12n
0722
(K12n
0722
)
A knot diagram
1
Linearized knot diagam
4 6 7 9 2 3 11 12 4 1 8 9
Solving Sequence
8,11
12 9 1
3,7
4 5 6 2 10
c
11
c
8
c
12
c
7
c
3
c
4
c
6
c
2
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
6
+ 3u
4
− u
3
− u
2
+ b + 2u + 1, −u
6
+ 3u
4
− u
3
− u
2
+ a + 2u,
u
9
+ u
8
− 5u
7
− 4u
6
+ 8u
5
+ 3u
4
− 5u
3
+ 2u
2
+ 3u + 1i
I
u
2
= h4u
21
+ 3u
20
+ ··· + b − 8u, u
21
− 2u
20
+ ··· + a + 6, u
22
+ 2u
21
+ ··· − 5u + 1i
I
u
3
= hb − 2u − 2, a − 2u − 1, u
2
− u − 1i
I
u
4
= hb + 2, a + u, u
2
− u − 1i
* 4 irreducible components of dim
C
= 0, with total 35 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
6
+3u
4
−u
3
−u
2
+b+2u+1, −u
6
+3u
4
−u
3
−u
2
+a+2u, u
9
+u
8
+· · ·+3u +1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
9
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
u
6
− 3u
4
+ u
3
+ u
2
− 2u
u
6
− 3u
4
+ u
3
+ u
2
− 2u − 1
a
7
=
u
u
a
4
=
u
6
− 3u
4
+ u
3
+ 2u
2
− 2u
u
6
− 3u
4
+ u
3
+ 2u
2
− 2u − 1
a
5
=
−u
8
+ 5u
6
− u
5
− 7u
4
+ 4u
3
+ 2u
2
− 4u − 1
−u
8
+ 5u
6
− u
5
− 7u
4
+ 3u
3
+ 2u
2
− 2u − 1
a
6
=
u
7
− 3u
5
+ u
4
+ u
3
− 2u
2
+ u
u
7
− 3u
5
+ u
4
+ u
3
− 2u
2
a
2
=
−u
8
+ 4u
6
− u
5
− 4u
4
+ 3u
3
− 2u
−u
8
+ 4u
6
− u
5
− 4u
4
+ 3u
3
+ u
2
− 2u − 1
a
10
=
−u
6
+ 3u
4
− 2u
2
+ 1
−u
8
+ 4u
6
− 4u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
8
− 2u
7
+ 24u
6
+ 6u
5
− 46u
4
+ 4u
3
+ 28u
2
− 18u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
9
− u
8
+ 7u
7
− 2u
6
+ 16u
5
+ 3u
4
+ 9u
3
+ 10u
2
+ 3u + 1
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
11
, c
12
u
9
+ u
8
− 5u
7
− 4u
6
+ 8u
5
+ 3u
4
− 5u
3
+ 2u
2
+ 3u + 1
c
4
, c
9
u
9
+ 5u
8
+ 10u
7
+ 9u
6
− u
5
− 15u
4
− 22u
3
− 16u
2
− 8u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
9
+ 13y
8
+ ··· − 11y −1
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
11
, c
12
y
9
− 11y
8
+ 49y
7
− 112y
6
+ 140y
5
− 105y
4
+ 69y
3
− 40y
2
+ 5y −1
c
4
, c
9
y
9
− 5y
8
+ 8y
7
+ 5y
6
− 25y
5
− 13y
4
+ 92y
3
− 24y
2
− 64y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.556651 + 0.655843I
a = −0.0328003 − 0.0846569I
b = −1.032800 − 0.084657I
4.56735 − 4.47297I −7.81258 + 6.23831I
u = 0.556651 − 0.655843I
a = −0.0328003 + 0.0846569I
b = −1.032800 + 0.084657I
4.56735 + 4.47297I −7.81258 − 6.23831I
u = −1.28665
a = −1.58604
b = −2.58604
−6.48693 −13.7120
u = 1.51165 + 0.13243I
a = −1.75672 + 1.70564I
b = −2.75672 + 1.70564I
−13.17090 − 3.99995I −16.3846 + 2.3960I
u = 1.51165 − 0.13243I
a = −1.75672 − 1.70564I
b = −2.75672 − 1.70564I
−13.17090 + 3.99995I −16.3846 − 2.3960I
u = −0.338768 + 0.252040I
a = 0.829715 − 0.547946I
b = −0.170285 − 0.547946I
−0.531790 + 0.852880I −9.17076 − 8.14648I
u = −0.338768 − 0.252040I
a = 0.829715 + 0.547946I
b = −0.170285 + 0.547946I
−0.531790 − 0.852880I −9.17076 + 8.14648I
u = −1.58621 + 0.20573I
a = −3.24718 − 1.53651I
b = −4.24718 − 1.53651I
−9.8278 + 10.8008I −14.7759 − 5.3771I
u = −1.58621 − 0.20573I
a = −3.24718 + 1.53651I
b = −4.24718 + 1.53651I
−9.8278 − 10.8008I −14.7759 + 5.3771I
5
II.
I
u
2
= h4u
21
+3u
20
+· · ·+b−8u, u
21
−2u
20
+· · ·+a+6, u
22
+2u
21
+· · ·−5u+1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
9
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
−u
21
+ 2u
20
+ ··· + 13u − 6
−4u
21
− 3u
20
+ ··· − 16u
2
+ 8u
a
7
=
u
u
a
4
=
2u
21
+ 4u
20
+ ··· + 5u − 5
−u
21
− u
20
+ ··· − 8u
2
+ 1
a
5
=
3u
20
+ 3u
19
+ ··· + 15u − 7
−3u
21
− 3u
20
+ ··· − 15u
2
+ 7u
a
6
=
−u
21
+ 11u
19
+ ··· − 17u
3
+ 9u
−2u
21
− 2u
20
+ ··· − 11u
2
+ 2u
a
2
=
−u
20
− u
19
+ ··· − 7u + 1
u
21
+ u
20
+ ··· + 6u
3
+ 8u
2
a
10
=
−u
6
+ 3u
4
− 2u
2
+ 1
−u
8
+ 4u
6
− 4u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 3u
20
+ 3u
19
− 32u
18
− 27u
17
+ 138u
16
+ 81u
15
− 317u
14
− 62u
13
+ 439u
12
− 123u
11
−
384u
10
+ 246u
9
+ 177u
8
− 178u
7
− 27u
6
+ 115u
5
+ 25u
4
− 37u
3
+ 20u
2
+ 18u − 11
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
22
− 4u
21
+ ··· − 11u − 1
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
11
, c
12
u
22
+ 2u
21
+ ··· − 5u + 1
c
4
, c
9
(u
11
− 2u
10
− 3u
9
+ 8u
8
− 8u
6
+ 9u
5
− 8u
4
− 7u
3
+ 12u
2
+ u − 2)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
22
+ 12y
21
+ ··· − 103y + 1
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
11
, c
12
y
22
− 24y
21
+ ··· − 27y + 1
c
4
, c
9
(y
11
− 10y
10
+ ··· + 49y −4)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.653871 + 0.639377I
a = 0.678912 + 0.802720I
b = 2.04567 − 0.21056I
−2.35449 − 7.64539I −11.71373 + 6.03391I
u = 0.653871 − 0.639377I
a = 0.678912 − 0.802720I
b = 2.04567 + 0.21056I
−2.35449 + 7.64539I −11.71373 − 6.03391I
u = 0.452757 + 0.672728I
a = −0.132038 − 0.926413I
b = 0.244470
4.87434 −6.59077 + 0.I
u = 0.452757 − 0.672728I
a = −0.132038 + 0.926413I
b = 0.244470
4.87434 −6.59077 + 0.I
u = 0.326778 + 0.705531I
a = −0.32042 + 2.18575I
b = 0.242416 + 0.347557I
−1.39120 + 3.13582I −9.76425 − 0.75545I
u = 0.326778 − 0.705531I
a = −0.32042 − 2.18575I
b = 0.242416 − 0.347557I
−1.39120 − 3.13582I −9.76425 + 0.75545I
u = −0.715155
a = 0.362929
b = 0.686958
−1.26486 −6.07510
u = −1.300610 + 0.077299I
a = −1.58364 + 0.11845I
b = −2.57660
−6.48450 −13.63121 + 0.I
u = −1.300610 − 0.077299I
a = −1.58364 − 0.11845I
b = −2.57660
−6.48450 −13.63121 + 0.I
u = −0.472498 + 0.509885I
a = −0.670269 − 0.742959I
b = 0.829406 + 0.775983I
−6.60747 + 1.76997I −13.10604 − 3.70025I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.472498 − 0.509885I
a = −0.670269 + 0.742959I
b = 0.829406 − 0.775983I
−6.60747 − 1.76997I −13.10604 + 3.70025I
u = 1.48308 + 0.04696I
a = 0.500256 − 1.109360I
b = 0.829406 − 0.775983I
−6.60747 − 1.76997I −13.10604 + 3.70025I
u = 1.48308 − 0.04696I
a = 0.500256 + 1.109360I
b = 0.829406 + 0.775983I
−6.60747 + 1.76997I −13.10604 − 3.70025I
u = −1.48082 + 0.20358I
a = 0.090121 − 0.149824I
b = 0.242416 + 0.347557I
−1.39120 + 3.13582I −9.76425 − 0.75545I
u = −1.48082 − 0.20358I
a = 0.090121 + 0.149824I
b = 0.242416 − 0.347557I
−1.39120 − 3.13582I −9.76425 + 0.75545I
u = −1.52066
a = 4.11109
b = 5.21838
−16.2219 −13.6940
u = −1.54155 + 0.21133I
a = 1.57352 + 0.56059I
b = 2.04567 + 0.21056I
−2.35449 + 7.64539I −11.71373 − 6.03391I
u = −1.54155 − 0.21133I
a = 1.57352 − 0.56059I
b = 2.04567 − 0.21056I
−2.35449 − 7.64539I −11.71373 + 6.03391I
u = 0.443905
a = 1.87359
b = −1.80820
−9.54474 0.158940
u = 1.63437
a = −1.53660
b = −1.80820
−9.54474 0.158940
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.68381
a = 4.31528
b = 5.21838
−16.2219 −13.6940
u = 0.231731
a = −2.39918
b = 0.686958
−1.26486 −6.07510
11
III. I
u
3
= hb − 2u − 2, a − 2u − 1, u
2
− u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u + 1
a
9
=
−u
−u − 1
a
1
=
−u
−u
a
3
=
2u + 1
2u + 2
a
7
=
u
u
a
4
=
u
u + 1
a
5
=
u
u + 1
a
6
=
−2u − 2
−3u − 2
a
2
=
−2u − 1
−3u − 1
a
10
=
−u
−u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −20
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
, c
10
u
2
+ u − 1
c
4
, c
9
u
2
c
5
, c
6
, c
11
c
12
u
2
− u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
4
, c
9
y
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −0.236068
b = 0.763932
−1.97392 −20.0000
u = 1.61803
a = 4.23607
b = 5.23607
−17.7653 −20.0000
15
IV. I
u
4
= hb + 2, a + u, u
2
− u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u + 1
a
9
=
−u
−u − 1
a
1
=
−u
−u
a
3
=
−u
−2
a
7
=
u
u
a
4
=
−u + 1
−1
a
5
=
−u + 1
−1
a
6
=
u − 1
−u + 2
a
2
=
−2
−2u + 1
a
10
=
−u
−u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −25
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
, c
10
u
2
+ u − 1
c
4
, c
9
u
2
c
5
, c
6
, c
11
c
12
u
2
− u − 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
4
, c
9
y
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.618034
b = −2.00000
−9.86960 −25.0000
u = 1.61803
a = −1.61803
b = −2.00000
−9.86960 −25.0000
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
10
((u
2
+ u − 1)
2
)(u
9
− u
8
+ ··· + 3u + 1)
· (u
22
− 4u
21
+ ··· − 11u − 1)
c
2
, c
3
, c
7
c
8
(u
2
+ u − 1)
2
(u
9
+ u
8
− 5u
7
− 4u
6
+ 8u
5
+ 3u
4
− 5u
3
+ 2u
2
+ 3u + 1)
· (u
22
+ 2u
21
+ ··· − 5u + 1)
c
4
, c
9
u
4
(u
9
+ 5u
8
+ 10u
7
+ 9u
6
− u
5
− 15u
4
− 22u
3
− 16u
2
− 8u − 4)
· (u
11
− 2u
10
− 3u
9
+ 8u
8
− 8u
6
+ 9u
5
− 8u
4
− 7u
3
+ 12u
2
+ u − 2)
2
c
5
, c
6
, c
11
c
12
(u
2
− u − 1)
2
(u
9
+ u
8
− 5u
7
− 4u
6
+ 8u
5
+ 3u
4
− 5u
3
+ 2u
2
+ 3u + 1)
· (u
22
+ 2u
21
+ ··· − 5u + 1)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
10
((y
2
− 3y + 1)
2
)(y
9
+ 13y
8
+ ··· − 11y − 1)
· (y
22
+ 12y
21
+ ··· − 103y + 1)
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
11
, c
12
(y
2
− 3y + 1)
2
· (y
9
− 11y
8
+ 49y
7
− 112y
6
+ 140y
5
− 105y
4
+ 69y
3
− 40y
2
+ 5y − 1)
· (y
22
− 24y
21
+ ··· − 27y + 1)
c
4
, c
9
y
4
(y
9
− 5y
8
+ 8y
7
+ 5y
6
− 25y
5
− 13y
4
+ 92y
3
− 24y
2
− 64y − 16)
· (y
11
− 10y
10
+ ··· + 49y − 4)
2
21
