12a
1214
(K12a
1214
)
A knot diagram
1
Linearized knot diagam
5 6 7 10 2 3 11 12 1 4 8 9
Solving Sequence
1,5
2 6 3
7,9
10 4 12 8 11
c
1
c
5
c
2
c
6
c
9
c
4
c
12
c
8
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, −u
6
+ 5u
4
+ u
3
− 6u
2
+ a − 2u, u
7
− 6u
5
− u
4
+ 10u
3
+ 3u
2
− 3u + 1i
I
u
2
= hu
17
− u
16
+ ··· + b + 1, u
17
− 3u
16
+ ··· + a + 5, u
18
− 2u
17
+ ··· + 5u + 1i
I
u
3
= hb + u, a + u, u
2
+ u − 1i
I
u
4
= hb − u − 1, a − u − 1, u
2
+ u − 1i
I
u
5
= hb + 1, a + 2, u − 1i
* 5 irreducible components of dim
C
= 0, with total 30 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
= hb +u, −u
6
+5u
4
+u
3
−6u
2
+a−2u, u
7
−6u
5
−u
4
+10u
3
+3u
2
−3u+1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
−u
−u
3
+ u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
7
=
u
3
− 2u
u
5
− 3u
3
+ u
a
9
=
u
6
− 5u
4
− u
3
+ 6u
2
+ 2u
−u
a
10
=
u
6
− 5u
4
− u
3
+ 6u
2
+ 3u
−u
a
4
=
u
4
− 3u
2
+ 1
u
6
− 4u
4
+ 3u
2
a
12
=
u
5
− 4u
3
− u
2
+ 3u
−u
2
a
8
=
−u
4
+ 3u
2
+ 2u
u
3
− u
a
11
=
−u
3
+ u
2
+ 3u
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
6
+ 4u
5
− 12u
4
− 22u
3
+ 18u
2
+ 30u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
9
, c
11
c
12
u
7
− 6u
5
+ u
4
+ 10u
3
− 3u
2
− 3u − 1
c
4
, c
10
u
7
− 4u
6
+ 11u
5
− 19u
4
+ 22u
3
− 20u
2
+ 8u − 4
3
(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
9
, c
11
c
12
y
7
− 12y
6
+ 56y
5
− 127y
4
+ 142y
3
− 67y
2
+ 3y −1
c
4
, c
10
y
7
+ 6y
6
+ 13y
5
− 21y
4
− 132y
3
− 200y
2
− 96y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.149610 + 0.279986I
a = 1.12909 + 0.89438I
b = 1.149610 − 0.279986I
−9.53872 + 4.72092I −18.8644 − 4.7284I
u = −1.149610 − 0.279986I
a = 1.12909 − 0.89438I
b = 1.149610 + 0.279986I
−9.53872 − 4.72092I −18.8644 + 4.7284I
u = 0.256916 + 0.244395I
a = 0.658925 + 1.198610I
b = −0.256916 − 0.244395I
−0.398617 − 0.781295I −9.36937 + 8.81210I
u = 0.256916 − 0.244395I
a = 0.658925 − 1.198610I
b = −0.256916 + 0.244395I
−0.398617 + 0.781295I −9.36937 − 8.81210I
u = −1.78027
a = 2.70935
b = 1.78027
13.9427 −18.6250
u = 1.78282 + 0.11231I
a = −2.14269 + 0.85665I
b = −1.78282 − 0.11231I
8.72326 − 8.52438I −19.4535 + 3.0874I
u = 1.78282 − 0.11231I
a = −2.14269 − 0.85665I
b = −1.78282 + 0.11231I
8.72326 + 8.52438I −19.4535 − 3.0874I
5
II.
I
u
2
= hu
17
− u
16
+ · · · + b + 1, u
17
− 3u
16
+ · · · + a + 5, u
18
− 2u
17
+ · · · + 5u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
−u
−u
3
+ u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
7
=
u
3
− 2u
u
5
− 3u
3
+ u
a
9
=
−u
17
+ 3u
16
+ ··· − 3u − 5
−u
17
+ u
16
+ ··· + 2u − 1
a
10
=
2u
16
− u
15
+ ··· − 5u − 4
−u
17
+ u
16
+ ··· + 2u − 1
a
4
=
u
4
− 3u
2
+ 1
u
6
− 4u
4
+ 3u
2
a
12
=
−u
16
+ u
15
+ ··· + 8u + 1
−u
16
+ u
15
+ ··· + 5u + 1
a
8
=
−u
10
+ 7u
8
− 16u
6
+ 2u
5
+ 13u
4
− 8u
3
− 3u
2
+ 6u − 1
−u
16
+ u
15
+ ··· + 4u + 1
a
11
=
−2u
17
+ 2u
16
+ ··· + 4u − 3
−2u
17
+ u
16
+ ··· + 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
16
−3u
15
−28u
14
+ 27u
13
+ 96u
12
−99u
11
−137u
10
+ 200u
9
+
34u
8
− 233u
7
+ 113u
6
+ 107u
5
− 118u
4
+ 26u
3
+ 38u
2
− 18u − 11
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
9
, c
11
c
12
u
18
+ 2u
17
+ ··· − 5u + 1
c
4
, c
10
(u
9
+ 2u
8
+ 7u
7
+ 10u
6
+ 15u
5
+ 16u
4
+ 8u
3
+ 6u
2
− 3u − 2)
2
7
(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
9
, c
11
c
12
y
18
− 24y
17
+ ··· − 39y + 1
c
4
, c
10
(y
9
+ 10y
8
+ 39y
7
+ 62y
6
− 13y
5
− 170y
4
− 178y
3
− 20y
2
+ 33y −4)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.099080 + 0.090870I
a = −0.384578 − 0.484468I
b = −0.375602 + 0.561098I
−4.71196 + 1.85169I −15.8408 − 4.0347I
u = −1.099080 − 0.090870I
a = −0.384578 + 0.484468I
b = −0.375602 − 0.561098I
−4.71196 − 1.85169I −15.8408 + 4.0347I
u = 0.404211 + 0.717214I
a = −0.242512 + 1.041630I
b = −1.76042 − 0.02141I
−15.1195 − 2.3160I −16.2080 + 2.7069I
u = 0.404211 − 0.717214I
a = −0.242512 − 1.041630I
b = −1.76042 + 0.02141I
−15.1195 + 2.3160I −16.2080 − 2.7069I
u = 1.18349
a = 2.74140
b = 1.71775
−14.6766 −17.6580
u = −1.187490 + 0.413479I
a = −1.70401 − 1.25443I
b = −1.77073 + 0.06860I
19.3766 + 6.2041I −18.9481 − 3.7555I
u = −1.187490 − 0.413479I
a = −1.70401 + 1.25443I
b = −1.77073 − 0.06860I
19.3766 − 6.2041I −18.9481 + 3.7555I
u = 0.703192
a = 0.770486
b = 0.187582
−1.23204 −6.62220
u = 0.375602 + 0.561098I
a = −0.227816 − 0.984272I
b = 1.099080 + 0.090870I
−4.71196 − 1.85169I −15.8408 + 4.0347I
u = 0.375602 − 0.561098I
a = −0.227816 + 0.984272I
b = 1.099080 − 0.090870I
−4.71196 + 1.85169I −15.8408 − 4.0347I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.394111
a = 2.82610
b = 1.63604
−9.50074 −2.72570
u = −1.63604
a = 0.680788
b = 0.394111
−9.50074 −2.72570
u = −1.71775
a = −1.88877
b = −1.18349
−14.6766 −17.6580
u = 1.76042 + 0.02141I
a = −0.478278 + 0.146184I
b = −0.404211 − 0.717214I
−15.1195 − 2.3160I −16.2080 + 2.7069I
u = 1.76042 − 0.02141I
a = −0.478278 − 0.146184I
b = −0.404211 + 0.717214I
−15.1195 + 2.3160I −16.2080 − 2.7069I
u = 1.77073 + 0.06860I
a = 1.41636 − 0.49822I
b = 1.187490 + 0.413479I
19.3766 − 6.2041I −18.9481 + 3.7555I
u = 1.77073 − 0.06860I
a = 1.41636 + 0.49822I
b = 1.187490 − 0.413479I
19.3766 + 6.2041I −18.9481 − 3.7555I
u = −0.187582
a = −2.88834
b = −0.703192
−1.23204 −6.62220
10
III. I
u
3
= hb + u, a + u, u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u + 1
a
6
=
−u
−u + 1
a
3
=
u
u
a
7
=
−1
0
a
9
=
−u
−u
a
10
=
0
−u
a
4
=
0
u
a
12
=
u
u − 1
a
8
=
−1
u − 1
a
11
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −20
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
, c
9
u
2
+ u − 1
c
4
, c
10
u
2
c
5
, c
6
, c
11
c
12
u
2
− u − 1
12
(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
9
, c
11
c
12
y
2
− 3y + 1
c
4
, c
10
y
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −0.618034
b = −0.618034
−1.97392 −20.0000
u = −1.61803
a = 1.61803
b = 1.61803
−17.7653 −20.0000
14
IV. I
u
4
= hb − u − 1, a − u − 1, u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u + 1
a
6
=
−u
−u + 1
a
3
=
u
u
a
7
=
−1
0
a
9
=
u + 1
u + 1
a
10
=
0
u + 1
a
4
=
0
u
a
12
=
−u − 1
−u − 2
a
8
=
−1
−u − 2
a
11
=
0
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −25
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
, c
9
u
2
+ u − 1
c
4
, c
10
u
2
c
5
, c
6
, c
11
c
12
u
2
− u − 1
16
(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
9
, c
11
c
12
y
2
− 3y + 1
c
4
, c
10
y
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.61803
b = 1.61803
−9.86960 −25.0000
u = −1.61803
a = −0.618034
b = −0.618034
−9.86960 −25.0000
18
V. I
u
5
= hb + 1, a + 2, u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
6
=
−1
0
a
3
=
0
1
a
7
=
−1
−1
a
9
=
−2
−1
a
10
=
−1
−1
a
4
=
−1
0
a
12
=
−1
−1
a
8
=
−1
0
a
11
=
−2
−1
(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
3
c
5
, c
6
, c
7
c
8
, c
9
, c
11
c
12
u + 1
c
4
, c
10
u − 1
20
(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
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −2.00000
b = −1.00000
−4.93480 −18.0000
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
, c
9
(u + 1)(u
2
+ u − 1)
2
(u
7
− 6u
5
+ u
4
+ 10u
3
− 3u
2
− 3u − 1)
· (u
18
+ 2u
17
+ ··· − 5u + 1)
c
4
, c
10
u
4
(u − 1)(u
7
− 4u
6
+ 11u
5
− 19u
4
+ 22u
3
− 20u
2
+ 8u − 4)
· (u
9
+ 2u
8
+ 7u
7
+ 10u
6
+ 15u
5
+ 16u
4
+ 8u
3
+ 6u
2
− 3u − 2)
2
c
5
, c
6
, c
11
c
12
(u + 1)(u
2
− u − 1)
2
(u
7
− 6u
5
+ u
4
+ 10u
3
− 3u
2
− 3u − 1)
· (u
18
+ 2u
17
+ ··· − 5u + 1)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
9
, c
11
c
12
(y −1)(y
2
− 3y + 1)
2
· (y
7
− 12y
6
+ 56y
5
− 127y
4
+ 142y
3
− 67y
2
+ 3y −1)
· (y
18
− 24y
17
+ ··· − 39y + 1)
c
4
, c
10
y
4
(y −1)(y
7
+ 6y
6
+ 13y
5
− 21y
4
− 132y
3
− 200y
2
− 96y −16)
· (y
9
+ 10y
8
+ 39y
7
+ 62y
6
− 13y
5
− 170y
4
− 178y
3
− 20y
2
+ 33y −4)
2
24
