12n
0576
(K12n
0576
)
A knot diagram
1
Linearized knot diagam
3 7 11 10 9 2 12 3 4 5 7 8
Solving Sequence
5,10 7,11
12 4 3 2 1 6 9 8
c
10
c
11
c
4
c
3
c
2
c
1
c
6
c
9
c
8
c
5
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
14
+ 3u
13
+ 9u
12
− 9u
11
− 20u
10
+ 4u
9
+ 22u
8
+ 18u
7
− 3u
6
− 24u
5
− 17u
4
+ 4u
3
+ 5u
2
+ b + 10u + 3,
− 3u
14
+ 5u
13
+ 12u
12
− 15u
11
− 24u
10
+ 8u
9
+ 24u
8
+ 25u
7
− 34u
5
− 24u
4
+ 5u
3
+ 5u
2
+ 2a + 17u + 4,
u
15
− 3u
14
− 2u
13
+ 11u
12
+ 2u
11
− 16u
10
− 6u
9
+ 7u
8
+ 14u
7
+ 8u
6
− 10u
5
− 13u
4
+ u
3
− u
2
+ 6u + 2i
I
u
2
= h−u
3
a − u
2
a − u
3
+ au − u
2
+ b − a + u, 2u
4
a + 2u
3
a + 2u
4
− 3u
2
a + 2u
3
+ a
2
− 2au − 3u
2
+ a − 3u + 1,
u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
I
u
3
= hu
3
+ 2u
2
+ b − 2u − 2, 2u
3
+ 3u
2
+ 3a − 3u − 3, u
4
− 3u
2
+ 3i
I
u
4
= hu
3
+ b, u
2
+ a + u − 1, u
4
− u
2
− 1i
I
v
1
= ha, b + 1, v + 1i
* 5 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
=
h−2u
14
+3u
13
+· · ·+b+3, −3u
14
+5u
13
+· · ·+2a+4, u
15
−3u
14
+· · ·+6u+2i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
7
=
3
2
u
14
−
5
2
u
13
+ ··· −
17
2
u − 2
2u
14
− 3u
13
+ ··· − 10u − 3
a
11
=
1
u
2
a
12
=
1
2
u
14
−
1
2
u
13
+ ··· −
5
2
u
2
−
7
2
u
2u
14
− 3u
13
+ ··· − 11u − 3
a
4
=
u
u
a
3
=
−u
3
+ 2u
−u
5
+ u
3
+ u
a
2
=
1
2
u
14
−
1
2
u
13
+ ··· −
3
2
u − 1
2u
14
− 3u
13
+ ··· − 10u − 3
a
1
=
3
2
u
14
−
3
2
u
13
+ ··· −
19
2
u − 3
7u
14
− 11u
13
+ ··· − 39u − 11
a
6
=
−u
5
+ 2u
3
− u
−u
5
+ u
3
+ u
a
9
=
−u
2
+ 1
−u
2
a
8
=
u
10
− 5u
8
+ 8u
6
− 3u
4
− 3u
2
+ 1
u
12
− 4u
10
+ 4u
8
+ 2u
6
− 3u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
12
+ 10u
10
+ 6u
9
− 18u
8
− 24u
7
+ 2u
6
+ 30u
5
+ 30u
4
+ 2u
3
− 22u
2
− 20u − 24
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
+ 27u
14
+ ··· + 21u + 1
c
2
, c
6
, c
7
c
11
, c
12
u
15
− u
14
+ ··· − 3u − 1
c
3
, c
5
u
15
+ 9u
14
+ ··· + 166u + 22
c
4
, c
9
, c
10
u
15
− 3u
14
+ ··· + 6u + 2
c
8
u
15
+ 3u
14
+ ··· + 326u + 178
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
− 91y
14
+ ··· + 125y −1
c
2
, c
6
, c
7
c
11
, c
12
y
15
− 27y
14
+ ··· + 21y −1
c
3
, c
5
y
15
+ 7y
14
+ ··· + 5336y −484
c
4
, c
9
, c
10
y
15
− 13y
14
+ ··· + 40y −4
c
8
y
15
− 73y
14
+ ··· − 18680y −31684
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.933456 + 0.545096I
a = 1.35570 − 0.53655I
b = 2.15361 − 1.54045I
−16.1665 − 1.0397I −16.7131 − 1.0217I
u = −0.933456 − 0.545096I
a = 1.35570 + 0.53655I
b = 2.15361 + 1.54045I
−16.1665 + 1.0397I −16.7131 + 1.0217I
u = −0.269872 + 0.870864I
a = −2.11208 + 1.63104I
b = 0.313757 + 0.153473I
−14.1086 + 6.0067I −14.4990 − 3.2830I
u = −0.269872 − 0.870864I
a = −2.11208 − 1.63104I
b = 0.313757 − 0.153473I
−14.1086 − 6.0067I −14.4990 + 3.2830I
u = −0.027957 + 0.721725I
a = 0.449087 − 1.012280I
b = 0.210760 + 0.156059I
2.42887 + 1.37514I −7.68727 − 5.21222I
u = −0.027957 − 0.721725I
a = 0.449087 + 1.012280I
b = 0.210760 − 0.156059I
2.42887 − 1.37514I −7.68727 + 5.21222I
u = −1.269950 + 0.268466I
a = −0.329692 − 0.304561I
b = −1.118860 − 0.755791I
−1.39715 + 2.17673I −12.08651 + 0.76556I
u = −1.269950 − 0.268466I
a = −0.329692 + 0.304561I
b = −1.118860 + 0.755791I
−1.39715 − 2.17673I −12.08651 − 0.76556I
u = 1.31650
a = −0.714575
b = −0.728053
−5.38119 −18.1900
u = 1.289830 + 0.310526I
a = 0.355754 − 0.895009I
b = 0.71766 − 1.39362I
−1.68505 − 5.12171I −12.9962 + 7.9827I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.289830 − 0.310526I
a = 0.355754 + 0.895009I
b = 0.71766 + 1.39362I
−1.68505 + 5.12171I −12.9962 − 7.9827I
u = 1.43481 + 0.35867I
a = 0.18363 + 1.96597I
b = 0.25436 + 4.36591I
−19.5375 − 10.4475I −18.0265 + 4.5376I
u = 1.43481 − 0.35867I
a = 0.18363 − 1.96597I
b = 0.25436 − 4.36591I
−19.5375 + 10.4475I −18.0265 − 4.5376I
u = 1.53735
a = 0.453716
b = −1.08285
14.6944 −19.9400
u = −0.300672
a = 0.456072
b = −0.251678
−0.497687 −19.8530
6
II. I
u
2
= h−u
3
a − u
2
a − u
3
+ au − u
2
+ b − a + u, 2u
4
a + 2u
4
+ · · · + a +
1, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
7
=
a
u
3
a + u
2
a + u
3
− au + u
2
+ a − u
a
11
=
1
u
2
a
12
=
−u
4
a − u
3
a + u
2
a − 2u
3
+ au − u
2
+ 3u + 1
−u
4
a + u
4
+ u
2
a − u
3
− u
2
+ 2u
a
4
=
u
u
a
3
=
−u
3
+ 2u
u
4
− u
3
− u
2
+ 2u − 1
a
2
=
u
3
a + u
2
a + u
3
− au + u
2
− u
u
3
a + u
2
a + u
3
− au + u
2
+ a − u
a
1
=
−u
4
a + u
3
a + 3u
2
a + 2u
3
− au + u
2
− 3u + 1
−2u
4
a + 2u
3
a − 2u
4
+ 4u
2
a + 2u
3
− 3au + 4u
2
+ 2a − 4u + 2
a
6
=
u
4
− u
2
− 1
u
4
− u
3
− u
2
+ 2u − 1
a
9
=
−u
2
+ 1
−u
2
a
8
=
−2u
2
+ 2
−2u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
+ 8u − 18
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
+ 13u
9
+ ··· + 536u + 49
c
2
, c
6
, c
7
c
11
, c
12
u
10
− u
9
− 6u
8
+ 4u
7
+ 18u
6
− 8u
5
− 31u
4
+ 13u
3
+ 28u
2
− 12u − 7
c
3
, c
5
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
4
, c
9
, c
10
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
8
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 9y
9
+ ··· − 137356y + 2401
c
2
, c
6
, c
7
c
11
, c
12
y
10
− 13y
9
+ ··· − 536y + 49
c
3
, c
5
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
c
4
, c
9
, c
10
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
8
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.21774
a = −0.591829
b = 0.253452
−5.69095 −15.4810
u = 1.21774
a = −1.53344
b = −2.63815
−5.69095 −15.4810
u = 0.309916 + 0.549911I
a = −1.031800 − 0.275887I
b = −1.067230 − 0.057202I
−3.61897 − 1.53058I −14.5151 + 4.4306I
u = 0.309916 + 0.549911I
a = 0.68255 + 2.69529I
b = −0.024468 + 0.261280I
−3.61897 − 1.53058I −14.5151 + 4.4306I
u = 0.309916 − 0.549911I
a = −1.031800 + 0.275887I
b = −1.067230 + 0.057202I
−3.61897 + 1.53058I −14.5151 − 4.4306I
u = 0.309916 − 0.549911I
a = 0.68255 − 2.69529I
b = −0.024468 − 0.261280I
−3.61897 + 1.53058I −14.5151 − 4.4306I
u = −1.41878 + 0.21917I
a = −0.310913 − 0.768355I
b = 0.556241 − 1.005760I
−9.16243 + 4.40083I −18.7443 − 3.4986I
u = −1.41878 + 0.21917I
a = 0.72280 + 1.60293I
b = 1.22781 + 3.58963I
−9.16243 + 4.40083I −18.7443 − 3.4986I
u = −1.41878 − 0.21917I
a = −0.310913 + 0.768355I
b = 0.556241 + 1.005760I
−9.16243 − 4.40083I −18.7443 + 3.4986I
u = −1.41878 − 0.21917I
a = 0.72280 − 1.60293I
b = 1.22781 − 3.58963I
−9.16243 − 4.40083I −18.7443 + 3.4986I
10
III. I
u
3
= hu
3
+ 2u
2
+ b − 2u − 2, 2u
3
+ 3u
2
+ 3a − 3u − 3, u
4
− 3u
2
+ 3i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
7
=
−
2
3
u
3
− u
2
+ u + 1
−u
3
− 2u
2
+ 2u + 2
a
11
=
1
u
2
a
12
=
2
3
u
3
+ u
2
− u
u
3
+ 3u
2
− 2u − 2
a
4
=
u
u
a
3
=
−u
3
+ 2u
−2u
3
+ 4u
a
2
=
−
1
3
u
3
+ u
2
+ u − 1
−u
3
+ 2u
2
+ 2u − 2
a
1
=
2
3
u
3
+ u
2
− u − 1
u
3
+ 2u
2
− 2u − 2
a
6
=
−u
3
+ 2u
−2u
3
+ 4u
a
9
=
−u
2
+ 1
−u
2
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 24
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
(u − 1)
4
c
3
, c
5
, c
8
u
4
+ 3u
2
+ 3
c
4
, c
9
, c
10
u
4
− 3u
2
+ 3
c
6
, c
11
, c
12
(u + 1)
4
12
(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)
4
c
3
, c
5
, c
8
(y
2
+ 3y + 3)
2
c
4
, c
9
, c
10
(y
2
− 3y + 3)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.271230 + 0.340625I
a = −0.30334 − 1.59997I
b = −0.06940 − 2.66266I
−3.28987 − 4.05977I −18.0000 + 3.4641I
u = 1.271230 − 0.340625I
a = −0.30334 + 1.59997I
b = −0.06940 + 2.66266I
−3.28987 + 4.05977I −18.0000 − 3.4641I
u = −1.271230 + 0.340625I
a = −0.696660 + 0.132080I
b = −1.93060 + 0.80145I
−3.28987 + 4.05977I −18.0000 − 3.4641I
u = −1.271230 − 0.340625I
a = −0.696660 − 0.132080I
b = −1.93060 − 0.80145I
−3.28987 − 4.05977I −18.0000 + 3.4641I
14
IV. I
u
4
= hu
3
+ b, u
2
+ a + u − 1, u
4
− u
2
− 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
7
=
−u
2
− u + 1
−u
3
a
11
=
1
u
2
a
12
=
−u
2
− u + 2
−u
3
+ u
2
a
4
=
u
u
a
3
=
−u
3
+ 2u
0
a
2
=
−u
3
− u
2
+ u + 1
−u
3
a
1
=
−u
2
− u + 1
−u
3
a
6
=
u
3
− 2u
0
a
9
=
−u
2
+ 1
−u
2
a
8
=
−1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 16
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
11
c
12
(u − 1)
4
c
2
, c
7
(u + 1)
4
c
3
, c
5
, c
8
u
4
+ u
2
− 1
c
4
, c
9
, c
10
u
4
− u
2
− 1
16
(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)
4
c
3
, c
5
, c
8
(y
2
+ y −1)
2
c
4
, c
9
, c
10
(y
2
− y −1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.786151I
a = 1.61803 − 0.78615I
b = 0.485868I
0.657974 −13.5280
u = − 0.786151I
a = 1.61803 + 0.78615I
b = − 0.485868I
0.657974 −13.5280
u = 1.27202
a = −1.89005
b = −2.05817
−7.23771 −22.4720
u = −1.27202
a = 0.653986
b = 2.05817
−7.23771 −22.4720
18
V. I
v
1
= ha, b + 1, v + 1i
(i) Arc colorings
a
5
=
−1
0
a
10
=
1
0
a
7
=
0
−1
a
11
=
1
0
a
12
=
1
1
a
4
=
−1
0
a
3
=
−1
0
a
2
=
−1
1
a
1
=
0
1
a
6
=
−1
0
a
9
=
1
0
a
8
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
c
6
, c
11
, c
12
u + 1
20
(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
21
(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
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
10
+ 13u
9
+ ··· + 536u + 49)(u
15
+ 27u
14
+ ··· + 21u + 1)
c
2
, c
7
(u − 1)
5
(u + 1)
4
· (u
10
− u
9
− 6u
8
+ 4u
7
+ 18u
6
− 8u
5
− 31u
4
+ 13u
3
+ 28u
2
− 12u − 7)
· (u
15
− u
14
+ ··· − 3u − 1)
c
3
, c
5
u(u
4
+ u
2
− 1)(u
4
+ 3u
2
+ 3)(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
· (u
15
+ 9u
14
+ ··· + 166u + 22)
c
4
, c
9
, c
10
u(u
4
− 3u
2
+ 3)(u
4
− u
2
− 1)(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
· (u
15
− 3u
14
+ ··· + 6u + 2)
c
6
, c
11
, c
12
(u − 1)
4
(u + 1)
5
· (u
10
− u
9
− 6u
8
+ 4u
7
+ 18u
6
− 8u
5
− 31u
4
+ 13u
3
+ 28u
2
− 12u − 7)
· (u
15
− u
14
+ ··· − 3u − 1)
c
8
u(u
4
+ u
2
− 1)(u
4
+ 3u
2
+ 3)(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
· (u
15
+ 3u
14
+ ··· + 326u + 178)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
10
− 9y
9
+ ··· − 137356y + 2401)
· (y
15
− 91y
14
+ ··· + 125y −1)
c
2
, c
6
, c
7
c
11
, c
12
((y −1)
9
)(y
10
− 13y
9
+ ··· − 536y + 49)(y
15
− 27y
14
+ ··· + 21y −1)
c
3
, c
5
y(y
2
+ y −1)
2
(y
2
+ 3y + 3)
2
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
· (y
15
+ 7y
14
+ ··· + 5336y −484)
c
4
, c
9
, c
10
y(y
2
− 3y + 3)
2
(y
2
− y −1)
2
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
· (y
15
− 13y
14
+ ··· + 40y −4)
c
8
y(y
2
+ y −1)
2
(y
2
+ 3y + 3)
2
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
· (y
15
− 73y
14
+ ··· − 18680y −31684)
24
