= ha, b − 1, v −1i
* 8 irreducible components of dim
C
= 0, with total 26 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
= hu
11
− 3u
10
+ · · · + 32b − 1, a − 1, u
13
− 3u
12
+ · · · − 3u + 1i
(i) Arc colorings
a
6
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
3
=
1
−0.0312500u
11
+ 0.0937500u
10
+ ··· − 0.0937500u + 0.0312500
a
7
=
−u
−u
3
+ u
a
8
=
0.0312500u
12
− 0.0937500u
11
+ ··· + 0.0937500u
2
− 2.03125u
7
32
u
12
−
11
16
u
11
+ ··· +
5
8
u +
1
32
a
9
=
−0.187500u
12
+ 0.593750u
11
+ ··· − 2.65625u − 0.0312500
7
32
u
12
−
11
16
u
11
+ ··· +
5
8
u +
1
32
a
2
=
1
−0.0312500u
11
+ 0.0937500u
10
+ ··· − 0.0937500u + 0.0312500
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
5
=
u
−0.0312500u
12
+ 0.0937500u
11
+ ··· − 0.0937500u
2
+ 1.03125u
a
10
=
9
16
u
12
−
31
16
u
11
+ ··· − 4u + 2
−0.812500u
12
+ 3.31250u
11
+ ··· + 7.50000u − 2.06250
a
4
=
−0.0312500u
12
+ 0.281250u
11
+ ··· + 0.781250u + 0.750000
−0.218750u
12
+ 1.09375u
11
+ ··· + 2.71875u − 0.812500
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
35
16
u
12
+
145
16
u
11
+
353
16
u
10
−
1973
16
u
9
−
21
2
u
8
+
4023
8
u
7
−
3425
8
u
6
−
643
8
u
5
+
3897
16
u
4
−
2063
16
u
3
+
161
16
u
2
+
651
16
u −
231
8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
13
+ 37u
12
+ ··· + 21u + 1
c
2
, c
5
, c
6
c
11
, c
12
u
13
+ 3u
12
+ ··· − 3u − 1
c
3
, c
7
u
13
− 5u
12
+ ··· + 36u + 26
c
4
, c
9
u
13
+ 3u
12
+ ··· + 8u + 2
c
8
, c
10
u
13
+ 5u
12
+ ··· + 32u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
13
− 237y
12
+ ··· + 173y −1
c
2
, c
5
, c
6
c
11
, c
12
y
13
− 37y
12
+ ··· + 21y −1
c
3
, c
7
y
13
− 65y
12
+ ··· − 5360y −676
c
4
, c
9
y
13
− 5y
12
+ ··· + 32y −4
c
8
, c
10
y
13
+ 7y
12
+ ··· + 480y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.584997 + 0.104914I
a = 1.00000
b = 1.224250 − 0.653734I
−0.88948 + 5.75156I −12.6714 − 7.2274I
u = −0.584997 − 0.104914I
a = 1.00000
b = 1.224250 + 0.653734I
−0.88948 − 5.75156I −12.6714 + 7.2274I
u = 0.140736 + 0.561263I
a = 1.00000
b = −0.773695 + 0.343562I
1.39701 − 2.29590I −8.94612 + 4.81765I
u = 0.140736 − 0.561263I
a = 1.00000
b = −0.773695 − 0.343562I
1.39701 + 2.29590I −8.94612 − 4.81765I
u = −0.571799
a = 1.00000
b = 1.29852
−4.92305 −18.0630
u = 0.530717 + 0.126593I
a = 1.00000
b = 0.900850 + 0.616546I
0.109607 − 0.527741I −11.42537 + 2.37191I
u = 0.530717 − 0.126593I
a = 1.00000
b = 0.900850 − 0.616546I
0.109607 + 0.527741I −11.42537 − 2.37191I
u = 0.297204
a = 1.00000
b = 0.282107
−0.561817 −17.7590
u = 2.49582 + 0.80059I
a = 1.00000
b = 3.34031 + 4.21214I
16.3066 − 9.5421I −15.4017 + 4.1459I
u = 2.49582 − 0.80059I
a = 1.00000
b = 3.34031 − 4.21214I
16.3066 + 9.5421I −15.4017 − 4.1459I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −2.63506 + 0.50378I
a = 1.00000
b = 4.40111 − 2.78969I
18.2542 + 3.1219I −13.76946 − 0.20883I
u = −2.63506 − 0.50378I
a = 1.00000
b = 4.40111 + 2.78969I
18.2542 − 3.1219I −13.76946 + 0.20883I
u = 3.38017
a = 1.00000
b = 9.23373
10.7959 −17.7500
6
II. I
u
2
= hb
4
− b
2
+ 2, a + 1, u − 1i
(i) Arc colorings
a
6
=
0
1
a
11
=
1
0
a
12
=
1
1
a
3
=
−1
b
a
7
=
−1
0
a
8
=
b − 1
−b
2
a
9
=
b
2
+ b − 1
−b
2
a
2
=
−1
b + 1
a
1
=
0
1
a
5
=
1
−b
a
10
=
b
3
− 1
−b
2
+ 2
a
4
=
b
2
− b − 1
−b
3
+ b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b
2
− 20
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
11
c
12
(u − 1)
4
c
2
, c
6
(u + 1)
4
c
3
, c
4
, c
7
c
9
u
4
− u
2
+ 2
c
8
(u
2
− u + 2)
2
c
10
(u
2
+ u + 2)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
11
, c
12
(y −1)
4
c
3
, c
4
, c
7
c
9
(y
2
− y + 2)
2
c
8
, c
10
(y
2
+ 3y + 4)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 0.978318 + 0.676097I
−2.46740 − 5.33349I −18.0000 + 5.2915I
u = 1.00000
a = −1.00000
b = 0.978318 − 0.676097I
−2.46740 + 5.33349I −18.0000 − 5.2915I
u = 1.00000
a = −1.00000
b = −0.978318 + 0.676097I
−2.46740 + 5.33349I −18.0000 − 5.2915I
u = 1.00000
a = −1.00000
b = −0.978318 − 0.676097I
−2.46740 − 5.33349I −18.0000 + 5.2915I
10
III. I
u
3
= hb − 1, a + 1, u − 1i
(i) Arc colorings
a
6
=
0
1
a
11
=
1
0
a
12
=
1
1
a
3
=
−1
1
a
7
=
−1
0
a
8
=
0
−1
a
9
=
1
−1
a
2
=
−1
2
a
1
=
0
1
a
5
=
1
−1
a
10
=
2
−1
a
4
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −24
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
8
, c
11
c
12
u − 1
c
2
, c
6
, c
7
c
9
, c
10
u + 1
12
(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
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 1.00000
−6.57974 −24.0000
14
IV. I
u
4
= hb + 1, a + 1, u − 1i
(i) Arc colorings
a
6
=
0
1
a
11
=
1
0
a
12
=
1
1
a
3
=
−1
−1
a
7
=
−1
0
a
8
=
−2
−1
a
9
=
−1
−1
a
2
=
−1
0
a
1
=
0
1
a
5
=
1
1
a
10
=
0
−1
a
4
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −24
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
7
c
8
, c
9
, c
11
c
12
u − 1
c
2
, c
3
, c
4
c
6
, c
10
u + 1
16
(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
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −1.00000
−6.57974 −24.0000
18
V. I
u
5
= hb − 1, a, u + 1i
(i) Arc colorings
a
6
=
0
−1
a
11
=
1
0
a
12
=
1
1
a
3
=
0
1
a
7
=
1
0
a
8
=
1
1
a
9
=
0
1
a
2
=
0
1
a
1
=
0
1
a
5
=
0
−1
a
10
=
1
−1
a
4
=
−1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = −18
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
c
3
, c
4
, c
6
c
7
, c
9
, c
11
c
12
u − 1
c
8
, c
10
u + 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
c
3
, c
4
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 1.00000
−4.93480 −18.0000
22
VI. I
u
6
= hb, a + 1, u + 1i
(i) Arc colorings
a
6
=
0
−1
a
11
=
1
0
a
12
=
1
1
a
3
=
−1
0
a
7
=
1
0
a
8
=
1
0
a
9
=
1
0
a
2
=
−1
1
a
1
=
0
1
a
5
=
−1
0
a
10
=
1
0
a
4
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
23
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u − 1
c
3
, c
4
, c
7
c
8
, c
9
, c
10
u
c
5
, c
11
, c
12
u + 1
24
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
11
, c
12
y −1
c
3
, c
4
, c
7
c
8
, c
9
, c
10
y
25
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
−3.28987 −12.0000
26
VII. I
u
7
= hb
4
+ 1, a + 1, u + 1i
(i) Arc colorings
a
6
=
0
−1
a
11
=
1
0
a
12
=
1
1
a
3
=
−1
b
a
7
=
1
0
a
8
=
−b + 1
b
2
a
9
=
−b
2
− b + 1
b
2
a
2
=
−1
b + 1
a
1
=
0
1
a
5
=
−1
b
a
10
=
b
3
− b
2
1
a
4
=
b
2
− b − 1
−b
3
+ b
(ii) Obstruction class = 1
(iii) Cusp Shapes = −16
27
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
(u − 1)
4
c
3
, c
4
, c
7
c
9
u
4
+ 1
c
5
, c
11
, c
12
(u + 1)
4
c
8
, c
10
(u
2
+ 1)
2
28
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
11
, c
12
(y −1)
4
c
3
, c
4
, c
7
c
9
(y
2
+ 1)
2
c
8
, c
10
(y + 1)
4
29
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0.707107 + 0.707107I
−1.64493 −16.0000
u = −1.00000
a = −1.00000
b = 0.707107 − 0.707107I
−1.64493 −16.0000
u = −1.00000
a = −1.00000
b = −0.707107 + 0.707107I
−1.64493 −16.0000
u = −1.00000
a = −1.00000
b = −0.707107 − 0.707107I
−1.64493 −16.0000
30
VIII. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
1
0
a
12
=
1
0
a
3
=
0
1
a
7
=
1
0
a
8
=
1
1
a
9
=
0
1
a
2
=
1
1
a
1
=
1
0
a
5
=
0
−1
a
10
=
1
−1
a
4
=
−1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = −18
31
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
, c
10
u + 1
c
2
, c
3
, c
4
c
5
, c
7
, c
9
u − 1
c
6
, c
11
, c
12
u
32
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
8
, c
9
, c
10
y −1
c
6
, c
11
, c
12
y
33
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
−4.93480 −18.0000
34
IX. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u(u − 1)
11
(u + 1)(u
13
+ 37u
12
+ ··· + 21u + 1)
c
2
, c
6
u(u − 1)
6
(u + 1)
6
(u
13
+ 3u
12
+ ··· − 3u − 1)
c
3
, c
7
u(u − 1)
3
(u + 1)(u
4
+ 1)(u
4
− u
2
+ 2)(u
13
− 5u
12
+ ··· + 36u + 26)
c
4
, c
9
u(u − 1)
3
(u + 1)(u
4
+ 1)(u
4
− u
2
+ 2)(u
13
+ 3u
12
+ ··· + 8u + 2)
c
5
, c
11
, c
12
u(u − 1)
7
(u + 1)
5
(u
13
+ 3u
12
+ ··· − 3u − 1)
c
8
u(u − 1)
2
(u + 1)
2
(u
2
+ 1)
2
(u
2
− u + 2)
2
(u
13
+ 5u
12
+ ··· + 32u + 4)
c
10
u(u + 1)
4
(u
2
+ 1)
2
(u
2
+ u + 2)
2
(u
13
+ 5u
12
+ ··· + 32u + 4)
35
X. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y(y −1)
12
(y
13
− 237y
12
+ ··· + 173y −1)
c
2
, c
5
, c
6
c
11
, c
12
y(y −1)
12
(y
13
− 37y
12
+ ··· + 21y −1)
c
3
, c
7
y(y −1)
4
(y
2
+ 1)
2
(y
2
− y + 2)
2
(y
13
− 65y
12
+ ··· − 5360y −676)
c
4
, c
9
y(y −1)
4
(y
2
+ 1)
2
(y
2
− y + 2)
2
(y
13
− 5y
12
+ ··· + 32y −4)
c
8
, c
10
y(y −1)
4
(y + 1)
4
(y
2
+ 3y + 4)
2
(y
13
+ 7y
12
+ ··· + 480y −16)
36
