12n
0040
(K12n
0040
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 12 11 4 6 7 10 9
Solving Sequence
2,6
5 3
1,10
9 4 12 7 11 8
c
5
c
2
c
1
c
9
c
4
c
12
c
6
c
11
c
7
c
3
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h34u
42
+ 241u
41
+ ··· + 32b + 95, 27u
42
+ 149u
41
+ ··· + 32a + 121, u
43
+ 7u
42
+ ··· + 8u + 1i
I
u
2
= h−au + 3b + 2a, a
6
− a
5
u − a
5
− 3a
4
u + 12a
3
u − 6a
3
− 9au + 18a − 27, u
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 55 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
= h34u
42
+ 241u
41
+ · · · + 32b + 95, 27u
42
+ 149u
41
+ · · · + 32a +
121, u
43
+ 7u
42
+ · · · + 8u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
10
=
−0.843750u
42
− 4.65625u
41
+ ··· − 25.8125u − 3.78125
−1.06250u
42
− 7.53125u
41
+ ··· − 14.0313u − 2.96875
a
9
=
−1.90625u
42
− 12.1875u
41
+ ··· − 39.8438u − 6.75000
−1.06250u
42
− 7.53125u
41
+ ··· − 14.0313u − 2.96875
a
4
=
−u
3
u
3
+ u
a
12
=
0.0312500u
41
+ 0.187500u
40
+ ··· + 0.218750u − 0.968750
−
1
32
u
42
−
7
32
u
41
+ ··· +
7
4
u −
1
32
a
7
=
−0.156250u
42
− 0.687500u
41
+ ··· + 0.0312500u + 1.31250
−0.281250u
42
− 2.09375u
41
+ ··· − 3.18750u − 0.406250
a
11
=
−0.125000u
42
− 1.53125u
41
+ ··· − 3.09375u − 0.843750
0.281250u
42
+ 2.15625u
41
+ ··· + 4.62500u + 0.468750
a
8
=
1.28125u
42
+ 9.50000u
41
+ ··· + 47.5313u + 7.81250
0.875000u
42
+ 5.59375u
41
+ ··· + 9.15625u + 2.03125
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
21
4
u
42
−
619
16
u
41
+ ··· −
887
8
u −
183
8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
43
+ 29u
42
+ ··· − 6u − 1
c
2
, c
5
u
43
+ 7u
42
+ ··· + 8u + 1
c
3
u
43
− 7u
42
+ ··· − 15u + 2
c
4
, c
8
u
43
− u
42
+ ··· + 4096u + 4096
c
6
u
43
− 9u
42
+ ··· + 301u − 32
c
7
, c
10
u
43
− 3u
42
+ ··· − 4u + 1
c
9
u
43
+ 3u
42
+ ··· + u
2
+ 1
c
11
u
43
− 19u
42
+ ··· − 2u − 1
c
12
u
43
− 13u
42
+ ··· − 396502u − 27289
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
43
− 23y
42
+ ··· − 70y −1
c
2
, c
5
y
43
+ 29y
42
+ ··· − 6y −1
c
3
y
43
− 75y
42
+ ··· + 109y −4
c
4
, c
8
y
43
− 65y
42
+ ··· + 150994944y −16777216
c
6
y
43
− 7y
42
+ ··· + 2985y −1024
c
7
, c
10
y
43
− 19y
42
+ ··· − 2y −1
c
9
y
43
− 59y
42
+ ··· − 2y −1
c
11
y
43
+ 13y
42
+ ··· − 54y −1
c
12
y
43
− 119y
42
+ ··· − 1101094234046y −744689521
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.108850 + 1.019930I
a = −2.54446 − 0.82403I
b = 0.887627 + 0.888202I
0.432070 − 0.070776I −2.00000 + 0.I
u = −0.108850 − 1.019930I
a = −2.54446 + 0.82403I
b = 0.887627 − 0.888202I
0.432070 + 0.070776I −2.00000 + 0.I
u = −1.03310
a = 1.81625
b = −1.75986
−5.16147 0.303180
u = 0.631242 + 0.697176I
a = −0.026925 + 1.068100I
b = 0.114579 − 0.360320I
1.44756 + 4.15054I −1.66670 − 7.98021I
u = 0.631242 − 0.697176I
a = −0.026925 − 1.068100I
b = 0.114579 + 0.360320I
1.44756 − 4.15054I −1.66670 + 7.98021I
u = 0.630221 + 0.873340I
a = 0.381127 − 0.768103I
b = −0.219981 + 0.236881I
0.968100 + 0.793721I 0
u = 0.630221 − 0.873340I
a = 0.381127 + 0.768103I
b = −0.219981 − 0.236881I
0.968100 − 0.793721I 0
u = −1.093460 + 0.077172I
a = 1.96976 − 0.17734I
b = −1.81042 + 0.05602I
−9.35130 + 7.40339I 0. − 4.50100I
u = −1.093460 − 0.077172I
a = 1.96976 + 0.17734I
b = −1.81042 − 0.05602I
−9.35130 − 7.40339I 0. + 4.50100I
u = 0.356528 + 0.829227I
a = −0.809224 − 0.565549I
b = 0.200199 + 0.315601I
−0.32126 + 1.54787I −2.12659 − 4.62507I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.356528 − 0.829227I
a = −0.809224 + 0.565549I
b = 0.200199 − 0.315601I
−0.32126 − 1.54787I −2.12659 + 4.62507I
u = −1.097840 + 0.043281I
a = −1.97365 + 0.09899I
b = 1.81083 − 0.03124I
−11.14880 + 1.85775I 0
u = −1.097840 − 0.043281I
a = −1.97365 − 0.09899I
b = 1.81083 + 0.03124I
−11.14880 − 1.85775I 0
u = −0.231681 + 1.107840I
a = −3.17472 − 0.52343I
b = 1.31188 + 0.91141I
−1.74448 − 7.62411I 0
u = −0.231681 − 1.107840I
a = −3.17472 + 0.52343I
b = 1.31188 − 0.91141I
−1.74448 + 7.62411I 0
u = 0.441020 + 1.047300I
a = −1.087770 + 0.250841I
b = 0.447924 + 0.048017I
−1.10611 + 1.42382I 0
u = 0.441020 − 1.047300I
a = −1.087770 − 0.250841I
b = 0.447924 − 0.048017I
−1.10611 − 1.42382I 0
u = −0.173823 + 1.139560I
a = 2.91930 + 0.38726I
b = −1.21169 − 0.76310I
−4.07750 − 2.66673I 0
u = −0.173823 − 1.139560I
a = 2.91930 − 0.38726I
b = −1.21169 + 0.76310I
−4.07750 + 2.66673I 0
u = 0.555429 + 1.053250I
a = 0.937490 − 0.560097I
b = −0.423819 + 0.102545I
−0.20886 + 5.67924I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.555429 − 1.053250I
a = 0.937490 + 0.560097I
b = −0.423819 − 0.102545I
−0.20886 − 5.67924I 0
u = −0.013036 + 1.234440I
a = 2.43278 + 0.00691I
b = −1.046990 − 0.429320I
−5.51855 − 0.23082I 0
u = −0.013036 − 1.234440I
a = 2.43278 − 0.00691I
b = −1.046990 + 0.429320I
−5.51855 + 0.23082I 0
u = 0.067968 + 1.257240I
a = −2.24159 + 0.11300I
b = 0.972223 + 0.322115I
−4.39423 + 4.95924I 0
u = 0.067968 − 1.257240I
a = −2.24159 − 0.11300I
b = 0.972223 − 0.322115I
−4.39423 − 4.95924I 0
u = −0.51561 + 1.34254I
a = 3.53483 − 0.69986I
b = −1.90537 − 0.30670I
−9.34262 − 5.50773I 0
u = −0.51561 − 1.34254I
a = 3.53483 + 0.69986I
b = −1.90537 + 0.30670I
−9.34262 + 5.50773I 0
u = −0.57404 + 1.34689I
a = 3.56336 − 0.81250I
b = −1.96829 − 0.24490I
−13.3037 − 13.3388I 0
u = −0.57404 − 1.34689I
a = 3.56336 + 0.81250I
b = −1.96829 + 0.24490I
−13.3037 + 13.3388I 0
u = −0.029590 + 0.530131I
a = 1.15627 + 1.68061I
b = 0.135978 − 0.751914I
1.74801 − 1.44634I 0.320350 + 0.404730I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.029590 − 0.530131I
a = 1.15627 − 1.68061I
b = 0.135978 + 0.751914I
1.74801 + 1.44634I 0.320350 − 0.404730I
u = −0.55869 + 1.36236I
a = −3.52852 + 0.79270I
b = 1.93730 + 0.24369I
−15.2690 − 7.7521I 0
u = −0.55869 − 1.36236I
a = −3.52852 − 0.79270I
b = 1.93730 − 0.24369I
−15.2690 + 7.7521I 0
u = −0.50713 + 1.39889I
a = −3.43056 + 0.73543I
b = 1.84983 + 0.24468I
−15.7130 − 3.8589I 0
u = −0.50713 − 1.39889I
a = −3.43056 − 0.73543I
b = 1.84983 − 0.24468I
−15.7130 + 3.8589I 0
u = −0.48108 + 1.40912I
a = 3.38956 − 0.70682I
b = −1.81208 − 0.24952I
−14.0896 + 1.7953I 0
u = −0.48108 − 1.40912I
a = 3.38956 + 0.70682I
b = −1.81208 + 0.24952I
−14.0896 − 1.7953I 0
u = 0.395813 + 0.305739I
a = 0.39533 + 1.71062I
b = 0.047086 − 0.461449I
1.64146 − 1.44258I 1.62393 + 1.37217I
u = 0.395813 − 0.305739I
a = 0.39533 − 1.71062I
b = 0.047086 + 0.461449I
1.64146 + 1.44258I 1.62393 − 1.37217I
u = −0.355224 + 0.327355I
a = 0.46521 + 1.45996I
b = 0.831377 − 0.614700I
0.49380 + 5.07125I −1.64078 − 7.14289I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.355224 − 0.327355I
a = 0.46521 − 1.45996I
b = 0.831377 + 0.614700I
0.49380 − 5.07125I −1.64078 + 7.14289I
u = −0.321614 + 0.162353I
a = −0.735715 − 0.820169I
b = −0.768263 + 0.294664I
−1.36971 + 0.61078I −6.09478 − 1.69888I
u = −0.321614 − 0.162353I
a = −0.735715 + 0.820169I
b = −0.768263 − 0.294664I
−1.36971 − 0.61078I −6.09478 + 1.69888I
9
II. I
u
2
= h−au + 3b + 2a, −a
5
u − 3a
4
u + · · · + 18a − 27, u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u − 1
a
3
=
u
u − 1
a
1
=
−1
0
a
10
=
a
1
3
au −
2
3
a
a
9
=
1
3
au +
1
3
a
1
3
au −
2
3
a
a
4
=
1
u − 1
a
12
=
−
1
3
a
2
− 1
−
1
3
a
2
u +
1
3
a
2
a
7
=
1
9
a
4
u +
1
3
a
2
u + ··· −
1
3
a
2
+ 1
−
1
9
a
4
u
a
11
=
2
9
a
4
u −
1
3
a
2
u + ··· +
1
3
a
2
− 1
−
1
9
a
4
u
a
8
=
1
3
au +
1
3
a
1
3
au −
2
3
a
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −
2
27
a
5
u +
1
27
a
5
−
4
9
a
4
u +
2
9
a
3
u −
4
9
a
3
+
5
3
a
2
u − a
2
−
2
3
au +
10
3
a − 5u + 2
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
6
c
2
(u
2
+ u + 1)
6
c
4
, c
8
u
12
c
6
, c
11
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
c
7
, c
9
, c
12
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
c
10
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
6
c
4
, c
8
y
12
c
6
, c
11
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
c
7
, c
9
, c
10
c
12
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.066864 + 1.367670I
b = −0.428243 − 0.664531I
1.89061 + 1.10558I 3.93112 − 2.76498I
u = 0.500000 + 0.866025I
a = 1.217870 − 0.625927I
b = −0.428243 + 0.664531I
1.89061 + 2.95419I 0.42156 − 3.46552I
u = 0.500000 + 0.866025I
a = −1.24734 − 1.31124I
b = 1.002190 + 0.295542I
−1.89061 + 1.10558I −7.50338 − 2.58970I
u = 0.500000 + 0.866025I
a = −1.75924 − 0.42461I
b = 1.002190 − 0.295542I
−1.89061 + 2.95419I −5.61650 − 4.08278I
u = 0.500000 + 0.866025I
a = 2.09482 + 0.09194I
b = −1.073950 + 0.558752I
− 3.66314I −4.13964 + 2.11509I
u = 0.500000 + 0.866025I
a = 1.12703 + 1.76820I
b = −1.073950 − 0.558752I
7.72290I −1.09315 − 8.26466I
u = 0.500000 − 0.866025I
a = 0.066864 − 1.367670I
b = −0.428243 + 0.664531I
1.89061 − 1.10558I 3.93112 + 2.76498I
u = 0.500000 − 0.866025I
a = 1.217870 + 0.625927I
b = −0.428243 − 0.664531I
1.89061 − 2.95419I 0.42156 + 3.46552I
u = 0.500000 − 0.866025I
a = −1.24734 + 1.31124I
b = 1.002190 − 0.295542I
−1.89061 − 1.10558I −7.50338 + 2.58970I
u = 0.500000 − 0.866025I
a = −1.75924 + 0.42461I
b = 1.002190 + 0.295542I
−1.89061 − 2.95419I −5.61650 + 4.08278I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 − 0.866025I
a = 2.09482 − 0.09194I
b = −1.073950 − 0.558752I
3.66314I −4.13964 − 2.11509I
u = 0.500000 − 0.866025I
a = 1.12703 − 1.76820I
b = −1.073950 + 0.558752I
− 7.72290I −1.09315 + 8.26466I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
6
)(u
43
+ 29u
42
+ ··· − 6u − 1)
c
2
((u
2
+ u + 1)
6
)(u
43
+ 7u
42
+ ··· + 8u + 1)
c
3
((u
2
− u + 1)
6
)(u
43
− 7u
42
+ ··· − 15u + 2)
c
4
, c
8
u
12
(u
43
− u
42
+ ··· + 4096u + 4096)
c
5
((u
2
− u + 1)
6
)(u
43
+ 7u
42
+ ··· + 8u + 1)
c
6
((u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
)(u
43
− 9u
42
+ ··· + 301u − 32)
c
7
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
43
− 3u
42
+ ··· − 4u + 1)
c
9
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
43
+ 3u
42
+ ··· + u
2
+ 1)
c
10
((u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
)(u
43
− 3u
42
+ ··· − 4u + 1)
c
11
((u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
)(u
43
− 19u
42
+ ··· − 2u − 1)
c
12
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
43
− 13u
42
+ ··· − 396502u − 27289)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
43
− 23y
42
+ ··· − 70y − 1)
c
2
, c
5
((y
2
+ y + 1)
6
)(y
43
+ 29y
42
+ ··· − 6y − 1)
c
3
((y
2
+ y + 1)
6
)(y
43
− 75y
42
+ ··· + 109y − 4)
c
4
, c
8
y
12
(y
43
− 65y
42
+ ··· + 1.50995 × 10
8
y − 1.67772 × 10
7
)
c
6
((y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
)(y
43
− 7y
42
+ ··· + 2985y − 1024)
c
7
, c
10
((y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
)(y
43
− 19y
42
+ ··· − 2y − 1)
c
9
((y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
)(y
43
− 59y
42
+ ··· − 2y − 1)
c
11
((y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
)(y
43
+ 13y
42
+ ··· − 54y − 1)
c
12
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
· (y
43
− 119y
42
+ ··· − 1101094234046y − 744689521)
16
