12a
0881
(K12a
0881
)
A knot diagram
1
Linearized knot diagam
4 6 7 10 3 2 11 12 1 5 8 9
Solving Sequence
8,12
9
1,4
2 10 5 11 7 3 6
c
8
c
12
c
1
c
9
c
4
c
11
c
7
c
3
c
6
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h30u
49
+ 63u
48
+ ··· + 4b + 19, 7u
48
+ 15u
47
+ ··· + 4a − 3, u
50
+ 4u
49
+ ··· − u + 1i
I
u
2
= hb + a, a
3
+ a
2
− 1, u
2
− u − 1i
* 2 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
=
h30u
49
+63u
48
+· · ·+4b+19, 7u
48
+15u
47
+· · ·+4a−3, u
50
+4u
49
+· · ·−u+1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
−
7
4
u
48
−
15
4
u
47
+ ··· −
7
2
u +
3
4
−
15
2
u
49
−
63
4
u
48
+ ··· + 6u −
19
4
a
2
=
1
4
u
49
+
3
4
u
48
+ ··· +
23
4
u + 1
−
1
4
u
49
−
3
4
u
48
+ ··· − 5u
2
+
1
4
u
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
5
=
−20u
49
−
185
4
u
48
+ ··· +
29
2
u −
43
4
51
2
u
49
+
235
4
u
48
+ ··· − 23u +
55
4
a
11
=
−u
u
a
7
=
−u
2
+ 1
u
2
a
3
=
−
111
4
u
49
− 65u
48
+ ··· +
85
4
u −
63
4
34.7500u
49
+ 81.2500u
48
+ ··· − 32.7500u + 19.5000
a
6
=
−
9
2
u
49
−
21
2
u
48
+ ··· + 4u − 2
21
4
u
49
+
51
4
u
48
+ ··· −
19
4
u + 3
(ii) Obstruction class = −1
(iii) Cusp Shapes = 25u
49
+ 59u
48
+ ··· −
83
2
u +
25
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
50
− 9u
49
+ ··· + 1832u + 113
c
2
, c
5
, c
6
u
50
− 3u
49
+ ··· − 10u + 1
c
3
u
50
+ 3u
49
+ ··· − 2392u + 241
c
4
, c
10
u
50
− u
49
+ ··· − 96u + 64
c
7
, c
8
, c
9
c
11
, c
12
u
50
− 4u
49
+ ··· + u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
50
+ 31y
49
+ ··· − 5984830y + 12769
c
2
, c
5
, c
6
y
50
+ 47y
49
+ ··· − 54y + 1
c
3
y
50
+ 11y
49
+ ··· − 2239214y + 58081
c
4
, c
10
y
50
− 35y
49
+ ··· − 54272y + 4096
c
7
, c
8
, c
9
c
11
, c
12
y
50
− 68y
49
+ ··· + 9y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.952202 + 0.085743I
a = 0.263868 + 0.811802I
b = 0.109466 + 1.005260I
1.99660 − 1.56126I 0. + 4.54210I
u = −0.952202 − 0.085743I
a = 0.263868 − 0.811802I
b = 0.109466 − 1.005260I
1.99660 + 1.56126I 0. − 4.54210I
u = −1.051590 + 0.116393I
a = −0.460087 − 1.130740I
b = 0.029530 − 1.139970I
7.55680 − 4.35419I 0
u = −1.051590 − 0.116393I
a = −0.460087 + 1.130740I
b = 0.029530 + 1.139970I
7.55680 + 4.35419I 0
u = 0.926369 + 0.110294I
a = −0.951862 + 0.190261I
b = 1.207080 − 0.506059I
4.83990 + 3.67287I 11.52712 − 5.42395I
u = 0.926369 − 0.110294I
a = −0.951862 − 0.190261I
b = 1.207080 + 0.506059I
4.83990 − 3.67287I 11.52712 + 5.42395I
u = 0.926358
a = 1.08741
b = −1.30017
1.12359 7.78030
u = 1.075460 + 0.335893I
a = 0.318078 + 0.440056I
b = 0.173538 + 1.100140I
5.70451 + 6.74184I 0
u = 1.075460 − 0.335893I
a = 0.318078 − 0.440056I
b = 0.173538 − 1.100140I
5.70451 − 6.74184I 0
u = 1.101720 + 0.265116I
a = −0.631574 − 0.369790I
b = 0.069190 − 0.623696I
6.52824 + 2.59024I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.101720 − 0.265116I
a = −0.631574 + 0.369790I
b = 0.069190 + 0.623696I
6.52824 − 2.59024I 0
u = 1.091080 + 0.380456I
a = −0.163597 − 0.625898I
b = −0.536851 − 1.297450I
11.5454 + 10.2862I 0
u = 1.091080 − 0.380456I
a = −0.163597 + 0.625898I
b = −0.536851 + 1.297450I
11.5454 − 10.2862I 0
u = −0.740193 + 0.295777I
a = −0.361219 − 0.137662I
b = −0.870973 − 0.536080I
4.15678 − 0.49764I 9.03893 + 1.33769I
u = −0.740193 − 0.295777I
a = −0.361219 + 0.137662I
b = −0.870973 + 0.536080I
4.15678 + 0.49764I 9.03893 − 1.33769I
u = 1.199130 + 0.267015I
a = 0.903623 + 0.631411I
b = 0.208467 + 0.051458I
13.08760 + 0.25307I 0
u = 1.199130 − 0.267015I
a = 0.903623 − 0.631411I
b = 0.208467 − 0.051458I
13.08760 − 0.25307I 0
u = −0.470779 + 0.592707I
a = 0.805640 − 0.073631I
b = −0.525112 + 1.088250I
7.73029 + 2.70190I 9.92902 − 0.03562I
u = −0.470779 − 0.592707I
a = 0.805640 + 0.073631I
b = −0.525112 − 1.088250I
7.73029 − 2.70190I 9.92902 + 0.03562I
u = −0.301413 + 0.651199I
a = −1.27352 + 0.89636I
b = −0.537119 − 0.669077I
7.21081 − 6.77625I 8.31083 + 6.21335I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.301413 − 0.651199I
a = −1.27352 − 0.89636I
b = −0.537119 + 0.669077I
7.21081 + 6.77625I 8.31083 − 6.21335I
u = −0.295790 + 0.583078I
a = 0.912564 − 0.926539I
b = 0.190072 + 0.467950I
1.43056 − 3.60804I 4.24958 + 7.07743I
u = −0.295790 − 0.583078I
a = 0.912564 + 0.926539I
b = 0.190072 − 0.467950I
1.43056 + 3.60804I 4.24958 − 7.07743I
u = −0.405535 + 0.509845I
a = −0.537103 + 0.460008I
b = 0.355378 − 0.582565I
1.81710 + 0.00783I 6.28682 + 0.01976I
u = −0.405535 − 0.509845I
a = −0.537103 − 0.460008I
b = 0.355378 + 0.582565I
1.81710 − 0.00783I 6.28682 − 0.01976I
u = −0.512306
a = 0.313006
b = 0.321368
0.766149 13.4410
u = −0.033821 + 0.416681I
a = −0.02102 + 1.94574I
b = 0.405990 + 0.159838I
2.06372 − 1.98561I 1.75080 + 4.13320I
u = −0.033821 − 0.416681I
a = −0.02102 − 1.94574I
b = 0.405990 − 0.159838I
2.06372 + 1.98561I 1.75080 − 4.13320I
u = 1.59708
a = −0.845107
b = 1.07699
8.19154 0
u = 1.61144 + 0.05910I
a = 1.25524 − 0.79623I
b = −1.80165 + 0.80672I
12.23120 + 1.76270I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.61144 − 0.05910I
a = 1.25524 + 0.79623I
b = −1.80165 − 0.80672I
12.23120 − 1.76270I 0
u = 0.267156 + 0.194860I
a = 0.14623 + 2.75307I
b = 0.727620 − 0.740672I
3.42049 + 3.22954I −0.37490 − 5.13369I
u = 0.267156 − 0.194860I
a = 0.14623 − 2.75307I
b = 0.727620 + 0.740672I
3.42049 − 3.22954I −0.37490 + 5.13369I
u = −1.71592
a = 0.726622
b = −0.423053
10.6526 0
u = −1.71698 + 0.02280I
a = −0.638945 − 0.755487I
b = 0.367724 + 1.082380I
14.3705 − 4.1609I 0
u = −1.71698 − 0.02280I
a = −0.638945 + 0.755487I
b = 0.367724 − 1.082380I
14.3705 + 4.1609I 0
u = 1.71834 + 0.01659I
a = −0.32264 + 2.56462I
b = 0.69393 − 4.29665I
11.59990 + 1.93326I 0
u = 1.71834 − 0.01659I
a = −0.32264 − 2.56462I
b = 0.69393 + 4.29665I
11.59990 − 1.93326I 0
u = 1.74020 + 0.02935I
a = 0.58408 − 2.98539I
b = −1.34410 + 5.19321I
17.6233 + 4.9552I 0
u = 1.74020 − 0.02935I
a = 0.58408 + 2.98539I
b = −1.34410 − 5.19321I
17.6233 − 4.9552I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.102654 + 0.234461I
a = −0.48752 − 2.52124I
b = −0.550472 + 0.258801I
−1.186530 + 0.460043I −6.06013 − 1.97452I
u = 0.102654 − 0.234461I
a = −0.48752 + 2.52124I
b = −0.550472 − 0.258801I
−1.186530 − 0.460043I −6.06013 + 1.97452I
u = −1.74331 + 0.08919I
a = −0.36769 + 2.41491I
b = 0.37187 − 4.07506I
15.7539 − 8.5170I 0
u = −1.74331 − 0.08919I
a = −0.36769 − 2.41491I
b = 0.37187 + 4.07506I
15.7539 + 8.5170I 0
u = −1.74805 + 0.07009I
a = 0.42579 − 1.94514I
b = −0.80402 + 3.33285I
16.7378 − 4.0067I 0
u = −1.74805 − 0.07009I
a = 0.42579 + 1.94514I
b = −0.80402 − 3.33285I
16.7378 + 4.0067I 0
u = −1.74828 + 0.10270I
a = 0.50895 − 2.70608I
b = −0.35501 + 4.68791I
−17.8318 − 12.3199I 0
u = −1.74828 − 0.10270I
a = 0.50895 + 2.70608I
b = −0.35501 − 4.68791I
−17.8318 + 12.3199I 0
u = −1.77321 + 0.06163I
a = −1.04826 + 1.68240I
b = 2.07789 − 3.22013I
−15.6447 − 1.6429I 0
u = −1.77321 − 0.06163I
a = −1.04826 − 1.68240I
b = 2.07789 + 3.22013I
−15.6447 + 1.6429I 0
9
II. I
u
2
= hb + a, a
3
+ a
2
− 1, u
2
− u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u − 1
a
1
=
u
−u − 1
a
4
=
a
−a
a
2
=
a
2
+ u
−a
2
− u − 1
a
10
=
−u
u
a
5
=
a
−a
a
11
=
−u
u
a
7
=
−u
u + 1
a
3
=
au + a
−au − 2a
a
6
=
a
2
u + a
2
+ a − u − 1
−a
2
u − 2a
2
− a + u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = a
2
u − 2a
2
− au − 5a − u + 5
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
+ u
2
− 1)
2
c
2
(u
3
− u
2
+ 2u − 1)
2
c
4
, c
10
u
6
c
5
, 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
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
(y
3
− y
2
+ 2y −1)
2
c
2
, c
5
, c
6
(y
3
+ 3y
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
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −0.877439 + 0.744862I
b = 0.877439 − 0.744862I
4.01109 + 2.82812I 8.89985 + 0.15818I
u = −0.618034
a = −0.877439 − 0.744862I
b = 0.877439 + 0.744862I
4.01109 − 2.82812I 8.89985 − 0.15818I
u = −0.618034
a = 0.754878
b = −0.754878
−0.126494 0.818320
u = 1.61803
a = −0.877439 + 0.744862I
b = 0.877439 − 0.744862I
11.90680 + 2.82812I 9.10673 − 4.43024I
u = 1.61803
a = −0.877439 − 0.744862I
b = 0.877439 + 0.744862I
11.90680 − 2.82812I 9.10673 + 4.43024I
u = 1.61803
a = 0.754878
b = −0.754878
7.76919 −1.83150
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
+ u
2
− 1)
2
)(u
50
− 9u
49
+ ··· + 1832u + 113)
c
2
((u
3
− u
2
+ 2u − 1)
2
)(u
50
− 3u
49
+ ··· − 10u + 1)
c
3
((u
3
+ u
2
− 1)
2
)(u
50
+ 3u
49
+ ··· − 2392u + 241)
c
4
, c
10
u
6
(u
50
− u
49
+ ··· − 96u + 64)
c
5
, c
6
((u
3
+ u
2
+ 2u + 1)
2
)(u
50
− 3u
49
+ ··· − 10u + 1)
c
7
, c
8
, c
9
((u
2
− u − 1)
3
)(u
50
− 4u
49
+ ··· + u + 1)
c
11
, c
12
((u
2
+ u − 1)
3
)(u
50
− 4u
49
+ ··· + u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
− y
2
+ 2y −1)
2
)(y
50
+ 31y
49
+ ··· − 5984830y + 12769)
c
2
, c
5
, c
6
((y
3
+ 3y
2
+ 2y −1)
2
)(y
50
+ 47y
49
+ ··· − 54y + 1)
c
3
((y
3
− y
2
+ 2y −1)
2
)(y
50
+ 11y
49
+ ··· − 2239214y + 58081)
c
4
, c
10
y
6
(y
50
− 35y
49
+ ··· − 54272y + 4096)
c
7
, c
8
, c
9
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
50
− 68y
49
+ ··· + 9y + 1)
15
