12a
0370
(K12a
0370
)
A knot diagram
1
Linearized knot diagam
3 6 9 10 2 11 12 1 4 5 8 7
Solving Sequence
2,5
6
3,10
11 7 1 4 9 8 12
c
5
c
2
c
10
c
6
c
1
c
4
c
9
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h1.57162 × 10
65
u
64
+ 6.57175 × 10
65
u
63
+ ··· + 2.92397 × 10
65
b − 2.95547 × 10
66
,
1.26210 × 10
66
u
64
+ 4.50966 × 10
66
u
63
+ ··· + 4.09356 × 10
66
a − 1.01879 × 10
67
, u
65
+ 4u
64
+ ··· − 51u + 7i
I
u
2
= hb, a
3
+ a
2
− 1, u + 1i
I
u
3
= hb
2
− 2, a
3
− a
2
+ 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 74 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
=
h1.57×10
65
u
64
+6.57×10
65
u
63
+· · ·+2.92×10
65
b−2.96×10
66
, 1.26×10
66
u
64
+
4.51 × 10
66
u
63
+ · · · + 4.09 × 10
66
a − 1.02 × 10
67
, u
65
+ 4u
64
+ · · · − 51u + 7i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
10
=
−0.308313u
64
− 1.10165u
63
+ ··· − 26.1352u + 2.48875
−0.537494u
64
− 2.24754u
63
+ ··· − 43.5603u + 10.1077
a
11
=
−0.845807u
64
− 3.34919u
63
+ ··· − 69.6955u + 12.5965
−0.537494u
64
− 2.24754u
63
+ ··· − 43.5603u + 10.1077
a
7
=
−0.896834u
64
− 4.06581u
63
+ ··· − 104.086u + 24.6201
−0.334251u
64
− 1.33325u
63
+ ··· − 27.2815u + 5.97496
a
1
=
u
3
u
5
− u
3
+ u
a
4
=
0.938529u
64
+ 4.21121u
63
+ ··· + 83.4299u − 19.2445
0.427727u
64
+ 1.79587u
63
+ ··· + 45.3628u − 9.76939
a
9
=
1.57556u
64
+ 6.38013u
63
+ ··· + 112.041u − 27.9236
0.190879u
64
+ 1.27293u
63
+ ··· + 6.17774u − 2.70329
a
8
=
1.20468u
64
+ 5.47814u
63
+ ··· + 111.316u − 28.2011
0.371889u
64
+ 1.74141u
63
+ ··· + 25.4181u − 6.03071
a
12
=
−0.221510u
64
− 0.993540u
63
+ ··· − 40.1704u + 6.43709
−0.326133u
64
− 1.36825u
63
+ ··· − 16.1337u + 4.29430
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0.860917u
64
+ 3.71890u
63
+ ··· + 8.19845u + 13.2317
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
65
+ 28u
64
+ ··· + 1621u + 49
c
2
, c
5
u
65
+ 4u
64
+ ··· − 51u + 7
c
3
, c
4
, c
9
c
10
u
65
− u
64
+ ··· − 8u − 8
c
6
, c
8
u
65
+ 2u
64
+ ··· + 2788u + 289
c
7
, c
11
, c
12
u
65
− 2u
64
+ ··· + 8u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
65
+ 28y
64
+ ··· + 644709y −2401
c
2
, c
5
y
65
− 28y
64
+ ··· + 1621y −49
c
3
, c
4
, c
9
c
10
y
65
− 79y
64
+ ··· + 1728y −64
c
6
, c
8
y
65
− 50y
64
+ ··· + 7602434y −83521
c
7
, c
11
, c
12
y
65
+ 54y
64
+ ··· + 98y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.612176 + 0.789701I
a = 0.561126 + 0.096922I
b = −0.952670 − 0.404352I
6.34514 + 1.30632I 13.77211 − 1.31410I
u = 0.612176 − 0.789701I
a = 0.561126 − 0.096922I
b = −0.952670 + 0.404352I
6.34514 − 1.30632I 13.77211 + 1.31410I
u = −0.747787 + 0.667407I
a = −0.360811 + 0.449280I
b = −0.034903 + 0.726729I
−0.65467 − 1.56191I 6.00000 + 0.I
u = −0.747787 − 0.667407I
a = −0.360811 − 0.449280I
b = −0.034903 − 0.726729I
−0.65467 + 1.56191I 6.00000 + 0.I
u = 0.537868 + 0.834733I
a = −0.578993 − 0.033369I
b = 0.891890 + 0.462986I
2.20984 + 5.53743I 9.36866 − 4.50579I
u = 0.537868 − 0.834733I
a = −0.578993 + 0.033369I
b = 0.891890 − 0.462986I
2.20984 − 5.53743I 9.36866 + 4.50579I
u = 0.879407 + 0.497658I
a = 1.025300 + 0.888790I
b = −0.712730 + 0.316213I
0.21641 − 3.44473I 7.87016 + 8.48336I
u = 0.879407 − 0.497658I
a = 1.025300 − 0.888790I
b = −0.712730 − 0.316213I
0.21641 + 3.44473I 7.87016 − 8.48336I
u = 0.698155 + 0.733599I
a = −0.513257 − 0.173198I
b = 1.036060 + 0.336652I
2.63490 − 2.89952I 9.88169 + 2.65135I
u = 0.698155 − 0.733599I
a = −0.513257 + 0.173198I
b = 1.036060 − 0.336652I
2.63490 + 2.89952I 9.88169 − 2.65135I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.04736
a = −0.0308709
b = 1.43162
3.32536 1.90800
u = −0.688518 + 0.611372I
a = −2.95116 + 0.90249I
b = 1.56873 + 0.00323I
2.81749 − 1.96826I 8.01195 + 0.26172I
u = −0.688518 − 0.611372I
a = −2.95116 − 0.90249I
b = 1.56873 − 0.00323I
2.81749 + 1.96826I 8.01195 − 0.26172I
u = −0.859363 + 0.654709I
a = 0.345067 − 0.440186I
b = 0.113262 − 0.740531I
2.97311 + 2.54501I 0
u = −0.859363 − 0.654709I
a = 0.345067 + 0.440186I
b = 0.113262 + 0.740531I
2.97311 − 2.54501I 0
u = −0.792220 + 0.742732I
a = 2.28438 − 0.98600I
b = −1.61762 − 0.02352I
8.72128 + 0.76781I 0
u = −0.792220 − 0.742732I
a = 2.28438 + 0.98600I
b = −1.61762 + 0.02352I
8.72128 − 0.76781I 0
u = −0.879226 + 0.208718I
a = −0.212799 + 0.396317I
b = −0.139382 + 0.414404I
−1.47345 + 0.81303I −1.06179 − 2.21079I
u = −0.879226 − 0.208718I
a = −0.212799 − 0.396317I
b = −0.139382 − 0.414404I
−1.47345 − 0.81303I −1.06179 + 2.21079I
u = 1.018090 + 0.446326I
a = −1.00410 − 1.10254I
b = 0.594982 − 0.430810I
−5.61394 − 4.98473I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.018090 − 0.446326I
a = −1.00410 + 1.10254I
b = 0.594982 + 0.430810I
−5.61394 + 4.98473I 0
u = −1.090720 + 0.327345I
a = 0.346639 − 0.382185I
b = 0.354858 − 0.503499I
−6.33395 + 1.70228I 0
u = −1.090720 − 0.327345I
a = 0.346639 + 0.382185I
b = 0.354858 + 0.503499I
−6.33395 − 1.70228I 0
u = 0.835134 + 0.207071I
a = 1.74135 + 0.81734I
b = −0.458541 + 0.146079I
−4.20314 + 2.29651I 7.92045 + 3.19787I
u = 0.835134 − 0.207071I
a = 1.74135 − 0.81734I
b = −0.458541 − 0.146079I
−4.20314 − 2.29651I 7.92045 − 3.19787I
u = −0.942425 + 0.650639I
a = −0.339182 + 0.434920I
b = −0.173665 + 0.754967I
−1.25054 + 6.68721I 0
u = −0.942425 − 0.650639I
a = −0.339182 − 0.434920I
b = −0.173665 − 0.754967I
−1.25054 − 6.68721I 0
u = −0.477249 + 1.048800I
a = 1.83392 − 0.00459I
b = −1.66803 + 0.12607I
11.03680 − 7.80758I 0
u = −0.477249 − 1.048800I
a = 1.83392 + 0.00459I
b = −1.66803 − 0.12607I
11.03680 + 7.80758I 0
u = 1.121100 + 0.284148I
a = 0.127434 + 0.111845I
b = −1.44392 − 0.12372I
−0.503562 + 0.442865I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.121100 − 0.284148I
a = 0.127434 − 0.111845I
b = −1.44392 + 0.12372I
−0.503562 − 0.442865I 0
u = −0.542642 + 1.041680I
a = −1.84192 + 0.12408I
b = 1.67890 − 0.10309I
15.4602 − 3.2500I 0
u = −0.542642 − 1.041680I
a = −1.84192 − 0.12408I
b = 1.67890 + 0.10309I
15.4602 + 3.2500I 0
u = −1.000100 + 0.618281I
a = 1.89737 − 1.82085I
b = −1.58542 − 0.10501I
1.82316 + 6.85461I 0
u = −1.000100 − 0.618281I
a = 1.89737 + 1.82085I
b = −1.58542 + 0.10501I
1.82316 − 6.85461I 0
u = −0.929923 + 0.728312I
a = −1.97076 + 1.33085I
b = 1.62385 + 0.07311I
8.30513 + 4.82290I 0
u = −0.929923 − 0.728312I
a = −1.97076 − 1.33085I
b = 1.62385 − 0.07311I
8.30513 − 4.82290I 0
u = 0.716718 + 0.381500I
a = −1.220570 − 0.577631I
b = 0.643611 − 0.098429I
0.785742 − 0.346166I 11.27090 + 0.19465I
u = 0.716718 − 0.381500I
a = −1.220570 + 0.577631I
b = 0.643611 + 0.098429I
0.785742 + 0.346166I 11.27090 − 0.19465I
u = −0.616760 + 1.016660I
a = 1.86366 − 0.26965I
b = −1.68444 + 0.07351I
12.05520 + 1.42142I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.616760 − 1.016660I
a = 1.86366 + 0.26965I
b = −1.68444 − 0.07351I
12.05520 − 1.42142I 0
u = 0.965461 + 0.702359I
a = 0.835684 + 0.911173I
b = −0.858855 + 0.509945I
1.85518 − 2.58936I 0
u = 0.965461 − 0.702359I
a = 0.835684 − 0.911173I
b = −0.858855 − 0.509945I
1.85518 + 2.58936I 0
u = −1.19939
a = 0.542136
b = 0.558137
0.215305 0
u = −1.223990 + 0.070272I
a = −0.568813 + 0.167355I
b = −0.598816 + 0.153922I
−3.79500 − 3.51729I 0
u = −1.223990 − 0.070272I
a = −0.568813 − 0.167355I
b = −0.598816 − 0.153922I
−3.79500 + 3.51729I 0
u = 1.036490 + 0.697733I
a = −0.809759 − 0.942093I
b = 0.818309 − 0.567364I
5.09312 − 6.92778I 0
u = 1.036490 − 0.697733I
a = −0.809759 + 0.942093I
b = 0.818309 + 0.567364I
5.09312 + 6.92778I 0
u = 1.084460 + 0.684896I
a = 0.789764 + 0.966569I
b = −0.784743 + 0.601883I
0.58292 − 11.22380I 0
u = 1.084460 − 0.684896I
a = 0.789764 − 0.966569I
b = −0.784743 − 0.601883I
0.58292 + 11.22380I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.113940 + 0.786858I
a = −1.41064 + 1.35680I
b = 1.66142 + 0.14314I
10.51140 + 5.10195I 0
u = −1.113940 − 0.786858I
a = −1.41064 − 1.35680I
b = 1.66142 − 0.14314I
10.51140 − 5.10195I 0
u = −1.163830 + 0.759847I
a = 1.29115 − 1.43417I
b = −1.65344 − 0.16744I
13.5310 + 9.7563I 0
u = −1.163830 − 0.759847I
a = 1.29115 + 1.43417I
b = −1.65344 + 0.16744I
13.5310 − 9.7563I 0
u = −1.192080 + 0.730572I
a = −1.21566 + 1.50565I
b = 1.64210 + 0.18253I
8.8189 + 14.2263I 0
u = −1.192080 − 0.730572I
a = −1.21566 − 1.50565I
b = 1.64210 − 0.18253I
8.8189 − 14.2263I 0
u = 1.42411
a = 0.159460
b = −1.60305
7.84976 0
u = 1.42552 + 0.07021I
a = −0.161047 − 0.012113I
b = 1.60335 + 0.03552I
3.90623 + 4.17049I 0
u = 1.42552 − 0.07021I
a = −0.161047 + 0.012113I
b = 1.60335 − 0.03552I
3.90623 − 4.17049I 0
u = 0.012164 + 0.540343I
a = 0.787036 − 0.425151I
b = −0.383178 − 0.436316I
−3.21514 + 1.52290I 5.38087 − 4.43310I
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.012164 − 0.540343I
a = 0.787036 + 0.425151I
b = −0.383178 + 0.436316I
−3.21514 − 1.52290I 5.38087 + 4.43310I
u = 0.377484 + 0.366802I
a = 0.07583 − 1.59486I
b = 1.322440 − 0.065855I
2.05865 − 3.23471I 10.89405 + 4.40187I
u = 0.377484 − 0.366802I
a = 0.07583 + 1.59486I
b = 1.322440 + 0.065855I
2.05865 + 3.23471I 10.89405 − 4.40187I
u = 0.348475
a = −2.14252
b = −1.35387
5.84839 16.9730
u = 0.260517
a = −1.67780
b = 0.360345
0.625974 16.0870
11
II. I
u
2
= hb, a
3
+ a
2
− 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
5
=
1
0
a
6
=
1
1
a
3
=
1
0
a
10
=
a
0
a
11
=
a
0
a
7
=
a
2
+ 1
1
a
1
=
−1
−1
a
4
=
1
0
a
9
=
a
0
a
8
=
2a
a
a
12
=
2a
2
+ a − 2
a
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2a
2
+ 2a + 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
9
c
10
u
3
c
5
(u + 1)
3
c
6
, c
8
u
3
− u
2
+ 1
c
7
u
3
+ u
2
+ 2u + 1
c
11
, c
12
u
3
− u
2
+ 2u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
3
c
3
, c
4
, c
9
c
10
y
3
c
6
, c
8
y
3
− y
2
+ 2y −1
c
7
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.877439 + 0.744862I
b = 0
−4.66906 − 2.82812I −0.18504 + 4.10401I
u = −1.00000
a = −0.877439 − 0.744862I
b = 0
−4.66906 + 2.82812I −0.18504 − 4.10401I
u = −1.00000
a = 0.754878
b = 0
−0.531480 2.37010
15
III. I
u
3
= hb
2
− 2, a
3
− a
2
+ 1, u − 1i
(i) Arc colorings
a
2
=
0
1
a
5
=
1
0
a
6
=
1
1
a
3
=
−1
0
a
10
=
a
b
a
11
=
b + a
b
a
7
=
ba + a
2
+ 1
ba + 1
a
1
=
1
1
a
4
=
ba + 1
2
a
9
=
−b − a
−b
a
8
=
−b − 2a
−b − a
a
12
=
−a
2
b − 2a
2
+ b + a + 2
−a
2
b − a
2
+ b + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a + 8
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
6
c
2
(u + 1)
6
c
3
, c
4
, c
9
c
10
(u
2
− 2)
3
c
6
, c
8
(u
3
+ u
2
− 1)
2
c
7
(u
3
− u
2
+ 2u − 1)
2
c
11
, c
12
(u
3
+ u
2
+ 2u + 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
9
c
10
(y −2)
6
c
6
, c
8
(y
3
− y
2
+ 2y −1)
2
c
7
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.877439 + 0.744862I
b = 1.41421
0.26574 + 2.82812I 4.49024 − 2.97945I
u = 1.00000
a = 0.877439 + 0.744862I
b = −1.41421
0.26574 + 2.82812I 4.49024 − 2.97945I
u = 1.00000
a = 0.877439 − 0.744862I
b = 1.41421
0.26574 − 2.82812I 4.49024 + 2.97945I
u = 1.00000
a = 0.877439 − 0.744862I
b = −1.41421
0.26574 − 2.82812I 4.49024 + 2.97945I
u = 1.00000
a = −0.754878
b = 1.41421
4.40332 11.0200
u = 1.00000
a = −0.754878
b = −1.41421
4.40332 11.0200
19
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
65
+ 28u
64
+ ··· + 1621u + 49)
c
2
((u − 1)
3
)(u + 1)
6
(u
65
+ 4u
64
+ ··· − 51u + 7)
c
3
, c
4
, c
9
c
10
u
3
(u
2
− 2)
3
(u
65
− u
64
+ ··· − 8u − 8)
c
5
((u − 1)
6
)(u + 1)
3
(u
65
+ 4u
64
+ ··· − 51u + 7)
c
6
, c
8
(u
3
− u
2
+ 1)(u
3
+ u
2
− 1)
2
(u
65
+ 2u
64
+ ··· + 2788u + 289)
c
7
((u
3
− u
2
+ 2u − 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
65
− 2u
64
+ ··· + 8u + 1)
c
11
, c
12
(u
3
− u
2
+ 2u − 1)(u
3
+ u
2
+ 2u + 1)
2
(u
65
− 2u
64
+ ··· + 8u + 1)
20
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
65
+ 28y
64
+ ··· + 644709y −2401)
c
2
, c
5
((y −1)
9
)(y
65
− 28y
64
+ ··· + 1621y −49)
c
3
, c
4
, c
9
c
10
y
3
(y −2)
6
(y
65
− 79y
64
+ ··· + 1728y −64)
c
6
, c
8
((y
3
− y
2
+ 2y −1)
3
)(y
65
− 50y
64
+ ··· + 7602434y −83521)
c
7
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
65
+ 54y
64
+ ··· + 98y −1)
21
