10
81
(K10a
7
)
A knot diagram
1
Linearized knot diagam
4 1 6 2 8 3 10 5 7 9
Solving Sequence
2,5
4 1
3,9
8 6 10 7
c
4
c
1
c
2
c
8
c
5
c
10
c
7
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.38501 × 10
15
u
47
+ 7.04930 × 10
15
u
46
+ ··· + 1.31625 × 10
15
b + 3.65379 × 10
15
,
1.00335 × 10
16
u
47
− 3.95579 × 10
16
u
46
+ ··· + 2.63249 × 10
15
a + 1.73053 × 10
16
, u
48
− 5u
47
+ ··· + 10u + 1i
I
u
2
= hb, −u
2
+ a + 2u − 1, u
3
− u
2
+ 1i
I
u
3
= ha
2
+ b + 2a + 1, a
3
+ 2a
2
+ a + 1, u + 1i
* 3 irreducible components of dim
C
= 0, with total 54 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
= h−1.39 × 10
15
u
47
+ 7.05 × 10
15
u
46
+ · · · + 1.32 × 10
15
b + 3.65 ×
10
15
, 1.00 × 10
16
u
47
− 3.96 × 10
16
u
46
+ · · · + 2.63 × 10
15
a + 1.73 ×
10
16
, u
48
− 5u
47
+ · · · + 10u + 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
4
=
1
−u
2
a
1
=
u
−u
3
+ u
a
3
=
−u
3
u
5
− u
3
+ u
a
9
=
−3.81140u
47
+ 15.0268u
46
+ ··· + 36.2869u − 6.57375
1.05225u
47
− 5.35561u
46
+ ··· − 25.7273u − 2.77592
a
8
=
−4.86364u
47
+ 20.3824u
46
+ ··· + 62.0142u − 3.79783
1.05225u
47
− 5.35561u
46
+ ··· − 25.7273u − 2.77592
a
6
=
0.828554u
47
− 0.00856335u
46
+ ··· + 28.0123u − 1.38653
−2.29509u
47
+ 7.50970u
46
+ ··· + 3.57759u + 0.200750
a
10
=
2.99681u
47
− 12.6887u
46
+ ··· − 38.0813u + 3.39699
0.302246u
47
− 0.355609u
46
+ ··· + 8.52273u + 0.974080
a
7
=
0.0322954u
47
+ 0.330979u
46
+ ··· + 8.02219u − 3.03662
−0.302246u
47
+ 0.355609u
46
+ ··· − 8.52273u − 0.974080
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
2572719667737379
1316245742897122
u
47
−
14240246736967463
1316245742897122
u
46
+ ··· −
66418667522566225
1316245742897122
u +
3102450479488985
658122871448561
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
48
− 5u
47
+ ··· + 10u + 1
c
2
u
48
+ 23u
47
+ ··· + 180u + 1
c
3
, c
6
u
48
− 2u
47
+ ··· + 28u − 8
c
5
, c
8
u
48
+ 2u
47
+ ··· − 28u − 8
c
7
, c
9
u
48
+ 5u
47
+ ··· − 10u + 1
c
10
u
48
− 23u
47
+ ··· − 180u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
c
9
y
48
− 23y
47
+ ··· − 180y + 1
c
2
, c
10
y
48
+ 9y
47
+ ··· − 29816y + 1
c
3
, c
5
, c
6
c
8
y
48
+ 24y
47
+ ··· − 464y + 64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351537 + 0.949778I
a = 1.006580 + 0.754616I
b = 0.695127 + 1.130530I
2.54767 + 8.71683I 2.88659 − 5.91299I
u = 0.351537 − 0.949778I
a = 1.006580 − 0.754616I
b = 0.695127 − 1.130530I
2.54767 − 8.71683I 2.88659 + 5.91299I
u = 0.935499 + 0.280058I
a = 0.258750 + 0.692688I
b = −0.32474 − 1.42072I
−4.55335 + 1.97419I −0.52111 + 3.87774I
u = 0.935499 − 0.280058I
a = 0.258750 − 0.692688I
b = −0.32474 + 1.42072I
−4.55335 − 1.97419I −0.52111 − 3.87774I
u = −0.958701 + 0.411863I
a = −1.12181 + 1.17224I
b = −0.845547 + 0.386680I
2.50599I 0. − 3.68111I
u = −0.958701 − 0.411863I
a = −1.12181 − 1.17224I
b = −0.845547 − 0.386680I
− 2.50599I 0. + 3.68111I
u = 1.027890 + 0.366302I
a = −0.558001 − 0.681766I
b = 0.02906 + 1.43386I
−5.20077 − 4.17900I −3.36906 + 7.53383I
u = 1.027890 − 0.366302I
a = −0.558001 + 0.681766I
b = 0.02906 − 1.43386I
−5.20077 + 4.17900I −3.36906 − 7.53383I
u = −0.852801 + 0.288192I
a = −2.58191 + 1.66058I
b = −0.332500 − 0.567513I
0.675636 + 0.515505I −2.57655 − 6.02720I
u = −0.852801 − 0.288192I
a = −2.58191 − 1.66058I
b = −0.332500 + 0.567513I
0.675636 − 0.515505I −2.57655 + 6.02720I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.516978 + 0.722434I
a = 0.871888 − 0.224038I
b = 0.549420 + 0.862669I
4.87326 + 0.03227I 4.84666 − 0.67896I
u = 0.516978 − 0.722434I
a = 0.871888 + 0.224038I
b = 0.549420 − 0.862669I
4.87326 − 0.03227I 4.84666 + 0.67896I
u = 0.425885 + 0.773654I
a = 1.69830 + 0.13159I
b = 0.950582 − 0.574763I
4.33954 + 2.65713I 5.08315 − 1.96927I
u = 0.425885 − 0.773654I
a = 1.69830 − 0.13159I
b = 0.950582 + 0.574763I
4.33954 − 2.65713I 5.08315 + 1.96927I
u = 0.295606 + 0.828875I
a = −0.631610 − 0.587022I
b = −0.544625 − 1.084280I
3.47198I 0. − 2.47118I
u = 0.295606 − 0.828875I
a = −0.631610 + 0.587022I
b = −0.544625 + 1.084280I
− 3.47198I 0. + 2.47118I
u = −1.116730 + 0.138646I
a = 1.10500 − 1.07815I
b = 0.704022 + 0.224888I
−0.675636 − 0.515505I 2.57655 + 6.02720I
u = −1.116730 − 0.138646I
a = 1.10500 + 1.07815I
b = 0.704022 − 0.224888I
−0.675636 + 0.515505I 2.57655 − 6.02720I
u = 0.992673 + 0.539998I
a = −0.601343 − 0.631046I
b = −1.060630 − 0.166744I
0.88639 − 2.97344I 0. + 2.64448I
u = 0.992673 − 0.539998I
a = −0.601343 + 0.631046I
b = −1.060630 + 0.166744I
0.88639 + 2.97344I 0. − 2.64448I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.628205 + 0.596659I
a = −1.064010 − 0.397214I
b = −0.829653 + 0.427683I
2.00090 − 1.53468I 2.12390 + 3.90788I
u = 0.628205 − 0.596659I
a = −1.064010 + 0.397214I
b = −0.829653 − 0.427683I
2.00090 + 1.53468I 2.12390 − 3.90788I
u = −0.547224 + 0.659382I
a = −0.71785 + 1.52298I
b = −0.434204 + 1.035090I
−0.88639 − 2.97344I −0.29359 + 2.64448I
u = −0.547224 − 0.659382I
a = −0.71785 − 1.52298I
b = −0.434204 − 1.035090I
−0.88639 + 2.97344I −0.29359 − 2.64448I
u = 0.747136 + 0.877281I
a = 0.933943 − 0.476702I
b = 0.485002 − 0.768666I
5.20077 − 4.17900I 3.36906 + 7.53383I
u = 0.747136 − 0.877281I
a = 0.933943 + 0.476702I
b = 0.485002 + 0.768666I
5.20077 + 4.17900I 3.36906 − 7.53383I
u = −0.833904
a = 0.880190
b = 0.275054
−1.20368 −8.97040
u = −1.079990 + 0.482069I
a = 1.72018 − 0.33297I
b = 0.365280 + 1.116600I
−4.33954 + 2.65713I −5.08315 + 0.I
u = −1.079990 − 0.482069I
a = 1.72018 + 0.33297I
b = 0.365280 − 1.116600I
−4.33954 − 2.65713I −5.08315 + 0.I
u = −1.035420 + 0.586063I
a = −2.05065 + 0.08569I
b = −0.591514 − 1.148530I
−2.34804 + 7.85171I 0. − 6.74189I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.035420 − 0.586063I
a = −2.05065 − 0.08569I
b = −0.591514 + 1.148530I
−2.34804 − 7.85171I 0. + 6.74189I
u = 1.046730 + 0.598380I
a = 1.36039 + 1.17731I
b = 0.378423 − 1.023550I
3.29646 − 5.08791I 0. + 5.66025I
u = 1.046730 − 0.598380I
a = 1.36039 − 1.17731I
b = 0.378423 + 1.023550I
3.29646 + 5.08791I 0. − 5.66025I
u = 0.964917 + 0.798665I
a = −0.046037 + 0.723287I
b = 0.325521 + 0.665488I
4.55335 − 1.97419I 0
u = 0.964917 − 0.798665I
a = −0.046037 − 0.723287I
b = 0.325521 − 0.665488I
4.55335 + 1.97419I 0
u = 1.098250 + 0.602404I
a = 0.575750 + 0.805276I
b = 1.111730 + 0.493637I
2.34804 − 7.85171I 0
u = 1.098250 − 0.602404I
a = 0.575750 − 0.805276I
b = 1.111730 − 0.493637I
2.34804 + 7.85171I 0
u = −1.249240 + 0.262371I
a = 0.509018 − 0.507195I
b = −0.233338 + 1.114770I
−4.87326 − 0.03227I 0
u = −1.249240 − 0.262371I
a = 0.509018 + 0.507195I
b = −0.233338 − 1.114770I
−4.87326 + 0.03227I 0
u = 1.158620 + 0.585509I
a = −1.46782 − 0.64788I
b = −0.576415 + 1.261620I
−2.54767 − 8.71683I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.158620 − 0.585509I
a = −1.46782 + 0.64788I
b = −0.576415 − 1.261620I
−2.54767 + 8.71683I 0
u = −1.332740 + 0.180321I
a = −0.022960 + 0.366381I
b = 0.509355 − 1.133010I
−3.29646 − 5.08791I 0
u = −1.332740 − 0.180321I
a = −0.022960 − 0.366381I
b = 0.509355 + 1.133010I
−3.29646 + 5.08791I 0
u = 1.187100 + 0.637571I
a = 1.69469 + 0.56994I
b = 0.73411 − 1.22507I
− 14.4927I 0
u = 1.187100 − 0.637571I
a = 1.69469 − 0.56994I
b = 0.73411 + 1.22507I
14.4927I 0
u = −0.249189 + 0.602859I
a = 0.538717 − 1.292580I
b = 0.050010 − 1.026510I
−2.00090 + 1.53468I −2.12390 − 3.90788I
u = −0.249189 − 0.602859I
a = 0.538717 + 1.292580I
b = 0.050010 + 1.026510I
−2.00090 − 1.53468I −2.12390 + 3.90788I
u = −0.0760954
a = −9.69862
b = −0.503995
1.20368 8.97040
9
II. I
u
2
= hb, −u
2
+ a + 2u − 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
4
=
1
−u
2
a
1
=
u
−u
2
+ u + 1
a
3
=
−u
2
+ 1
−u
2
a
9
=
u
2
− 2u + 1
0
a
8
=
u
2
− 2u + 1
0
a
6
=
1
0
a
10
=
u
2
− u + 1
−u
2
+ u + 1
a
7
=
−u
u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
+ 8u − 4
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ u
2
− 1
c
2
, c
6
u
3
+ u
2
+ 2u + 1
c
3
u
3
− u
2
+ 2u − 1
c
4
u
3
− u
2
+ 1
c
5
, c
8
u
3
c
7
(u + 1)
3
c
9
, c
10
(u − 1)
3
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
3
− y
2
+ 2y −1
c
2
, c
3
, c
6
y
3
+ 3y
2
+ 2y −1
c
5
, c
8
y
3
c
7
, c
9
, c
10
(y −1)
3
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.539798 − 0.182582I
b = 0
4.66906 − 2.82812I 2.80443 + 4.65175I
u = 0.877439 − 0.744862I
a = −0.539798 + 0.182582I
b = 0
4.66906 + 2.82812I 2.80443 − 4.65175I
u = −0.754878
a = 3.07960
b = 0
0.531480 −10.6090
13
III. I
u
3
= ha
2
+ b + 2a + 1, a
3
+ 2a
2
+ a + 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
5
=
1
0
a
4
=
1
−1
a
1
=
−1
0
a
3
=
1
−1
a
9
=
a
−a
2
− 2a − 1
a
8
=
a
2
+ 3a + 1
−a
2
− 2a − 1
a
6
=
a
2
+ a − 1
−a
2
− a + 1
a
10
=
−2
−a
2
− a + 1
a
7
=
a
2
+ a − 1
−a
2
− a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = a
2
− 6a − 3
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
3
c
2
, c
4
(u + 1)
3
c
3
, c
6
u
3
c
5
u
3
+ u
2
+ 2u + 1
c
7
u
3
− u
2
+ 1
c
8
, c
10
u
3
− u
2
+ 2u − 1
c
9
u
3
+ u
2
− 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
6
y
3
c
5
, c
8
, c
10
y
3
+ 3y
2
+ 2y −1
c
7
, c
9
y
3
− y
2
+ 2y −1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.122561 + 0.744862I
b = −0.215080 − 1.307140I
−4.66906 + 2.82812I −2.80443 − 4.65175I
u = −1.00000
a = −0.122561 − 0.744862I
b = −0.215080 + 1.307140I
−4.66906 − 2.82812I −2.80443 + 4.65175I
u = −1.00000
a = −1.75488
b = −0.569840
−0.531480 10.6090
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
3
+ u
2
− 1)(u
48
− 5u
47
+ ··· + 10u + 1)
c
2
((u + 1)
3
)(u
3
+ u
2
+ 2u + 1)(u
48
+ 23u
47
+ ··· + 180u + 1)
c
3
u
3
(u
3
− u
2
+ 2u − 1)(u
48
− 2u
47
+ ··· + 28u − 8)
c
4
((u + 1)
3
)(u
3
− u
2
+ 1)(u
48
− 5u
47
+ ··· + 10u + 1)
c
5
u
3
(u
3
+ u
2
+ 2u + 1)(u
48
+ 2u
47
+ ··· − 28u − 8)
c
6
u
3
(u
3
+ u
2
+ 2u + 1)(u
48
− 2u
47
+ ··· + 28u − 8)
c
7
((u + 1)
3
)(u
3
− u
2
+ 1)(u
48
+ 5u
47
+ ··· − 10u + 1)
c
8
u
3
(u
3
− u
2
+ 2u − 1)(u
48
+ 2u
47
+ ··· − 28u − 8)
c
9
((u − 1)
3
)(u
3
+ u
2
− 1)(u
48
+ 5u
47
+ ··· − 10u + 1)
c
10
((u − 1)
3
)(u
3
− u
2
+ 2u − 1)(u
48
− 23u
47
+ ··· − 180u + 1)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
c
9
((y −1)
3
)(y
3
− y
2
+ 2y −1)(y
48
− 23y
47
+ ··· − 180y + 1)
c
2
, c
10
((y −1)
3
)(y
3
+ 3y
2
+ 2y −1)(y
48
+ 9y
47
+ ··· − 29816y + 1)
c
3
, c
5
, c
6
c
8
y
3
(y
3
+ 3y
2
+ 2y −1)(y
48
+ 24y
47
+ ··· − 464y + 64)
19
