12a
0580
(K12a
0580
)
A knot diagram
1
Linearized knot diagam
3 7 9 10 11 2 1 12 4 5 6 8
Solving Sequence
5,10
11 6 12 4 9 3 8 1 2 7
c
10
c
5
c
11
c
4
c
9
c
3
c
8
c
12
c
1
c
7
c
2
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
33
− 23u
31
+ ··· − 3u
2
+ 1i
I
u
2
= hu − 1i
* 2 irreducible components of dim
C
= 0, with total 34 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
33
− 23u
31
+ · · · − 3u
2
+ 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
−u
u
a
9
=
−u
2
+ 1
u
2
a
3
=
u
3
− 2u
−u
3
+ u
a
8
=
u
8
− 5u
6
+ 7u
4
− 4u
2
+ 1
−u
10
+ 6u
8
− 11u
6
+ 6u
4
+ u
2
a
1
=
−u
14
+ 9u
12
− 30u
10
+ 47u
8
− 38u
6
+ 16u
4
− 4u
2
+ 1
u
16
− 10u
14
+ 38u
12
− 68u
10
+ 56u
8
− 14u
6
− 2u
4
− 2u
2
a
2
=
−u
22
+ 15u
20
+ ··· − 6u
2
+ 1
u
22
− 14u
20
+ ··· − 12u
4
− u
2
a
7
=
u
20
− 13u
18
+ ··· − 7u
2
+ 1
−u
22
+ 14u
20
+ ··· + 12u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
−4u
31
+88u
29
−856u
27
+4848u
25
−4u
24
−17720u
23
+68u
22
+43792u
21
−492u
20
−74544u
19
+
1976u
18
+ 87444u
17
−4820u
16
−69912u
15
+ 7344u
14
+ 37972u
13
−6924u
12
−15152u
11
+
3884u
10
+ 5628u
9
−1284u
8
−1920u
7
+ 364u
6
+ 380u
5
−124u
4
−64u
3
+ 12u
2
+ 16u + 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
33
+ 18u
32
+ ··· + 6u + 1
c
2
, c
6
u
33
− 9u
31
+ ··· − 2u + 1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
33
− 23u
31
+ ··· − 3u
2
+ 1
c
7
, c
8
, c
12
u
33
− 3u
32
+ ··· + 66u − 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
33
− 6y
32
+ ··· − 10y −1
c
2
, c
6
y
33
− 18y
32
+ ··· + 6y −1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y
33
− 46y
32
+ ··· + 6y −1
c
7
, c
8
, c
12
y
33
+ 33y
32
+ ··· + 1818y −81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.078530 + 0.260345I
−3.08176 + 0.16732I 6.09584 + 0.50750I
u = −1.078530 − 0.260345I
−3.08176 − 0.16732I 6.09584 − 0.50750I
u = 1.110390 + 0.241958I
0.82041 + 4.14448I 10.07295 − 3.71315I
u = 1.110390 − 0.241958I
0.82041 − 4.14448I 10.07295 + 3.71315I
u = −1.128410 + 0.263489I
−2.55219 − 8.92303I 6.95631 + 6.78209I
u = −1.128410 − 0.263489I
−2.55219 + 8.92303I 6.95631 − 6.78209I
u = −1.166230 + 0.050255I
5.91535 − 0.56111I 14.6924 + 0.1759I
u = −1.166230 − 0.050255I
5.91535 + 0.56111I 14.6924 − 0.1759I
u = 1.171240 + 0.120548I
4.72112 + 4.72894I 11.48738 − 6.71856I
u = 1.171240 − 0.120548I
4.72112 − 4.72894I 11.48738 + 6.71856I
u = 0.375181 + 0.517251I
−7.29542 + 6.26206I 2.41928 − 6.99571I
u = 0.375181 − 0.517251I
−7.29542 − 6.26206I 2.41928 + 6.99571I
u = 0.320762 + 0.524473I
−7.45714 − 2.83006I 1.68987 − 0.19716I
u = 0.320762 − 0.524473I
−7.45714 + 2.83006I 1.68987 + 0.19716I
u = −0.346601 + 0.496167I
−3.76561 − 1.64301I 5.31561 + 3.87855I
u = −0.346601 − 0.496167I
−3.76561 + 1.64301I 5.31561 − 3.87855I
u = −0.450226 + 0.303526I
−0.48021 − 3.32430I 7.27824 + 9.56553I
u = −0.450226 − 0.303526I
−0.48021 + 3.32430I 7.27824 − 9.56553I
u = 0.443763 + 0.077336I
0.744274 + 0.060227I 13.86304 − 1.24873I
u = 0.443763 − 0.077336I
0.744274 − 0.060227I 13.86304 + 1.24873I
u = −0.129735 + 0.357401I
−1.43361 + 1.06619I 0.406261 − 0.591044I
u = −0.129735 − 0.357401I
−1.43361 − 1.06619I 0.406261 + 0.591044I
u = −1.74291
11.6408 0
u = 1.74733 + 0.06051I
7.06644 + 1.13358I 0
u = 1.74733 − 0.06051I
7.06644 − 1.13358I 0
u = −1.75819 + 0.05873I
11.17330 − 5.39711I 0
u = −1.75819 − 0.05873I
11.17330 + 5.39711I 0
u = 1.76174 + 0.06588I
7.87109 + 10.31650I 0
u = 1.76174 − 0.06588I
7.87109 − 10.31650I 0
u = 1.77307 + 0.01258I
16.6439 + 0.8345I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.77307 − 0.01258I
16.6439 − 0.8345I 0
u = −1.77410 + 0.02777I
15.4601 − 5.3571I 0
u = −1.77410 − 0.02777I
15.4601 + 5.3571I 0
6
II. I
u
2
= hu − 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
1
0
a
11
=
1
−1
a
6
=
1
0
a
12
=
0
−1
a
4
=
−1
1
a
9
=
0
1
a
3
=
−1
0
a
8
=
0
1
a
1
=
0
−1
a
2
=
1
−1
a
7
=
0
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u + 1
c
2
, c
3
, c
4
c
5
, c
6
, c
9
c
10
, c
11
u − 1
c
7
, c
8
, c
12
u
8
(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
9
, c
10
, c
11
y −1
c
7
, c
8
, c
12
y
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
1.64493 6.00000
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
33
+ 18u
32
+ ··· + 6u + 1)
c
2
, c
6
(u − 1)(u
33
− 9u
31
+ ··· − 2u + 1)
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(u − 1)(u
33
− 23u
31
+ ··· − 3u
2
+ 1)
c
7
, c
8
, c
12
u(u
33
− 3u
32
+ ··· + 66u − 9)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
33
− 6y
32
+ ··· − 10y −1)
c
2
, c
6
(y −1)(y
33
− 18y
32
+ ··· + 6y −1)
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(y −1)(y
33
− 46y
32
+ ··· + 6y −1)
c
7
, c
8
, c
12
y(y
33
+ 33y
32
+ ··· + 1818y −81)
12
