12a
0236
(K12a
0236
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 5 11 12 1 4 8 9
Solving Sequence
8,11
12 9 1
4,7
3 10 5 6 2
c
11
c
8
c
12
c
7
c
3
c
10
c
4
c
6
c
2
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h15u
53
+ 20u
52
+ ··· + 2b − 8, 49u
53
+ 76u
52
+ ··· + 4a − 12, u
54
+ 3u
53
+ ··· + 3u − 1i
I
u
2
= hb, a
3
− a
2
u + a
2
− 2au + 4a − 2u + 3, u
2
− u − 1i
I
u
3
= hb + 1, a, u + 1i
* 3 irreducible components of dim
C
= 0, with total 61 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
=
h15u
53
+20u
52
+· · ·+2b−8, 49u
53
+76u
52
+· · ·+4a−12, u
54
+3u
53
+· · ·+3u−1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
9
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
−
49
4
u
53
− 19u
52
+ ··· −
105
4
u + 3
−
15
2
u
53
− 10u
52
+ ··· −
27
2
u + 4
a
7
=
u
u
a
3
=
−
7
4
u
53
−
17
4
u
52
+ ··· −
23
4
u −
9
4
3u
53
+
19
4
u
52
+ ··· + 7u −
5
4
a
10
=
u
3
− 2u
u
5
− 3u
3
+ u
a
5
=
9
4
u
53
−
327
4
u
51
+ ··· +
9
4
u − 5
19
2
u
53
+ 12u
52
+ ··· +
39
2
u − 5
a
6
=
1
4
u
53
+
3
4
u
52
+ ··· +
23
4
u +
5
4
−u
16
+ 10u
14
+ ··· − 6u
3
− 4u
2
a
2
=
−
1
4
u
52
−
1
2
u
51
+ ··· −
9
2
u −
1
4
1
4
u
53
+
1
2
u
52
+ ··· +
11
2
u
2
+
5
4
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −8u
53
− 6u
52
+ ··· − 19u +
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
54
+ 18u
53
+ ··· + 28u + 1
c
2
, c
5
u
54
+ 2u
53
+ ··· − 14u
2
+ 1
c
3
u
54
− 4u
53
+ ··· − 2672u + 433
c
4
, c
10
u
54
+ 2u
53
+ ··· − 224u − 64
c
7
, c
8
, c
9
c
11
, c
12
u
54
+ 3u
53
+ ··· + 3u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
54
+ 38y
53
+ ··· − 252y + 1
c
2
, c
5
y
54
− 18y
53
+ ··· − 28y + 1
c
3
y
54
− 22y
53
+ ··· − 7170760y + 187489
c
4
, c
10
y
54
+ 36y
53
+ ··· − 1024y + 4096
c
7
, c
8
, c
9
c
11
, c
12
y
54
− 73y
53
+ ··· − 33y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.996240 + 0.174836I
a = −0.046917 − 0.171089I
b = −1.041700 + 0.270047I
−0.93612 + 5.33064I 0
u = −0.996240 − 0.174836I
a = −0.046917 + 0.171089I
b = −1.041700 − 0.270047I
−0.93612 − 5.33064I 0
u = −0.921952 + 0.190975I
a = 0.120199 + 0.149841I
b = 0.923078 − 0.334826I
−0.122335 + 0.095210I −8.00000 + 0.I
u = −0.921952 − 0.190975I
a = 0.120199 − 0.149841I
b = 0.923078 + 0.334826I
−0.122335 − 0.095210I −8.00000 + 0.I
u = 1.064210 + 0.209793I
a = −0.63295 − 1.61348I
b = 0.282300 − 1.168280I
−4.75278 − 2.80286I 0
u = 1.064210 − 0.209793I
a = −0.63295 + 1.61348I
b = 0.282300 + 1.168280I
−4.75278 + 2.80286I 0
u = 1.030910 + 0.348284I
a = −0.95535 − 1.48256I
b = 0.541961 − 1.230030I
−3.03671 − 5.51505I 0
u = 1.030910 − 0.348284I
a = −0.95535 + 1.48256I
b = 0.541961 + 1.230030I
−3.03671 + 5.51505I 0
u = 1.050340 + 0.376575I
a = 0.96873 + 1.41375I
b = −0.57619 + 1.29485I
−4.25463 − 11.21320I 0
u = 1.050340 − 0.376575I
a = 0.96873 − 1.41375I
b = −0.57619 − 1.29485I
−4.25463 + 11.21320I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.851855 + 0.026742I
a = −0.20081 − 2.78618I
b = 0.052036 − 0.745442I
1.33981 − 2.97841I −16.2532 + 3.7844I
u = 0.851855 − 0.026742I
a = −0.20081 + 2.78618I
b = 0.052036 + 0.745442I
1.33981 + 2.97841I −16.2532 − 3.7844I
u = 1.113960 + 0.306274I
a = 0.77810 + 1.39695I
b = −0.384474 + 1.331420I
−9.35844 − 4.90130I 0
u = 1.113960 − 0.306274I
a = 0.77810 − 1.39695I
b = −0.384474 − 1.331420I
−9.35844 + 4.90130I 0
u = 1.175560 + 0.167588I
a = 0.43398 + 1.38174I
b = −0.143925 + 1.276040I
−6.63511 + 1.61390I 0
u = 1.175560 − 0.167588I
a = 0.43398 − 1.38174I
b = −0.143925 − 1.276040I
−6.63511 − 1.61390I 0
u = −0.519490 + 0.531321I
a = −0.713490 − 0.165731I
b = −0.208766 + 1.145180I
−1.05139 − 3.92853I −12.01559 + 2.10078I
u = −0.519490 − 0.531321I
a = −0.713490 + 0.165731I
b = −0.208766 − 1.145180I
−1.05139 + 3.92853I −12.01559 − 2.10078I
u = −0.723166
a = 0.147422
b = 0.462202
−1.27288 −6.75210
u = −0.542660 + 0.430784I
a = 0.615074 + 0.066547I
b = 0.268273 − 0.941014I
−0.103530 + 1.138840I −10.54250 − 3.54234I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.542660 − 0.430784I
a = 0.615074 − 0.066547I
b = 0.268273 + 0.941014I
−0.103530 − 1.138840I −10.54250 + 3.54234I
u = −0.364228 + 0.577449I
a = −0.934188 − 0.148020I
b = 0.138461 + 1.174430I
−4.71225 + 1.89294I −15.7177 − 3.8918I
u = −0.364228 − 0.577449I
a = −0.934188 + 0.148020I
b = 0.138461 − 1.174430I
−4.71225 − 1.89294I −15.7177 + 3.8918I
u = −0.250036 + 0.633419I
a = −1.104300 − 0.206805I
b = 0.417478 + 1.191560I
−0.22294 + 7.78581I −9.62304 − 7.92359I
u = −0.250036 − 0.633419I
a = −1.104300 + 0.206805I
b = 0.417478 − 1.191560I
−0.22294 − 7.78581I −9.62304 + 7.92359I
u = −0.225695 + 0.589084I
a = 1.140710 + 0.141782I
b = −0.419400 − 1.081010I
0.86035 + 2.32908I −7.32309 − 3.24267I
u = −0.225695 − 0.589084I
a = 1.140710 − 0.141782I
b = −0.419400 + 1.081010I
0.86035 − 2.32908I −7.32309 + 3.24267I
u = 1.55391 + 0.02757I
a = −0.098217 − 0.452334I
b = −0.140479 − 0.824642I
−6.99883 − 2.48049I 0
u = 1.55391 − 0.02757I
a = −0.098217 + 0.452334I
b = −0.140479 + 0.824642I
−6.99883 + 2.48049I 0
u = −0.281495 + 0.312842I
a = 0.916828 − 0.383169I
b = −0.123921 − 0.686515I
−0.459739 + 0.937129I −7.88969 − 7.07124I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.281495 − 0.312842I
a = 0.916828 + 0.383169I
b = −0.123921 + 0.686515I
−0.459739 − 0.937129I −7.88969 + 7.07124I
u = 0.135244 + 0.362504I
a = 2.08092 + 0.05555I
b = −0.644171 − 0.344545I
3.08884 + 1.84557I −1.95761 − 1.92890I
u = 0.135244 − 0.362504I
a = 2.08092 − 0.05555I
b = −0.644171 + 0.344545I
3.08884 − 1.84557I −1.95761 + 1.92890I
u = 0.208405 + 0.314817I
a = −2.34049 − 0.17879I
b = 0.644426 + 0.215596I
2.77868 − 3.61490I −2.24333 + 5.08342I
u = 0.208405 − 0.314817I
a = −2.34049 + 0.17879I
b = 0.644426 − 0.215596I
2.77868 + 3.61490I −2.24333 − 5.08342I
u = 1.63895
a = −0.245628
b = −0.507248
−9.62995 0
u = −1.69921 + 0.00586I
a = 0.06322 − 2.33836I
b = −0.045647 − 1.071130I
−7.84998 + 3.09710I 0
u = −1.69921 − 0.00586I
a = 0.06322 + 2.33836I
b = −0.045647 + 1.071130I
−7.84998 − 3.09710I 0
u = 1.70201 + 0.03703I
a = −0.494003 − 0.097988I
b = −1.107300 − 0.281751I
−9.45410 − 0.92395I 0
u = 1.70201 − 0.03703I
a = −0.494003 + 0.097988I
b = −1.107300 + 0.281751I
−9.45410 + 0.92395I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.72317 + 0.04214I
a = 0.547011 + 0.098121I
b = 1.263400 + 0.305417I
−10.67310 − 6.19112I 0
u = 1.72317 − 0.04214I
a = 0.547011 − 0.098121I
b = 1.263400 − 0.305417I
−10.67310 + 6.19112I 0
u = 1.72764
a = 0.546125
b = 1.28414
−14.7580 0
u = −1.72994 + 0.09259I
a = 0.38653 − 1.80892I
b = −0.64312 − 1.34017I
−12.8350 + 7.3215I 0
u = −1.72994 − 0.09259I
a = 0.38653 + 1.80892I
b = −0.64312 + 1.34017I
−12.8350 − 7.3215I 0
u = −1.73510 + 0.10150I
a = −0.37724 + 1.76762I
b = 0.69739 + 1.37855I
−14.1294 + 13.1902I 0
u = −1.73510 − 0.10150I
a = −0.37724 − 1.76762I
b = 0.69739 − 1.37855I
−14.1294 − 13.1902I 0
u = −1.73884 + 0.05594I
a = 0.26375 − 1.92837I
b = −0.39367 − 1.37924I
−14.8035 + 3.9179I 0
u = −1.73884 − 0.05594I
a = 0.26375 + 1.92837I
b = −0.39367 + 1.37924I
−14.8035 − 3.9179I 0
u = −1.75253 + 0.07823I
a = −0.27989 + 1.81978I
b = 0.53764 + 1.48090I
−19.6252 + 6.5170I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.75253 − 0.07823I
a = −0.27989 − 1.81978I
b = 0.53764 − 1.48090I
−19.6252 − 6.5170I 0
u = −1.75789 + 0.04187I
a = −0.16460 + 1.90001I
b = 0.29057 + 1.50492I
−17.1812 − 0.7272I 0
u = −1.75789 − 0.04187I
a = −0.16460 − 1.90001I
b = 0.29057 − 1.50492I
−17.1812 + 0.7272I 0
u = 0.168072
a = −3.39312
b = 0.392381
−1.32970 −6.13160
10
II. I
u
2
= hb, a
3
− a
2
u + a
2
− 2au + 4a − 2u + 3, u
2
− u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u + 1
a
9
=
−u
−u − 1
a
1
=
−u
−u
a
4
=
a
0
a
7
=
u
u
a
3
=
−au
−au − a
a
10
=
1
0
a
5
=
a
0
a
6
=
a
2
u + u
u
a
2
=
−a
2
u − a
2
− u
−2a
2
u − a
2
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −10a
2
u − 9a
2
+ 6au + a − 3u − 21
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
10
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
, c
9
(u
2
+ u − 1)
3
c
11
, c
12
(u
2
− u − 1)
3
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
4
, c
10
y
6
c
7
, c
8
, c
9
c
11
, c
12
(y
2
− 3y + 1)
3
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −0.922021
b = 0
−2.10041 −19.0460
u = −0.618034
a = −0.34801 + 2.11500I
b = 0
2.03717 + 2.82812I −5.93195 − 1.57712I
u = −0.618034
a = −0.34801 − 2.11500I
b = 0
2.03717 − 2.82812I −5.93195 + 1.57712I
u = 1.61803
a = 0.132927 + 0.807858I
b = 0
−5.85852 − 2.82812I −8.44207 + 3.24268I
u = 1.61803
a = 0.132927 − 0.807858I
b = 0
−5.85852 + 2.82812I −8.44207 − 3.24268I
u = 1.61803
a = 0.352181
b = 0
−9.99610 −25.2060
14
III. I
u
3
= hb + 1, a, u + 1i
(i) Arc colorings
a
8
=
0
−1
a
11
=
1
0
a
12
=
1
1
a
9
=
1
0
a
1
=
0
1
a
4
=
0
−1
a
7
=
−1
−1
a
3
=
−1
−2
a
10
=
1
1
a
5
=
−1
−2
a
6
=
0
1
a
2
=
−1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −18
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
9
, c
11
c
12
u + 1
c
4
, c
10
u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = −1.00000
−4.93480 −18.0000
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
3
− u
2
+ 2u − 1)
2
(u
54
+ 18u
53
+ ··· + 28u + 1)
c
2
(u + 1)(u
3
+ u
2
− 1)
2
(u
54
+ 2u
53
+ ··· − 14u
2
+ 1)
c
3
(u + 1)(u
3
− u
2
+ 2u − 1)
2
(u
54
− 4u
53
+ ··· − 2672u + 433)
c
4
, c
10
u
6
(u − 1)(u
54
+ 2u
53
+ ··· − 224u − 64)
c
5
(u + 1)(u
3
− u
2
+ 1)
2
(u
54
+ 2u
53
+ ··· − 14u
2
+ 1)
c
6
(u + 1)(u
3
+ u
2
+ 2u + 1)
2
(u
54
+ 18u
53
+ ··· + 28u + 1)
c
7
, c
8
, c
9
(u + 1)(u
2
+ u − 1)
3
(u
54
+ 3u
53
+ ··· + 3u − 1)
c
11
, c
12
(u + 1)(u
2
− u − 1)
3
(u
54
+ 3u
53
+ ··· + 3u − 1)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y −1)(y
3
+ 3y
2
+ 2y −1)
2
(y
54
+ 38y
53
+ ··· − 252y + 1)
c
2
, c
5
(y −1)(y
3
− y
2
+ 2y −1)
2
(y
54
− 18y
53
+ ··· − 28y + 1)
c
3
(y −1)(y
3
+ 3y
2
+ 2y −1)
2
(y
54
− 22y
53
+ ··· − 7170760y + 187489)
c
4
, c
10
y
6
(y −1)(y
54
+ 36y
53
+ ··· − 1024y + 4096)
c
7
, c
8
, c
9
c
11
, c
12
(y −1)(y
2
− 3y + 1)
3
(y
54
− 73y
53
+ ··· − 33y + 1)
20
