12n
0112
(K12n
0112
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 4 12 11 6 7 8 10
Solving Sequence
7,12 4,8
3 6 11 10 1 5 2 9
c
7
c
3
c
6
c
11
c
10
c
12
c
5
c
2
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h1077612353882483u
46
+ 6277936033129932u
45
+ ··· + 3412880928878836b − 505154446936889,
− 1.84787 × 10
15
u
46
− 8.27233 × 10
15
u
45
+ ··· + 3.41288 × 10
15
a − 4.21925 × 10
16
,
u
47
+ 4u
46
+ ··· + 19u + 1i
I
u
2
= hb + u, u
2
+ a − u + 3, u
3
− u
2
+ 2u − 1i
I
u
3
= h−2u
2
a − au − u
2
+ 5b − 3a − 3u + 1, a
2
+ 2u
2
+ a + 2, u
3
− u
2
+ 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 56 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
= h1.08 × 10
15
u
46
+ 6.28 × 10
15
u
45
+ · · · + 3.41 × 10
15
b − 5.05 ×
10
14
, −1.85 × 10
15
u
46
− 8.27 × 10
15
u
45
+ · · · + 3.41 × 10
15
a − 4.22 ×
10
16
, u
47
+ 4u
46
+ · · · + 19u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
0.541439u
46
+ 2.42386u
45
+ ··· + 24.5319u + 12.3627
−0.315749u
46
− 1.83948u
45
+ ··· − 7.06067u + 0.148014
a
8
=
1
u
2
a
3
=
0.225691u
46
+ 0.584374u
45
+ ··· + 17.4712u + 12.5108
−0.315749u
46
− 1.83948u
45
+ ··· − 7.06067u + 0.148014
a
6
=
−0.280491u
46
− 1.05919u
45
+ ··· − 11.4369u − 5.46550
−0.0627768u
46
− 0.766991u
45
+ ··· + 0.136172u − 0.280491
a
11
=
u
u
3
+ u
a
10
=
u
3
+ 2u
u
3
+ u
a
1
=
−u
7
− 4u
5
− 4u
3
−u
7
− 3u
5
− 2u
3
+ u
a
5
=
−0.348010u
46
− 0.729833u
45
+ ··· − 18.8545u − 5.94845
−0.0657486u
46
+ 0.160517u
45
+ ··· − 0.310668u − 0.351986
a
2
=
0.421710u
46
+ 1.79404u
45
+ ··· + 25.1802u + 9.15766
0.0657486u
46
− 0.160517u
45
+ ··· + 0.310668u + 0.351986
a
9
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
1830100709424405
1706440464439418
u
46
−
8067984018653329
1706440464439418
u
45
+ ···−
99643634186005859
3412880928878836
u −
34355223541311473
3412880928878836
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
47
+ 30u
46
+ ··· + 15u + 1
c
2
, c
4
u
47
− 4u
46
+ ··· − 7u − 1
c
3
, c
6
u
47
− 4u
46
+ ··· + 5u − 1
c
5
, c
9
u
47
− 3u
46
+ ··· − 1920u
2
+ 512
c
7
, c
8
, c
11
u
47
− 4u
46
+ ··· + 19u − 1
c
10
u
47
+ 4u
46
+ ··· + 847u − 49
c
12
u
47
− 22u
46
+ ··· + 8436145u + 61891
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
47
− 22y
46
+ ··· + 563y −1
c
2
, c
4
y
47
− 30y
46
+ ··· + 15y −1
c
3
, c
6
y
47
+ 6y
46
+ ··· + 15y −1
c
5
, c
9
y
47
+ 49y
46
+ ··· + 1966080y −262144
c
7
, c
8
, c
11
y
47
+ 38y
46
+ ··· + 299y −1
c
10
y
47
− 50y
46
+ ··· + 706923y −2401
c
12
y
47
− 78y
46
+ ··· + 77502601952059y −3830495881
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.990601
a = 0.833676
b = 0.664625
−5.24020 −20.8000
u = 0.786688 + 0.584594I
a = −0.577297 + 0.239191I
b = −0.688477 − 0.244449I
−4.21325 − 2.69895I −16.8130 + 7.1920I
u = 0.786688 − 0.584594I
a = −0.577297 − 0.239191I
b = −0.688477 + 0.244449I
−4.21325 + 2.69895I −16.8130 − 7.1920I
u = −0.937187 + 0.140092I
a = −1.77830 + 0.44317I
b = −1.08476 + 1.02775I
−11.2714 + 9.6299I −12.05023 − 5.56960I
u = −0.937187 − 0.140092I
a = −1.77830 − 0.44317I
b = −1.08476 − 1.02775I
−11.2714 − 9.6299I −12.05023 + 5.56960I
u = −0.909124 + 0.020827I
a = −1.61166 − 0.74978I
b = −1.07056 − 1.08227I
−11.11700 + 1.76651I −12.65951 − 0.89834I
u = −0.909124 − 0.020827I
a = −1.61166 + 0.74978I
b = −1.07056 + 1.08227I
−11.11700 − 1.76651I −12.65951 + 0.89834I
u = −0.884070 + 0.070567I
a = 1.80575 − 0.63524I
b = 1.10158 − 1.05658I
−6.89137 + 3.99655I −10.09960 − 2.77536I
u = −0.884070 − 0.070567I
a = 1.80575 + 0.63524I
b = 1.10158 + 1.05658I
−6.89137 − 3.99655I −10.09960 + 2.77536I
u = 0.066162 + 1.148650I
a = 0.853937 + 0.565919I
b = 0.908296 + 0.182917I
1.43778 − 0.23607I −8.00000 + 0.I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.066162 − 1.148650I
a = 0.853937 − 0.565919I
b = 0.908296 − 0.182917I
1.43778 + 0.23607I −8.00000 + 0.I
u = 0.193279 + 1.168140I
a = 0.93146 + 1.53635I
b = 0.201370 − 0.733758I
0.26621 − 2.20233I −8.00000 + 0.I
u = 0.193279 − 1.168140I
a = 0.93146 − 1.53635I
b = 0.201370 + 0.733758I
0.26621 + 2.20233I −8.00000 + 0.I
u = −0.165090 + 1.196660I
a = 1.36966 + 0.81390I
b = 0.37394 − 1.40206I
5.59945 + 4.78062I −8.00000 + 0.I
u = −0.165090 − 1.196660I
a = 1.36966 − 0.81390I
b = 0.37394 + 1.40206I
5.59945 − 4.78062I −8.00000 + 0.I
u = −0.083039 + 1.251340I
a = −1.050290 − 0.729908I
b = −0.032297 + 1.293670I
6.38366 − 1.10832I 0
u = −0.083039 − 1.251340I
a = −1.050290 + 0.729908I
b = −0.032297 − 1.293670I
6.38366 + 1.10832I 0
u = −0.528989 + 1.159830I
a = −0.276035 − 0.628526I
b = −1.099140 − 0.886724I
−8.15630 − 4.45150I 0
u = −0.528989 − 1.159830I
a = −0.276035 + 0.628526I
b = −1.099140 + 0.886724I
−8.15630 + 4.45150I 0
u = −0.431113 + 1.216070I
a = 0.183252 + 0.675767I
b = 1.17095 + 0.89948I
−3.36307 + 0.70782I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.431113 − 1.216070I
a = 0.183252 − 0.675767I
b = 1.17095 − 0.89948I
−3.36307 − 0.70782I 0
u = 0.268673 + 1.267170I
a = −1.155430 + 0.312483I
b = −0.561623 − 0.268923I
2.34688 − 3.35469I 0
u = 0.268673 − 1.267170I
a = −1.155430 − 0.312483I
b = −0.561623 + 0.268923I
2.34688 + 3.35469I 0
u = 0.195960 + 1.323790I
a = −1.87522 − 1.08808I
b = −0.483549 + 0.192110I
1.76552 − 3.06033I 0
u = 0.195960 − 1.323790I
a = −1.87522 + 1.08808I
b = −0.483549 − 0.192110I
1.76552 + 3.06033I 0
u = 0.660594
a = −1.60556
b = −0.440165
−1.60120 −4.93400
u = −0.442721 + 1.267860I
a = −1.46962 − 0.89724I
b = −0.94426 + 1.16097I
−7.25263 + 3.05957I 0
u = −0.442721 − 1.267860I
a = −1.46962 + 0.89724I
b = −0.94426 − 1.16097I
−7.25263 − 3.05957I 0
u = 0.492547 + 1.264850I
a = 0.648333 − 0.236834I
b = 0.695100 + 0.386206I
−1.35004 − 5.25916I 0
u = 0.492547 − 1.264850I
a = 0.648333 + 0.236834I
b = 0.695100 − 0.386206I
−1.35004 + 5.25916I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.431930 + 1.300870I
a = −0.097552 − 0.602796I
b = −1.16017 − 0.97433I
−7.00155 + 6.55990I 0
u = −0.431930 − 1.300870I
a = −0.097552 + 0.602796I
b = −1.16017 + 0.97433I
−7.00155 − 6.55990I 0
u = 0.113592 + 1.383630I
a = −0.110146 − 0.914625I
b = 0.324059 + 0.771489I
4.86422 − 2.82003I 0
u = 0.113592 − 1.383630I
a = −0.110146 + 0.914625I
b = 0.324059 − 0.771489I
4.86422 + 2.82003I 0
u = −0.403708 + 1.330620I
a = 1.43895 + 0.94313I
b = 1.01510 − 1.16548I
−2.50308 + 8.61469I 0
u = −0.403708 − 1.330620I
a = 1.43895 − 0.94313I
b = 1.01510 + 1.16548I
−2.50308 − 8.61469I 0
u = −0.41853 + 1.38032I
a = −1.39448 − 0.94283I
b = −1.03069 + 1.11636I
−6.4847 + 14.4827I 0
u = −0.41853 − 1.38032I
a = −1.39448 + 0.94283I
b = −1.03069 − 1.11636I
−6.4847 − 14.4827I 0
u = 0.532405 + 0.124560I
a = −0.76238 − 2.70239I
b = −0.221659 + 0.420216I
−2.77991 − 0.46414I −9.29269 − 10.51983I
u = 0.532405 − 0.124560I
a = −0.76238 + 2.70239I
b = −0.221659 − 0.420216I
−2.77991 + 0.46414I −9.29269 + 10.51983I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.380325 + 0.322466I
a = 0.088091 − 0.742546I
b = 0.519940 + 0.346223I
−0.572975 − 1.134450I −6.55719 + 6.13916I
u = 0.380325 − 0.322466I
a = 0.088091 + 0.742546I
b = 0.519940 − 0.346223I
−0.572975 + 1.134450I −6.55719 − 6.13916I
u = 0.20471 + 1.49097I
a = −0.255203 + 0.595995I
b = −0.547663 − 0.662712I
2.62514 − 6.12394I 0
u = 0.20471 − 1.49097I
a = −0.255203 − 0.595995I
b = −0.547663 + 0.662712I
2.62514 + 6.12394I 0
u = −0.394822 + 0.084126I
a = 0.10651 + 2.29856I
b = 0.240504 + 1.222520I
2.33626 − 2.65352I 2.71092 + 0.17214I
u = −0.394822 − 0.084126I
a = 0.10651 − 2.29856I
b = 0.240504 − 1.222520I
2.33626 + 2.65352I 2.71092 − 0.17214I
u = −0.0592454
a = 10.7472
b = 0.523564
−1.19029 −8.22650
9
II. I
u
2
= hb + u, u
2
+ a − u + 3, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−u
2
+ u − 3
−u
a
8
=
1
u
2
a
3
=
−u
2
− 3
−u
a
6
=
−u
−u
2
a
11
=
u
u
2
− u + 1
a
10
=
u
2
+ 1
u
2
− u + 1
a
1
=
−1
0
a
5
=
−u
−u
2
a
2
=
−u
2
− u − 2
−u
2
a
9
=
u
2
+ 1
u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12u
2
+ 11u − 24
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
8
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
4
, c
10
, c
12
u
3
− u
2
+ 1
c
5
, c
9
u
3
c
6
, c
11
u
3
+ u
2
+ 2u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
11
y
3
+ 3y
2
+ 2y −1
c
2
, c
4
, c
10
c
12
y
3
− y
2
+ 2y −1
c
5
, c
9
y
3
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −1.122560 + 0.744862I
b = −0.215080 − 1.307140I
6.04826 − 5.65624I −1.68581 + 7.63120I
u = 0.215080 − 1.307140I
a = −1.122560 − 0.744862I
b = −0.215080 + 1.307140I
6.04826 + 5.65624I −1.68581 − 7.63120I
u = 0.569840
a = −2.75488
b = −0.569840
−2.22691 −21.6280
13
III.
I
u
3
= h−2u
2
a − au − u
2
+ 5b − 3a − 3u + 1, a
2
+ 2u
2
+ a + 2, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
a
2
5
u
2
a +
1
5
u
2
+ ··· +
3
5
a −
1
5
a
8
=
1
u
2
a
3
=
2
5
u
2
a +
1
5
u
2
+ ··· +
8
5
a −
1
5
2
5
u
2
a +
1
5
u
2
+ ··· +
3
5
a −
1
5
a
6
=
1
5
u
2
a +
8
5
u
2
+ ··· +
4
5
a +
17
5
−
1
5
u
2
a +
2
5
u
2
+ ··· +
1
5
a +
8
5
a
11
=
u
u
2
− u + 1
a
10
=
u
2
+ 1
u
2
− u + 1
a
1
=
−1
0
a
5
=
1
5
u
2
a +
8
5
u
2
+ ··· +
4
5
a +
17
5
−
1
5
u
2
a +
2
5
u
2
+ ··· +
1
5
a +
8
5
a
2
=
2u
2
+ a − u + 3
−
1
5
u
2
a +
2
5
u
2
+ ··· +
1
5
a +
8
5
a
9
=
u
2
+ 1
u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
17
5
u
2
a +
24
5
au −
21
5
u
2
−
28
5
a +
2
5
u −
74
5
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
8
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
10
, c
12
(u
3
− u
2
+ 1)
2
c
5
, c
9
u
6
c
6
, c
11
(u
3
+ u
2
+ 2u + 1)
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
, c
10
c
12
(y
3
− y
2
+ 2y −1)
2
c
5
, c
9
y
6
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = 0.824718 − 0.424452I
b = −0.215080 + 1.307140I
6.04826 −4.98605 + 1.29886I
u = 0.215080 + 1.307140I
a = −1.82472 + 0.42445I
b = −0.569840
1.91067 − 2.82812I −11.5625 − 9.3388I
u = 0.215080 − 1.307140I
a = 0.824718 + 0.424452I
b = −0.215080 − 1.307140I
6.04826 −4.98605 − 1.29886I
u = 0.215080 − 1.307140I
a = −1.82472 − 0.42445I
b = −0.569840
1.91067 + 2.82812I −11.5625 + 9.3388I
u = 0.569840
a = −0.50000 + 1.54901I
b = −0.215080 + 1.307140I
1.91067 + 2.82812I −13.9515 − 6.1477I
u = 0.569840
a = −0.50000 − 1.54901I
b = −0.215080 − 1.307140I
1.91067 − 2.82812I −13.9515 + 6.1477I
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
3
)(u
47
+ 30u
46
+ ··· + 15u + 1)
c
2
((u
3
+ u
2
− 1)
3
)(u
47
− 4u
46
+ ··· − 7u − 1)
c
3
((u
3
− u
2
+ 2u − 1)
3
)(u
47
− 4u
46
+ ··· + 5u − 1)
c
4
((u
3
− u
2
+ 1)
3
)(u
47
− 4u
46
+ ··· − 7u − 1)
c
5
, c
9
u
9
(u
47
− 3u
46
+ ··· − 1920u
2
+ 512)
c
6
((u
3
+ u
2
+ 2u + 1)
3
)(u
47
− 4u
46
+ ··· + 5u − 1)
c
7
, c
8
((u
3
− u
2
+ 2u − 1)
3
)(u
47
− 4u
46
+ ··· + 19u − 1)
c
10
((u
3
− u
2
+ 1)
3
)(u
47
+ 4u
46
+ ··· + 847u − 49)
c
11
((u
3
+ u
2
+ 2u + 1)
3
)(u
47
− 4u
46
+ ··· + 19u − 1)
c
12
((u
3
− u
2
+ 1)
3
)(u
47
− 22u
46
+ ··· + 8436145u + 61891)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
+ 3y
2
+ 2y −1)
3
)(y
47
− 22y
46
+ ··· + 563y −1)
c
2
, c
4
((y
3
− y
2
+ 2y −1)
3
)(y
47
− 30y
46
+ ··· + 15y −1)
c
3
, c
6
((y
3
+ 3y
2
+ 2y −1)
3
)(y
47
+ 6y
46
+ ··· + 15y −1)
c
5
, c
9
y
9
(y
47
+ 49y
46
+ ··· + 1966080y −262144)
c
7
, c
8
, c
11
((y
3
+ 3y
2
+ 2y −1)
3
)(y
47
+ 38y
46
+ ··· + 299y −1)
c
10
((y
3
− y
2
+ 2y −1)
3
)(y
47
− 50y
46
+ ··· + 706923y −2401)
c
12
(y
3
− y
2
+ 2y −1)
3
· (y
47
− 78y
46
+ ··· + 77502601952059y −3830495881)
19
