12n
0468
(K12n
0468
)
A knot diagram
1
Linearized knot diagam
3 6 9 10 2 11 12 5 4 6 7 10
Solving Sequence
6,10
11 7
3,12
2 1 5 4 9 8
c
10
c
6
c
11
c
2
c
1
c
5
c
4
c
9
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h50017u
23
− 192212u
22
+ ··· + 807182b − 1440535,
− 887697u
23
+ 1683868u
22
+ ··· + 403591a + 7501915, u
24
− 2u
23
+ ··· − 14u + 1i
I
u
2
= hb, a − u − 1, u
2
+ u − 1i
I
u
3
= hb
2
− 2, a + u − 1, u
2
− u − 1i
* 3 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
= h5.00 × 10
4
u
23
− 1.92 × 10
5
u
22
+ · · · + 8.07 × 10
5
b − 1.44 × 10
6
, −8.88 ×
10
5
u
23
+1.68×10
6
u
22
+· · ·+4.04×10
5
a+7.50×10
6
, u
24
−2u
23
+· · ·−14u+1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
3
=
2.19950u
23
− 4.17221u
22
+ ··· + 80.3744u − 18.5879
−0.0619650u
23
+ 0.238127u
22
+ ··· − 4.26658u + 1.78465
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
2.19950u
23
− 4.17221u
22
+ ··· + 80.3744u − 18.5879
−0.155552u
23
+ 0.463063u
22
+ ··· − 5.24199u + 2.01143
a
1
=
u
4
− 3u
2
+ 1
u
4
− 2u
2
a
5
=
−1.28329u
23
+ 2.79375u
22
+ ··· − 48.2229u + 16.0757
0.0668610u
23
− 0.539826u
22
+ ··· + 6.83004u − 2.15414
a
4
=
−1.21643u
23
+ 2.25393u
22
+ ··· − 41.3929u + 13.9216
0.0668610u
23
− 0.539826u
22
+ ··· + 6.83004u − 2.15414
a
9
=
−2.01143u
23
+ 3.86730u
22
+ ··· − 83.0096u + 22.9180
0.557673u
23
− 0.761132u
22
+ ··· + 14.1472u − 3.26037
a
8
=
u
3
− 2u
−u
5
+ 3u
3
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
456499
403591
u
23
+
452261
403591
u
22
+ ··· −
8480209
403591
u −
5152499
403591
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
24
+ 5u
23
+ ··· + 53u + 1
c
2
, c
5
u
24
+ 3u
23
+ ··· + 7u + 1
c
3
, c
4
, c
9
u
24
− u
23
+ ··· − 4u − 4
c
6
, c
7
, c
10
c
11
u
24
+ 2u
23
+ ··· + 14u + 1
c
8
u
24
+ 3u
23
+ ··· − 4u − 4
c
12
u
24
+ 20u
23
+ ··· − 7204u + 113
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
+ 35y
23
+ ··· − 2365y + 1
c
2
, c
5
y
24
− 5y
23
+ ··· − 53y + 1
c
3
, c
4
, c
9
y
24
− 19y
23
+ ··· − 144y + 16
c
6
, c
7
, c
10
c
11
y
24
− 36y
23
+ ··· − 124y + 1
c
8
y
24
+ 41y
23
+ ··· − 272y + 16
c
12
y
24
− 108y
23
+ ··· − 36653012y + 12769
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.964868
a = 1.34648
b = −0.375511
−3.09721 −1.73870
u = 0.789115
a = −0.656714
b = −1.73162
−4.32235 2.54540
u = −1.195640 + 0.370602I
a = −0.572553 − 0.900506I
b = −0.06466 − 1.81298I
3.39573 − 7.66310I −0.37088 + 5.98262I
u = −1.195640 − 0.370602I
a = −0.572553 + 0.900506I
b = −0.06466 + 1.81298I
3.39573 + 7.66310I −0.37088 − 5.98262I
u = 0.394218 + 0.611999I
a = −0.26532 − 1.46494I
b = −0.096993 − 0.945243I
−1.63157 + 4.24750I −3.56364 − 6.51398I
u = 0.394218 − 0.611999I
a = −0.26532 + 1.46494I
b = −0.096993 + 0.945243I
−1.63157 − 4.24750I −3.56364 + 6.51398I
u = −1.300810 + 0.139494I
a = −0.624407 + 0.553888I
b = −0.341664 + 0.962592I
4.01026 − 1.38355I 0.744304 + 1.165315I
u = −1.300810 − 0.139494I
a = −0.624407 − 0.553888I
b = −0.341664 − 0.962592I
4.01026 + 1.38355I 0.744304 − 1.165315I
u = −0.567026 + 0.327854I
a = 0.027057 + 0.847883I
b = −0.318541 + 0.587410I
1.105110 − 0.832342I 4.13142 + 2.88592I
u = −0.567026 − 0.327854I
a = 0.027057 − 0.847883I
b = −0.318541 − 0.587410I
1.105110 + 0.832342I 4.13142 − 2.88592I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.330220 + 0.199162I
a = −0.460296 + 0.718151I
b = 0.01595 + 1.48977I
7.39054 + 2.83153I 3.91368 − 2.93748I
u = 1.330220 − 0.199162I
a = −0.460296 − 0.718151I
b = 0.01595 − 1.48977I
7.39054 − 2.83153I 3.91368 + 2.93748I
u = 0.394897 + 0.488286I
a = 1.365600 − 0.212541I
b = 0.148526 − 0.193691I
−1.60715 − 0.53576I −3.93456 − 0.12149I
u = 0.394897 − 0.488286I
a = 1.365600 + 0.212541I
b = 0.148526 + 0.193691I
−1.60715 + 0.53576I −3.93456 + 0.12149I
u = −1.60569
a = −0.0892641
b = 1.28709
3.92860 2.04700
u = 0.322044
a = 2.66696
b = 0.304688
−1.11472 −13.2200
u = 1.68408
a = −0.772565
b = −0.260162
6.28088 −2.91920
u = 1.79002 + 0.10661I
a = 0.653626 − 0.552899I
b = 0.45171 − 2.56882I
14.1732 + 9.8238I 0. − 4.62190I
u = 1.79002 − 0.10661I
a = 0.653626 + 0.552899I
b = 0.45171 + 2.56882I
14.1732 − 9.8238I 0. + 4.62190I
u = 1.82183 + 0.02838I
a = 0.439745 + 0.654915I
b = 0.79204 + 2.27718I
15.6632 + 2.1276I 1.47628 − 0.94453I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.82183 − 0.02838I
a = 0.439745 − 0.654915I
b = 0.79204 − 2.27718I
15.6632 − 2.1276I 1.47628 + 0.94453I
u = −1.82860 + 0.05006I
a = 0.568562 + 0.623923I
b = 0.63662 + 2.42066I
19.1541 − 4.0381I 3.21227 + 2.25463I
u = −1.82860 − 0.05006I
a = 0.568562 − 0.623923I
b = 0.63662 − 2.42066I
19.1541 + 4.0381I 3.21227 − 2.25463I
u = 0.0970532
a = −10.7589
b = 1.32951
−6.54674 −14.1170
7
II. I
u
2
= hb, a − u − 1, u
2
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u − 1
a
7
=
u
−u + 1
a
3
=
u + 1
0
a
12
=
u
−u
a
2
=
u + 1
−u
a
1
=
0
−u
a
5
=
u + 1
0
a
4
=
u + 1
0
a
9
=
1
0
a
8
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
8
c
9
u
2
c
5
(u + 1)
2
c
6
, c
7
u
2
− u − 1
c
10
, c
11
, c
12
u
2
+ u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
2
c
3
, c
4
, c
8
c
9
y
2
c
6
, c
7
, c
10
c
11
, c
12
y
2
− 3y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.61803
b = 0
−0.657974 6.00000
u = −1.61803
a = −0.618034
b = 0
7.23771 6.00000
11
III. I
u
3
= hb
2
− 2, a + u − 1, u
2
− u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u − 1
a
7
=
u
−u − 1
a
3
=
−u + 1
b
a
12
=
−u
u
a
2
=
−u + 1
b + u
a
1
=
0
u
a
5
=
u − 1
−b
a
4
=
−b + u − 1
−b
a
9
=
bu − b − 1
−2
a
8
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
8
c
9
(u
2
− 2)
2
c
6
, c
7
, c
12
(u
2
+ u − 1)
2
c
10
, c
11
(u
2
− u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
8
c
9
(y −2)
4
c
6
, c
7
, c
10
c
11
, c
12
(y
2
− 3y + 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 1.61803
b = 1.41421
−5.59278 −4.00000
u = −0.618034
a = 1.61803
b = −1.41421
−5.59278 −4.00000
u = 1.61803
a = −0.618034
b = 1.41421
2.30291 −4.00000
u = 1.61803
a = −0.618034
b = −1.41421
2.30291 −4.00000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
24
+ 5u
23
+ ··· + 53u + 1)
c
2
((u − 1)
2
)(u + 1)
4
(u
24
+ 3u
23
+ ··· + 7u + 1)
c
3
, c
4
, c
9
u
2
(u
2
− 2)
2
(u
24
− u
23
+ ··· − 4u − 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
24
+ 3u
23
+ ··· + 7u + 1)
c
6
, c
7
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
24
+ 2u
23
+ ··· + 14u + 1)
c
8
u
2
(u
2
− 2)
2
(u
24
+ 3u
23
+ ··· − 4u − 4)
c
10
, c
11
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
24
+ 2u
23
+ ··· + 14u + 1)
c
12
((u
2
+ u − 1)
3
)(u
24
+ 20u
23
+ ··· − 7204u + 113)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
24
+ 35y
23
+ ··· − 2365y + 1)
c
2
, c
5
((y −1)
6
)(y
24
− 5y
23
+ ··· − 53y + 1)
c
3
, c
4
, c
9
y
2
(y −2)
4
(y
24
− 19y
23
+ ··· − 144y + 16)
c
6
, c
7
, c
10
c
11
((y
2
− 3y + 1)
3
)(y
24
− 36y
23
+ ··· − 124y + 1)
c
8
y
2
(y −2)
4
(y
24
+ 41y
23
+ ··· − 272y + 16)
c
12
((y
2
− 3y + 1)
3
)(y
24
− 108y
23
+ ··· − 3.66530 × 10
7
y + 12769)
17
