12n
0466
(K12n
0466
)
A knot diagram
1
Linearized knot diagam
3 6 9 8 2 11 12 3 4 6 7 10
Solving Sequence
6,11
7 12
3,8
9 2 1 5 4 10
c
6
c
11
c
7
c
8
c
2
c
1
c
5
c
4
c
10
c
3
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−296878516u
31
− 276724358u
30
+ ··· + 441171721b − 537239537,
− 397220095u
31
− 1740337107u
30
+ ··· + 882343442a − 7870777298, u
32
+ 2u
31
+ ··· + 6u + 1i
I
u
2
= hb − 1, a
2
+ 2a − 2u − 3, u
2
+ u − 1i
I
u
3
= hb + 1, a − 1, u
2
− u − 1i
* 3 irreducible components of dim
C
= 0, with total 38 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−2.97 × 10
8
u
31
− 2.77 × 10
8
u
30
+ · · · + 4.41 × 10
8
b − 5.37 × 10
8
, −3.97 ×
10
8
u
31
−1.74×10
9
u
30
+· · · +8.82×10
8
a−7.87×10
9
, u
32
+2u
31
+· · · +6u +1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
12
=
u
−u
3
+ u
a
3
=
0.450188u
31
+ 1.97240u
30
+ ··· − 8.67806u + 8.92031
0.672932u
31
+ 0.627249u
30
+ ··· + 0.391427u + 1.21776
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
9
=
−2.40634u
31
− 3.60527u
30
+ ··· + 10.8122u − 12.0712
0.218460u
31
− 0.291561u
30
+ ··· + 2.47123u − 1.19894
a
2
=
1.12312u
31
+ 2.59965u
30
+ ··· − 8.28663u + 10.1381
0.672932u
31
+ 0.627249u
30
+ ··· + 0.391427u + 1.21776
a
1
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
a
5
=
−1.26155u
31
− 2.69290u
30
+ ··· + 10.8624u − 9.38644
−1.03259u
31
− 0.976097u
30
+ ··· − 1.09592u − 1.51802
a
4
=
−0.145728u
31
− 1.66532u
30
+ ··· + 12.4977u − 7.36530
−1.43787u
31
− 1.24356u
30
+ ··· − 2.16974u − 1.82405
a
10
=
−u
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
619975605
441171721
u
31
−
1193144219
441171721
u
30
+ ··· +
5092498689
441171721
u +
1186500205
441171721
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
32
+ 37u
31
+ ··· + 305u + 1
c
2
, c
5
u
32
+ 3u
31
+ ··· − 7u − 1
c
3
, c
8
, c
9
u
32
+ u
31
+ ··· − 4u + 4
c
4
u
32
− 3u
31
+ ··· + 12u − 4
c
6
, c
7
, c
10
c
11
u
32
+ 2u
31
+ ··· + 6u + 1
c
12
u
32
+ 4u
31
+ ··· − 20u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
32
− 77y
31
+ ··· − 68569y + 1
c
2
, c
5
y
32
− 37y
31
+ ··· − 305y + 1
c
3
, c
8
, c
9
y
32
− 27y
31
+ ··· − 208y + 16
c
4
y
32
+ 33y
31
+ ··· − 336y + 16
c
6
, c
7
, c
10
c
11
y
32
− 36y
31
+ ··· − 56y + 1
c
12
y
32
+ 36y
31
+ ··· − 568y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.647923 + 0.702860I
a = −0.11556 − 1.50516I
b = −1.60683 + 0.23584I
−5.40621 + 7.69916I 6.58632 − 5.74078I
u = 0.647923 − 0.702860I
a = −0.11556 + 1.50516I
b = −1.60683 − 0.23584I
−5.40621 − 7.69916I 6.58632 + 5.74078I
u = −0.934535
a = 0.920748
b = −1.27591
0.214319 11.1130
u = −0.528252 + 0.752905I
a = 0.219575 − 1.040740I
b = 1.65258 + 0.07144I
−9.84928 − 2.50165I 2.86477 + 2.84418I
u = −0.528252 − 0.752905I
a = 0.219575 + 1.040740I
b = 1.65258 − 0.07144I
−9.84928 + 2.50165I 2.86477 − 2.84418I
u = 0.383863 + 0.766338I
a = −0.382365 − 0.514382I
b = −1.62188 − 0.11406I
−6.19158 − 2.80814I 5.10038 + 0.76938I
u = 0.383863 − 0.766338I
a = −0.382365 + 0.514382I
b = −1.62188 + 0.11406I
−6.19158 + 2.80814I 5.10038 − 0.76938I
u = −0.750792
a = −1.84633
b = 0.310110
5.68749 17.7080
u = 0.505153 + 0.538065I
a = 0.446790 + 1.310620I
b = 0.571805 − 0.732824I
1.93515 + 4.07265I 8.92952 − 7.04568I
u = 0.505153 − 0.538065I
a = 0.446790 − 1.310620I
b = 0.571805 + 0.732824I
1.93515 − 4.07265I 8.92952 + 7.04568I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.395138 + 0.481331I
a = −0.551243 − 0.194239I
b = 0.642691 + 0.571386I
1.68006 − 0.53375I 7.96422 − 0.32995I
u = 0.395138 − 0.481331I
a = −0.551243 + 0.194239I
b = 0.642691 − 0.571386I
1.68006 + 0.53375I 7.96422 + 0.32995I
u = −1.40514 + 0.25786I
a = 0.848313 + 0.771560I
b = −1.59035 − 0.06318I
−0.512682 − 0.890588I 8.15263 + 0.I
u = −1.40514 − 0.25786I
a = 0.848313 − 0.771560I
b = −1.59035 + 0.06318I
−0.512682 + 0.890588I 8.15263 + 0.I
u = 1.44428 + 0.09174I
a = 0.42149 − 1.57076I
b = −0.611773 + 0.763659I
4.34598 + 2.60375I 7.93484 − 3.36675I
u = 1.44428 − 0.09174I
a = 0.42149 + 1.57076I
b = −0.611773 − 0.763659I
4.34598 − 2.60375I 7.93484 + 3.36675I
u = 1.44949
a = 0.265813
b = 1.38846
8.83218 9.93110
u = −1.45288 + 0.07180I
a = −0.88013 + 1.22133I
b = 0.795382 − 0.670343I
7.57700 − 1.16778I 11.36341 + 0.54162I
u = −1.45288 − 0.07180I
a = −0.88013 − 1.22133I
b = 0.795382 + 0.670343I
7.57700 + 1.16778I 11.36341 − 0.54162I
u = −1.51955 + 0.16796I
a = −0.08132 − 1.78171I
b = 0.464019 + 0.904411I
8.62955 − 6.62987I 12.50269 + 5.26876I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.51955 − 0.16796I
a = −0.08132 + 1.78171I
b = 0.464019 − 0.904411I
8.62955 + 6.62987I 12.50269 − 5.26876I
u = −0.299017 + 0.362824I
a = −0.106458 + 1.400860I
b = −0.751935 − 0.325766I
−1.32881 − 1.04587I 0.13410 + 4.26985I
u = −0.299017 − 0.362824I
a = −0.106458 − 1.400860I
b = −0.751935 + 0.325766I
−1.32881 + 1.04587I 0.13410 − 4.26985I
u = 1.52451 + 0.26507I
a = −0.93070 + 1.22794I
b = 1.61492 − 0.22198I
−3.16368 + 6.23347I 6.00000 − 3.63332I
u = 1.52451 − 0.26507I
a = −0.93070 − 1.22794I
b = 1.61492 + 0.22198I
−3.16368 − 6.23347I 6.00000 + 3.63332I
u = 0.451835
a = 0.432993
b = 0.202201
0.642131 15.9520
u = −1.58469
a = −0.466896
b = 0.692717
7.78318 17.5750
u = −1.58841 + 0.23470I
a = 0.92703 + 1.56136I
b = −1.56211 − 0.33651I
2.01653 − 11.20740I 0
u = −1.58841 − 0.23470I
a = 0.92703 − 1.56136I
b = −1.56211 + 0.33651I
2.01653 + 11.20740I 0
u = 1.61270
a = −0.848466
b = −0.0619566
13.8091 18.0020
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.68667
a = 1.46963
b = −1.31101
9.57646 6.00000
u = −0.145893
a = 9.44166
b = 1.06237
3.33910 1.65970
8
II. I
u
2
= hb − 1, a
2
+ 2a − 2u − 3, u
2
+ u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u − 1
a
12
=
u
−u + 1
a
3
=
a
1
a
8
=
u
−u
a
9
=
au − 2
−au − 2u
a
2
=
a + 1
1
a
1
=
1
0
a
5
=
−a
−1
a
4
=
−au − u + 1
au − a + u − 2
a
10
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
9
(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
10
(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
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.28825
b = 1.00000
4.27683 12.0000
u = 0.618034
a = −3.28825
b = 1.00000
4.27683 12.0000
u = −1.61803
a = −0.125968
b = 1.00000
12.1725 12.0000
u = −1.61803
a = −1.87403
b = 1.00000
12.1725 12.0000
12
III. I
u
3
= hb + 1, a − 1, u
2
− u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u − 1
a
12
=
u
−u − 1
a
3
=
1
−1
a
8
=
−u
u
a
9
=
−u
u
a
2
=
0
−1
a
1
=
−1
0
a
5
=
1
−1
a
4
=
1
−1
a
10
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2
13
(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
14
(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
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 1.00000
b = −1.00000
−0.657974 2.00000
u = 1.61803
a = 1.00000
b = −1.00000
7.23771 2.00000
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
32
+ 37u
31
+ ··· + 305u + 1)
c
2
((u − 1)
2
)(u + 1)
4
(u
32
+ 3u
31
+ ··· − 7u − 1)
c
3
, c
8
, c
9
u
2
(u
2
− 2)
2
(u
32
+ u
31
+ ··· − 4u + 4)
c
4
u
2
(u
2
− 2)
2
(u
32
− 3u
31
+ ··· + 12u − 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
32
+ 3u
31
+ ··· − 7u − 1)
c
6
, c
7
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
32
+ 2u
31
+ ··· + 6u + 1)
c
10
, c
11
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
32
+ 2u
31
+ ··· + 6u + 1)
c
12
((u
2
+ u − 1)
3
)(u
32
+ 4u
31
+ ··· − 20u + 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
32
− 77y
31
+ ··· − 68569y + 1)
c
2
, c
5
((y −1)
6
)(y
32
− 37y
31
+ ··· − 305y + 1)
c
3
, c
8
, c
9
y
2
(y −2)
4
(y
32
− 27y
31
+ ··· − 208y + 16)
c
4
y
2
(y −2)
4
(y
32
+ 33y
31
+ ··· − 336y + 16)
c
6
, c
7
, c
10
c
11
((y
2
− 3y + 1)
3
)(y
32
− 36y
31
+ ··· − 56y + 1)
c
12
((y
2
− 3y + 1)
3
)(y
32
+ 36y
31
+ ··· − 568y + 1)
18
