12n
0624
(K12n
0624
)
A knot diagram
1
Linearized knot diagam
3 8 11 8 3 11 2 12 4 7 9 5
Solving Sequence
8,12 3,9
2 1 7 11 4 5 6 10
c
8
c
2
c
1
c
7
c
11
c
3
c
4
c
5
c
10
c
6
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
16
− u
15
+ 6u
14
− 5u
13
+ 13u
12
− 10u
11
+ 8u
10
− 9u
9
− 9u
8
− 4u
7
− 9u
6
+ 8u
4
+ 2u
3
+ 9u
2
+ b + u − 1,
u
16
− 3u
15
+ ··· + a + 5, u
18
− 2u
17
+ ··· − 8u + 1i
I
u
2
= hu
11
+ 2u
10
+ 6u
9
+ 8u
8
+ 12u
7
+ 12u
6
+ 9u
5
+ 6u
4
− u
2
+ b − 2u − 2,
− u
11
− 3u
10
− 10u
9
− 18u
8
− 30u
7
− 35u
6
− 33u
5
− 23u
4
− 6u
3
+ 4u
2
+ a + 9u + 6,
u
12
+ 3u
11
+ 9u
10
+ 16u
9
+ 25u
8
+ 30u
7
+ 28u
6
+ 21u
5
+ 8u
4
− u
3
− 5u
2
− 5u − 1i
* 2 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
= hu
16
− u
15
+ · · · + b − 1, u
16
− 3u
15
+ · · · + a + 5, u
18
− 2u
17
+ · · · − 8u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
3
=
−u
16
+ 3u
15
+ ··· + 18u − 5
−u
16
+ u
15
+ ··· − u + 1
a
9
=
1
−u
2
a
2
=
−2u
16
+ 4u
15
+ ··· + 17u − 4
−u
16
+ u
15
+ ··· − u + 1
a
1
=
−u
15
+ u
14
+ ··· − 10u + 1
u
16
− u
15
+ ··· + 3u
3
+ 10u
2
a
7
=
−u
17
− 5u
15
+ ··· + 5u + 1
−u
16
+ u
15
+ ··· − 11u
2
+ u
a
11
=
−u
u
3
+ u
a
4
=
u
17
− 2u
16
+ ··· + 19u − 5
−u
17
+ u
16
+ ··· − 9u + 2
a
5
=
2u
17
− 3u
16
+ ··· + 28u − 7
−u
17
+ u
16
+ ··· − 9u + 2
a
6
=
u
16
− 2u
15
+ ··· − 6u − 1
u
17
− u
16
+ ··· + 7u − 1
a
10
=
−u
17
+ u
16
+ ··· − 4u + 1
−u
16
+ u
15
+ ··· + 7u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
17
−3u
16
+ 19u
15
−23u
14
+ 70u
13
−72u
12
+ 124u
11
−114u
10
+
89u
9
− 89u
8
− 15u
7
− 26u
6
− 27u
5
− 2u
4
+ 51u
3
− 15u
2
+ 50u − 24
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
18
+ 14u
17
+ ··· + 21504u + 4096
c
2
, c
7
u
18
− 16u
17
+ ··· + 224u − 64
c
3
, c
6
, c
10
u
18
+ 2u
17
+ ··· − 2u − 1
c
4
, c
12
u
18
+ 3u
17
+ ··· − 2u − 1
c
5
u
18
− 5u
17
+ ··· − 4u + 1
c
8
, c
11
u
18
+ 2u
17
+ ··· + 8u + 1
c
9
u
18
− u
17
+ ··· − 120u − 61
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
18
+ 82y
17
+ ··· − 437256192y + 16777216
c
2
, c
7
y
18
− 14y
17
+ ··· − 21504y + 4096
c
3
, c
6
, c
10
y
18
+ 28y
17
+ ··· − 18y + 1
c
4
, c
12
y
18
+ 35y
17
+ ··· + 20y + 1
c
5
y
18
− 45y
17
+ ··· − 20y + 1
c
8
, c
11
y
18
+ 14y
17
+ ··· − 32y + 1
c
9
y
18
+ 33y
17
+ ··· + 24762y + 3721
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.957035 + 0.038927I
a = 1.29444 − 2.14220I
b = −1.52646 + 1.38753I
12.15720 + 5.50571I −6.58833 − 2.14870I
u = 0.957035 − 0.038927I
a = 1.29444 + 2.14220I
b = −1.52646 − 1.38753I
12.15720 − 5.50571I −6.58833 + 2.14870I
u = −0.292699 + 1.060130I
a = 0.130715 − 0.442861I
b = −0.090901 + 0.653348I
−0.84836 − 1.94165I −6.35139 + 2.74428I
u = −0.292699 − 1.060130I
a = 0.130715 + 0.442861I
b = −0.090901 − 0.653348I
−0.84836 + 1.94165I −6.35139 − 2.74428I
u = 0.074993 + 1.132910I
a = 0.751791 + 0.803103I
b = 0.794717 − 0.290190I
−3.55462 + 1.02828I −12.61847 − 1.28084I
u = 0.074993 − 1.132910I
a = 0.751791 − 0.803103I
b = 0.794717 + 0.290190I
−3.55462 − 1.02828I −12.61847 + 1.28084I
u = −0.796441 + 0.291195I
a = 0.151574 + 0.889968I
b = −0.775836 − 0.472537I
1.24471 − 1.98200I −5.60868 + 9.18970I
u = −0.796441 − 0.291195I
a = 0.151574 − 0.889968I
b = −0.775836 + 0.472537I
1.24471 + 1.98200I −5.60868 − 9.18970I
u = 0.147448 + 1.249270I
a = −1.46732 − 1.39599I
b = −1.62783 + 0.04234I
−11.88260 + 2.10947I −15.7034 − 5.8224I
u = 0.147448 − 1.249270I
a = −1.46732 + 1.39599I
b = −1.62783 − 0.04234I
−11.88260 − 2.10947I −15.7034 + 5.8224I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.490413 + 1.272180I
a = 1.140910 + 0.046544I
b = −1.42593 − 1.40587I
8.34549 − 0.36039I −9.48611 − 0.73252I
u = 0.490413 − 1.272180I
a = 1.140910 − 0.046544I
b = −1.42593 + 1.40587I
8.34549 + 0.36039I −9.48611 + 0.73252I
u = 0.458123 + 1.327850I
a = −0.80670 − 2.00900I
b = −1.59701 + 1.33472I
7.89248 + 10.55150I −10.01468 − 4.70572I
u = 0.458123 − 1.327850I
a = −0.80670 + 2.00900I
b = −1.59701 − 1.33472I
7.89248 − 10.55150I −10.01468 + 4.70572I
u = −0.35310 + 1.38486I
a = −0.684948 + 0.964946I
b = −1.194280 − 0.456860I
−4.04162 − 6.18627I −15.0481 + 5.4779I
u = −0.35310 − 1.38486I
a = −0.684948 − 0.964946I
b = −1.194280 + 0.456860I
−4.04162 + 6.18627I −15.0481 − 5.4779I
u = 0.443581
a = 1.93499
b = −1.59894
−8.07946 −1.20520
u = 0.184865
a = −1.95591
b = 0.485984
−0.676311 −14.9570
6
II.
I
u
2
= hu
11
+2u
10
+· · ·+b−2, −u
11
−3u
10
+· · ·+a+6, u
12
+3u
11
+· · ·−5u−1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
3
=
u
11
+ 3u
10
+ ··· − 9u − 6
−u
11
− 2u
10
− 6u
9
− 8u
8
− 12u
7
− 12u
6
− 9u
5
− 6u
4
+ u
2
+ 2u + 2
a
9
=
1
−u
2
a
2
=
u
10
+ 4u
9
+ 10u
8
+ 18u
7
+ 23u
6
+ 24u
5
+ 17u
4
+ 6u
3
− 3u
2
− 7u − 4
−u
11
− 2u
10
− 6u
9
− 8u
8
− 12u
7
− 12u
6
− 9u
5
− 6u
4
+ u
2
+ 2u + 2
a
1
=
−2u
11
− 6u
10
+ ··· + 14u + 10
−u
10
− 2u
9
− 5u
8
− 5u
7
− 5u
6
− 2u
5
+ 2u
4
+ 3u
3
+ 4u
2
+ u − 2
a
7
=
−u
11
− 5u
10
+ ··· + 15u + 8
−u
10
− 2u
9
− 5u
8
− 6u
7
− 6u
6
− 5u
5
+ u
3
+ 3u
2
+ 2u − 2
a
11
=
−u
u
3
+ u
a
4
=
u
11
+ 4u
10
+ ··· − 10u − 6
u
10
+ 2u
9
+ 5u
8
+ 6u
7
+ 6u
6
+ 5u
5
− u
3
− 3u
2
− 2u + 1
a
5
=
u
11
+ 3u
10
+ ··· − 8u − 7
u
10
+ 2u
9
+ 5u
8
+ 6u
7
+ 6u
6
+ 5u
5
− u
3
− 3u
2
− 2u + 1
a
6
=
−u
11
− 4u
10
+ ··· + 14u + 8
u
11
+ 2u
10
+ 6u
9
+ 8u
8
+ 12u
7
+ 12u
6
+ 9u
5
+ 6u
4
− u
2
− 2u − 3
a
10
=
3u
11
+ 8u
10
+ ··· − 5u − 4
−u
10
− 4u
9
+ ··· + 7u + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 7u
11
+ 15u
10
+ 48u
9
+ 70u
8
+ 112u
7
+ 121u
6
+ 105u
5
+ 81u
4
+ 18u
3
− 2u
2
− 21u − 29
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
− 14u
11
+ ··· − 67u + 9
c
2
u
12
+ 2u
11
+ ··· − u + 3
c
3
, c
10
u
12
− u
11
− 2u
10
− u
9
+ 2u
8
+ 4u
7
− 2u
6
+ 3u
5
− 3u
4
+ 4u
3
− 2u
2
+ u − 1
c
4
, c
12
u
12
+ 2u
11
− u
10
− u
9
+ 6u
8
− 3u
7
+ u
6
+ 5u
5
− 5u
4
+ 7u
3
− 5u
2
+ 3u − 1
c
5
u
12
+ 14u
11
+ ··· + 3u + 1
c
6
u
12
+ u
11
− 2u
10
+ u
9
+ 2u
8
− 4u
7
− 2u
6
− 3u
5
− 3u
4
− 4u
3
− 2u
2
− u − 1
c
7
u
12
− 2u
11
+ ··· + u + 3
c
8
u
12
+ 3u
11
+ ··· − 5u − 1
c
9
u
12
− 2u
10
− 6u
9
+ 13u
8
+ u
7
+ 2u
6
− 9u
5
− 8u
4
+ 9u
3
− 4u
2
+ 5u − 3
c
11
u
12
− 3u
11
+ ··· + 5u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
− 22y
11
+ ··· + 29y + 81
c
2
, c
7
y
12
− 14y
11
+ ··· − 67y + 9
c
3
, c
6
, c
10
y
12
− 5y
11
+ ··· + 3y + 1
c
4
, c
12
y
12
− 6y
11
+ ··· + y + 1
c
5
y
12
− 30y
11
+ ··· + 21y + 1
c
8
, c
11
y
12
+ 9y
11
+ ··· − 15y + 1
c
9
y
12
− 4y
11
+ ··· − y + 9
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.945559 + 0.176012I
a = 0.384985 + 0.661564I
b = −0.934205 − 0.378092I
1.63828 − 1.45323I 1.054104 + 0.917158I
u = −0.945559 − 0.176012I
a = 0.384985 − 0.661564I
b = −0.934205 + 0.378092I
1.63828 + 1.45323I 1.054104 − 0.917158I
u = −0.083486 + 1.161420I
a = 1.36019 − 1.88104I
b = 1.71485 + 0.22346I
−11.61070 − 1.25522I −12.48094 − 2.13033I
u = −0.083486 − 1.161420I
a = 1.36019 + 1.88104I
b = 1.71485 − 0.22346I
−11.61070 + 1.25522I −12.48094 + 2.13033I
u = 0.262959 + 1.174910I
a = −0.73282 + 1.30130I
b = 0.886932 − 0.447310I
−8.28259 + 3.18829I −11.15779 − 3.99090I
u = 0.262959 − 1.174910I
a = −0.73282 − 1.30130I
b = 0.886932 + 0.447310I
−8.28259 − 3.18829I −11.15779 + 3.99090I
u = −0.451022 + 1.172390I
a = −0.0134217 + 0.1288200I
b = −0.609088 + 0.508190I
−1.38760 − 3.47878I −6.82613 + 4.40874I
u = −0.451022 − 1.172390I
a = −0.0134217 − 0.1288200I
b = −0.609088 − 0.508190I
−1.38760 + 3.47878I −6.82613 − 4.40874I
u = 0.612728
a = −2.21290
b = 0.702300
−4.82802 −5.83830
u = −0.45888 + 1.40396I
a = −0.414406 + 0.891813I
b = −1.172970 − 0.328171I
−3.32828 − 6.54831I −6.31944 + 10.84380I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.45888 − 1.40396I
a = −0.414406 − 0.891813I
b = −1.172970 + 0.328171I
−3.32828 + 6.54831I −6.31944 − 10.84380I
u = −0.260752
a = −3.95615
b = 1.52667
−8.44787 −23.7010
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
12
− 14u
11
+ ··· − 67u + 9)(u
18
+ 14u
17
+ ··· + 21504u + 4096)
c
2
(u
12
+ 2u
11
+ ··· − u + 3)(u
18
− 16u
17
+ ··· + 224u − 64)
c
3
, c
10
(u
12
− u
11
− 2u
10
− u
9
+ 2u
8
+ 4u
7
− 2u
6
+ 3u
5
− 3u
4
+ 4u
3
− 2u
2
+ u − 1)
· (u
18
+ 2u
17
+ ··· − 2u − 1)
c
4
, c
12
(u
12
+ 2u
11
− u
10
− u
9
+ 6u
8
− 3u
7
+ u
6
+ 5u
5
− 5u
4
+ 7u
3
− 5u
2
+ 3u − 1)
· (u
18
+ 3u
17
+ ··· − 2u − 1)
c
5
(u
12
+ 14u
11
+ ··· + 3u + 1)(u
18
− 5u
17
+ ··· − 4u + 1)
c
6
(u
12
+ u
11
− 2u
10
+ u
9
+ 2u
8
− 4u
7
− 2u
6
− 3u
5
− 3u
4
− 4u
3
− 2u
2
− u − 1)
· (u
18
+ 2u
17
+ ··· − 2u − 1)
c
7
(u
12
− 2u
11
+ ··· + u + 3)(u
18
− 16u
17
+ ··· + 224u − 64)
c
8
(u
12
+ 3u
11
+ ··· − 5u − 1)(u
18
+ 2u
17
+ ··· + 8u + 1)
c
9
(u
12
− 2u
10
− 6u
9
+ 13u
8
+ u
7
+ 2u
6
− 9u
5
− 8u
4
+ 9u
3
− 4u
2
+ 5u − 3)
· (u
18
− u
17
+ ··· − 120u − 61)
c
11
(u
12
− 3u
11
+ ··· + 5u − 1)(u
18
+ 2u
17
+ ··· + 8u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
12
− 22y
11
+ ··· + 29y + 81)
· (y
18
+ 82y
17
+ ··· − 437256192y + 16777216)
c
2
, c
7
(y
12
− 14y
11
+ ··· − 67y + 9)(y
18
− 14y
17
+ ··· − 21504y + 4096)
c
3
, c
6
, c
10
(y
12
− 5y
11
+ ··· + 3y + 1)(y
18
+ 28y
17
+ ··· − 18y + 1)
c
4
, c
12
(y
12
− 6y
11
+ ··· + y + 1)(y
18
+ 35y
17
+ ··· + 20y + 1)
c
5
(y
12
− 30y
11
+ ··· + 21y + 1)(y
18
− 45y
17
+ ··· − 20y + 1)
c
8
, c
11
(y
12
+ 9y
11
+ ··· − 15y + 1)(y
18
+ 14y
17
+ ··· − 32y + 1)
c
9
(y
12
− 4y
11
+ ··· − y + 9)(y
18
+ 33y
17
+ ··· + 24762y + 3721)
13
