12n
0582
(K12n
0582
)
A knot diagram
1
Linearized knot diagam
3 7 12 11 10 2 12 1 3 5 4 8
Solving Sequence
3,12 4,8
1 9 7 2 6 11 5 10
c
3
c
12
c
8
c
7
c
2
c
6
c
11
c
4
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
7
+ u
6
− 6u
5
+ 4u
4
− 9u
3
+ 3u
2
+ 2b − 2u, −u
8
− 7u
6
− 2u
5
− 13u
4
− 10u
3
− 3u
2
+ 4a − 10u + 2,
u
9
− u
8
+ 9u
7
− 5u
6
+ 25u
5
− 3u
4
+ 23u
3
+ 7u
2
+ 6u + 2i
I
u
2
= hb − u − 1, 3a − u, u
2
+ 3i
I
u
3
= hb + u + 1, a + u, u
2
+ 1i
I
v
1
= ha, b − 1, v −1i
* 4 irreducible components of dim
C
= 0, with total 14 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
7
+ u
6
− 6u
5
+ 4u
4
− 9u
3
+ 3u
2
+ 2b − 2u, −u
8
− 7u
6
+ · · · +
4a + 2, u
9
− u
8
+ · · · + 6u + 2i
(i) Arc colorings
a
3
=
1
0
a
12
=
0
u
a
4
=
1
u
2
a
8
=
1
4
u
8
+
7
4
u
6
+ ··· +
5
2
u −
1
2
1
2
u
7
−
1
2
u
6
+ ··· −
3
2
u
2
+ u
a
1
=
−
1
4
u
8
−
7
4
u
6
+ ··· −
3
2
u −
1
2
−
1
4
u
8
−
5
4
u
6
+ ··· −
5
4
u
2
−
1
2
a
9
=
u
5
+ 4u
3
+ 3u
u
5
+ 3u
3
+ u
a
7
=
1
4
u
8
+
7
4
u
6
+ ··· +
5
2
u −
1
2
−
1
4
u
8
−
9
4
u
6
+ ··· − u −
1
2
a
2
=
−
1
2
u
6
+
1
2
u
5
+ ··· −
7
2
u
2
−
3
2
u
−
1
4
u
8
−
5
4
u
6
+ ··· −
5
4
u
2
−
1
2
a
6
=
−u
4
− 3u
2
− 1
−u
6
− 4u
4
− 3u
2
a
11
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
10
=
u
3
+ 2u
u
5
+ 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
8
+ 2u
7
− 18u
6
+ 10u
5
− 50u
4
+ 6u
3
− 46u
2
− 12u − 12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
− 4u
8
+ ··· + 145u + 64
c
2
, c
6
u
9
− 2u
8
+ 4u
7
− 6u
6
+ 18u
5
− 2u
4
+ 8u
3
− 6u
2
− 7u + 8
c
3
, c
4
, c
5
c
10
, c
11
u
9
− u
8
+ 9u
7
− 5u
6
+ 25u
5
− 3u
4
+ 23u
3
+ 7u
2
+ 6u + 2
c
7
, c
8
, c
12
u
9
+ 2u
8
− 8u
7
− 18u
6
+ 18u
5
+ 50u
4
+ 4u
3
− 34u
2
+ 9u + 8
c
9
u
9
− 19u
8
+ ··· − 654u + 82
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
+ 40y
8
+ ··· + 6177y −4096
c
2
, c
6
y
9
+ 4y
8
+ ··· + 145y −64
c
3
, c
4
, c
5
c
10
, c
11
y
9
+ 17y
8
+ ··· + 8y −4
c
7
, c
8
, c
12
y
9
− 20y
8
+ ··· + 625y −64
c
9
y
9
+ 17y
8
+ ··· − 49688y −6724
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.286210 + 1.260340I
a = 0.755885 − 0.774032I
b = 0.258411 − 0.633815I
6.78979 + 0.44094I 1.257807 − 0.497446I
u = 0.286210 − 1.260340I
a = 0.755885 + 0.774032I
b = 0.258411 + 0.633815I
6.78979 − 0.44094I 1.257807 + 0.497446I
u = −0.064698 + 0.563024I
a = −0.501501 + 1.062470I
b = 0.419693 − 0.038751I
0.80178 + 1.43893I −0.93515 − 5.88586I
u = −0.064698 − 0.563024I
a = −0.501501 − 1.062470I
b = 0.419693 + 0.038751I
0.80178 − 1.43893I −0.93515 + 5.88586I
u = −0.312120
a = −1.25225
b = −0.623552
−0.900968 −13.4400
u = 0.30797 + 1.73568I
a = −0.412318 + 1.116980I
b = −0.65742 + 2.01585I
17.2800 − 2.5995I 1.01236 + 1.23711I
u = 0.30797 − 1.73568I
a = −0.412318 − 1.116980I
b = −0.65742 − 2.01585I
17.2800 + 2.5995I 1.01236 − 1.23711I
u = 0.12658 + 1.95644I
a = −0.215942 − 1.161020I
b = 0.29109 − 2.91331I
−7.97178 − 5.85424I 0.38489 + 1.87688I
u = 0.12658 − 1.95644I
a = −0.215942 + 1.161020I
b = 0.29109 + 2.91331I
−7.97178 + 5.85424I 0.38489 − 1.87688I
5
II. I
u
2
= hb − u − 1, 3a − u, u
2
+ 3i
(i) Arc colorings
a
3
=
1
0
a
12
=
0
u
a
4
=
1
−3
a
8
=
1
3
u
u + 1
a
1
=
−
1
3
u
−1
a
9
=
0
u
a
7
=
1
3
u
1
a
2
=
−
1
3
u + 1
−1
a
6
=
1
0
a
11
=
u
−2u
a
5
=
−2
3
a
10
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
(u − 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
2
+ 3
c
6
, c
7
, c
8
(u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
8
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(y + 3)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.73205I
a = 0.577350I
b = 1.00000 + 1.73205I
13.1595 0
u = − 1.73205I
a = − 0.577350I
b = 1.00000 − 1.73205I
13.1595 0
9
III. I
u
3
= hb + u + 1, a + u, u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
12
=
0
u
a
4
=
1
−1
a
8
=
−u
−u − 1
a
1
=
−u
−1
a
9
=
0
−u
a
7
=
−u
−1
a
2
=
−u + 1
−1
a
6
=
−1
0
a
11
=
u
0
a
5
=
0
−1
a
10
=
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
6
, c
7
c
8
(u − 1)
2
c
2
, c
12
(u + 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
2
+ 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
8
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(y + 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = − 1.000000I
b = −1.00000 − 1.00000I
3.28987 0
u = − 1.000000I
a = 1.000000I
b = −1.00000 + 1.00000I
3.28987 0
13
IV. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
3
=
1
0
a
12
=
1
0
a
4
=
1
0
a
8
=
0
1
a
1
=
1
−1
a
9
=
1
0
a
7
=
−1
1
a
2
=
2
−1
a
6
=
1
0
a
11
=
1
0
a
5
=
1
0
a
10
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
u − 1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
c
6
, c
7
, c
8
u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
8
, c
12
y −1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
9
− 4u
8
+ ··· + 145u + 64)
c
2
(u − 1)
3
(u + 1)
2
· (u
9
− 2u
8
+ 4u
7
− 6u
6
+ 18u
5
− 2u
4
+ 8u
3
− 6u
2
− 7u + 8)
c
3
, c
4
, c
5
c
10
, c
11
u(u
2
+ 1)(u
2
+ 3)(u
9
− u
8
+ ··· + 6u + 2)
c
6
(u − 1)
2
(u + 1)
3
· (u
9
− 2u
8
+ 4u
7
− 6u
6
+ 18u
5
− 2u
4
+ 8u
3
− 6u
2
− 7u + 8)
c
7
, c
8
(u − 1)
2
(u + 1)
3
· (u
9
+ 2u
8
− 8u
7
− 18u
6
+ 18u
5
+ 50u
4
+ 4u
3
− 34u
2
+ 9u + 8)
c
9
u(u
2
+ 1)(u
2
+ 3)(u
9
− 19u
8
+ ··· − 654u + 82)
c
12
(u − 1)
3
(u + 1)
2
· (u
9
+ 2u
8
− 8u
7
− 18u
6
+ 18u
5
+ 50u
4
+ 4u
3
− 34u
2
+ 9u + 8)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
9
+ 40y
8
+ ··· + 6177y −4096)
c
2
, c
6
((y −1)
5
)(y
9
+ 4y
8
+ ··· + 145y −64)
c
3
, c
4
, c
5
c
10
, c
11
y(y + 1)
2
(y + 3)
2
(y
9
+ 17y
8
+ ··· + 8y −4)
c
7
, c
8
, c
12
((y −1)
5
)(y
9
− 20y
8
+ ··· + 625y −64)
c
9
y(y + 1)
2
(y + 3)
2
(y
9
+ 17y
8
+ ··· − 49688y −6724)
19
