10
66
(K10a
40
)
A knot diagram
1
Linearized knot diagam
6 7 10 8 3 2 9 5 1 4
Solving Sequence
4,8
5
9,10
1 3 6 7 2
c
4
c
8
c
10
c
3
c
5
c
7
c
2
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, u
15
+ u
14
− 2u
13
− 3u
12
+ 4u
11
+ 6u
10
− u
9
− 6u
8
+ 2u
7
+ 5u
6
+ 3u
5
− 3u
4
+ 2u
3
+ 2u
2
+ 2a + 3u,
u
16
+ u
15
− 3u
14
− 4u
13
+ 6u
12
+ 9u
11
− 5u
10
− 12u
9
+ 3u
8
+ 11u
7
+ u
6
− 8u
5
− u
4
+ 5u
3
+ 3u
2
− 2u − 1i
I
u
2
= h11603u
23
+ 6022u
22
+ ··· + 8177b + 4273, 3426u
23
− 2155u
22
+ ··· + 8177a − 28435,
u
24
+ u
23
+ ··· + 4u + 1i
I
u
3
= hb − 1, a
2
− 4a + 2, u + 1i
I
u
4
= hb + 1, a + 2, u − 1i
* 4 irreducible components of dim
C
= 0, with total 43 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
= hb − u, u
15
+ u
14
+ · · · + 2a + 3u, u
16
+ u
15
+ · · · − 2u − 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
5
=
1
u
2
a
9
=
−u
−u
3
+ u
a
10
=
−
1
2
u
15
−
1
2
u
14
+ ··· − u
2
−
3
2
u
u
a
1
=
−
1
2
u
15
−
1
2
u
14
+ ··· − u
2
−
5
2
u
u
a
3
=
1
2
u
14
+
1
2
u
13
+ ··· + u +
3
2
−u
2
a
6
=
−
1
2
u
15
−
3
2
u
14
+ ··· −
9
2
u − 1
1
2
u
14
+
1
2
u
13
+ ··· + u +
1
2
a
7
=
u
3
u
5
− u
3
+ u
a
2
=
u
10
− u
8
+ 2u
6
+ u
4
+ u
2
+ 1
−
1
2
u
14
−
1
2
u
13
+ ··· − u −
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
15
+ u
14
− 4u
13
− 3u
12
+ 12u
11
+ 8u
10
− 19u
9
− 12u
8
+ 22u
7
+
17u
6
− 13u
5
− 13u
4
+ 6u
3
+ 12u
2
− u − 16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
16
+ 3u
15
+ ··· − 2u − 2
c
3
, c
4
, c
8
c
10
u
16
+ u
15
+ ··· − 2u − 1
c
5
u
16
− 9u
15
+ ··· − 34u + 14
c
7
, c
9
u
16
+ 7u
15
+ ··· + 10u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y
16
− 15y
15
+ ··· − 20y + 4
c
3
, c
4
, c
8
c
10
y
16
− 7y
15
+ ··· − 10y + 1
c
5
y
16
− 3y
15
+ ··· − 1156y + 196
c
7
, c
9
y
16
+ 9y
15
+ ··· − 38y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.788317 + 0.682807I
a = −0.862130 − 0.839659I
b = −0.788317 + 0.682807I
−0.17586 + 4.85157I −10.18415 − 6.53900I
u = −0.788317 − 0.682807I
a = −0.862130 + 0.839659I
b = −0.788317 − 0.682807I
−0.17586 − 4.85157I −10.18415 + 6.53900I
u = 0.591599 + 0.705742I
a = 0.502397 − 0.588564I
b = 0.591599 + 0.705742I
3.06515 − 1.13134I −4.88295 + 2.50814I
u = 0.591599 − 0.705742I
a = 0.502397 + 0.588564I
b = 0.591599 − 0.705742I
3.06515 + 1.13134I −4.88295 − 2.50814I
u = −0.403938 + 0.782402I
a = −0.331306 − 0.329211I
b = −0.403938 + 0.782402I
−1.21964 − 2.39915I −9.20728 + 0.67092I
u = −0.403938 − 0.782402I
a = −0.331306 + 0.329211I
b = −0.403938 − 0.782402I
−1.21964 + 2.39915I −9.20728 − 0.67092I
u = 1.043770 + 0.418403I
a = 1.76067 − 2.04191I
b = 1.043770 + 0.418403I
−8.80698 − 2.79176I −16.7106 + 5.2072I
u = 1.043770 − 0.418403I
a = 1.76067 + 2.04191I
b = 1.043770 − 0.418403I
−8.80698 + 2.79176I −16.7106 − 5.2072I
u = −1.034800 + 0.560504I
a = −1.60194 − 1.34258I
b = −1.034800 + 0.560504I
−1.63698 + 4.78532I −12.50670 − 3.64348I
u = −1.034800 − 0.560504I
a = −1.60194 + 1.34258I
b = −1.034800 − 0.560504I
−1.63698 − 4.78532I −12.50670 + 3.64348I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.123030 + 0.603482I
a = 1.88319 − 1.11133I
b = 1.123030 + 0.603482I
−0.34351 − 9.16484I −10.75715 + 8.12303I
u = 1.123030 − 0.603482I
a = 1.88319 + 1.11133I
b = 1.123030 − 0.603482I
−0.34351 + 9.16484I −10.75715 − 8.12303I
u = 0.703289
a = −2.07989
b = 0.703289
−6.93855 −11.2730
u = −1.184280 + 0.595800I
a = −2.08419 − 1.05231I
b = −1.184280 + 0.595800I
−5.9872 + 13.0293I −14.9902 − 8.3428I
u = −1.184280 − 0.595800I
a = −2.08419 + 1.05231I
b = −1.184280 − 0.595800I
−5.9872 − 13.0293I −14.9902 + 8.3428I
u = −0.397419
a = 0.546503
b = −0.397419
−0.684897 −14.2490
6
II. I
u
2
= h11603u
23
+ 6022u
22
+ · · · + 8177b + 4273, 3426u
23
− 2155u
22
+
· · · + 8177a − 28435, u
24
+ u
23
+ · · · + 4u + 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
5
=
1
u
2
a
9
=
−u
−u
3
+ u
a
10
=
−0.418980u
23
+ 0.263544u
22
+ ··· + 0.0328972u + 3.47744
−1.41898u
23
− 0.736456u
22
+ ··· − 5.96710u − 0.522563
a
1
=
u
23
+ u
22
+ ··· + 6u + 4
−1.41898u
23
− 0.736456u
22
+ ··· − 5.96710u − 0.522563
a
3
=
1.20509u
23
− 0.359300u
22
+ ··· + 1.96025u − 2.45787
0.682524u
23
+ 0.537116u
22
+ ··· + 5.15336u + 0.418980
a
6
=
−0.203987u
23
+ 1.29118u
22
+ ··· + 4.42057u + 5.51266
−0.997187u
23
− 0.285190u
22
+ ··· − 4.02678u + 0.362236
a
7
=
u
3
u
5
− u
3
+ u
a
2
=
1.12511u
23
− 0.0760670u
22
+ ··· + 0.287025u − 3.10578
0.00195671u
23
+ 0.323346u
22
+ ··· − 0.322979u − 0.791488
(ii) Obstruction class = −1
(iii) Cusp Shapes =
19748
8177
u
23
+
17088
8177
u
22
+ ··· +
29544
8177
u −
105438
8177
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
(u
12
− u
11
− 5u
10
+ 4u
9
+ 9u
8
− 4u
7
− 6u
6
− 2u
5
+ 3u
3
+ u
2
+ 1)
2
c
3
, c
4
, c
8
c
10
u
24
+ u
23
+ ··· + 4u + 1
c
5
(u
12
+ 3u
11
+ ··· + 4u + 1)
2
c
7
, c
9
u
24
+ 13u
23
+ ··· + 4u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
(y
12
− 11y
11
+ ··· + 2y + 1)
2
c
3
, c
4
, c
8
c
10
y
24
− 13y
23
+ ··· − 4y + 1
c
5
(y
12
+ y
11
+ ··· − 2y + 1)
2
c
7
, c
9
y
24
− 5y
23
+ ··· + 48y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.961597 + 0.331697I
a = −2.11926 + 0.49208I
b = −1.189900 + 0.171507I
−3.28987 − 1.20211I −12.00000 + 5.63740I
u = 0.961597 − 0.331697I
a = −2.11926 − 0.49208I
b = −1.189900 − 0.171507I
−3.28987 + 1.20211I −12.00000 − 5.63740I
u = −0.778724 + 0.569322I
a = 0.272376 − 0.021441I
b = −0.564477 − 0.633261I
−0.174773 + 0.093609I −10.00912 + 0.76204I
u = −0.778724 − 0.569322I
a = 0.272376 + 0.021441I
b = −0.564477 + 0.633261I
−0.174773 − 0.093609I −10.00912 − 0.76204I
u = −0.285725 + 0.889847I
a = −0.777424 + 0.420961I
b = −1.104540 − 0.597792I
−3.28987 − 7.58818I −12.00000 + 5.13539I
u = −0.285725 − 0.889847I
a = −0.777424 − 0.420961I
b = −1.104540 + 0.597792I
−3.28987 + 7.58818I −12.00000 − 5.13539I
u = 0.384175 + 0.809134I
a = 0.520131 + 0.408228I
b = 0.998981 − 0.600305I
1.84911 + 3.88480I −7.19439 − 4.17140I
u = 0.384175 − 0.809134I
a = 0.520131 − 0.408228I
b = 0.998981 + 0.600305I
1.84911 − 3.88480I −7.19439 + 4.17140I
u = −0.564477 + 0.633261I
a = 0.005650 + 0.310630I
b = −0.778724 − 0.569322I
−0.174773 − 0.093609I −10.00912 − 0.76204I
u = −0.564477 − 0.633261I
a = 0.005650 − 0.310630I
b = −0.778724 + 0.569322I
−0.174773 + 0.093609I −10.00912 + 0.76204I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.057630 + 0.470734I
a = 2.07384 + 0.60989I
b = 1.284660 + 0.258642I
−8.42885 + 3.88480I −16.8056 − 4.1714I
u = −1.057630 − 0.470734I
a = 2.07384 − 0.60989I
b = 1.284660 − 0.258642I
−8.42885 − 3.88480I −16.8056 + 4.1714I
u = 0.998981 + 0.600305I
a = −0.351273 − 0.367190I
b = 0.384175 − 0.809134I
1.84911 − 3.88480I −7.19439 + 4.17140I
u = 0.998981 − 0.600305I
a = −0.351273 + 0.367190I
b = 0.384175 + 0.809134I
1.84911 + 3.88480I −7.19439 − 4.17140I
u = 1.165410 + 0.089633I
a = −1.166860 − 0.270592I
b = −0.313835 − 0.336199I
−6.40496 + 0.09361I −13.99088 + 0.76204I
u = 1.165410 − 0.089633I
a = −1.166860 + 0.270592I
b = −0.313835 + 0.336199I
−6.40496 − 0.09361I −13.99088 − 0.76204I
u = −1.189900 + 0.171507I
a = 1.78490 + 0.45036I
b = 0.961597 + 0.331697I
−3.28987 − 1.20211I −12.00000 + 5.63740I
u = −1.189900 − 0.171507I
a = 1.78490 − 0.45036I
b = 0.961597 − 0.331697I
−3.28987 + 1.20211I −12.00000 − 5.63740I
u = −1.104540 + 0.597792I
a = 0.414520 − 0.510865I
b = −0.285725 − 0.889847I
−3.28987 + 7.58818I −12.00000 − 5.13539I
u = −1.104540 − 0.597792I
a = 0.414520 + 0.510865I
b = −0.285725 + 0.889847I
−3.28987 − 7.58818I −12.00000 + 5.13539I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.284660 + 0.258642I
a = −1.80572 + 0.62135I
b = −1.057630 + 0.470734I
−8.42885 + 3.88480I −16.8056 − 4.1714I
u = 1.284660 − 0.258642I
a = −1.80572 − 0.62135I
b = −1.057630 − 0.470734I
−8.42885 − 3.88480I −16.8056 + 4.1714I
u = −0.313835 + 0.336199I
a = 2.64911 + 1.49979I
b = 1.165410 − 0.089633I
−6.40496 − 0.09361I −13.99088 − 0.76204I
u = −0.313835 − 0.336199I
a = 2.64911 − 1.49979I
b = 1.165410 + 0.089633I
−6.40496 + 0.09361I −13.99088 + 0.76204I
12
III. I
u
3
= hb − 1, a
2
− 4a + 2, u + 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
−1
a
5
=
1
1
a
9
=
1
0
a
10
=
a
1
a
1
=
a − 1
1
a
3
=
−a + 1
−1
a
6
=
−a + 1
−a + 3
a
7
=
−1
−1
a
2
=
−1
a − 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −20
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
2
− 2
c
3
, c
7
, c
8
c
9
(u − 1)
2
c
4
, c
10
(u + 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
(y −2)
2
c
3
, c
4
, c
7
c
8
, c
9
, c
10
(y −1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.585786
b = 1.00000
−8.22467 −20.0000
u = −1.00000
a = 3.41421
b = 1.00000
−8.22467 −20.0000
16
IV. I
u
4
= hb + 1, a + 2, u − 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
1
a
5
=
1
1
a
9
=
−1
0
a
10
=
−2
−1
a
1
=
−1
−1
a
3
=
−1
−1
a
6
=
1
1
a
7
=
1
1
a
2
=
−1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
c
3
, c
8
u + 1
c
4
, c
7
, c
9
c
10
u − 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
y
c
3
, c
4
, c
7
c
8
, c
9
, c
10
y −1
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −2.00000
b = −1.00000
−3.28987 −12.0000
20
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u(u
2
− 2)(u
12
− u
11
+ ··· + u
2
+ 1)
2
· (u
16
+ 3u
15
+ ··· − 2u − 2)
c
3
, c
8
((u − 1)
2
)(u + 1)(u
16
+ u
15
+ ··· − 2u − 1)(u
24
+ u
23
+ ··· + 4u + 1)
c
4
, c
10
(u − 1)(u + 1)
2
(u
16
+ u
15
+ ··· − 2u − 1)(u
24
+ u
23
+ ··· + 4u + 1)
c
5
u(u
2
− 2)(u
12
+ 3u
11
+ ··· + 4u + 1)
2
(u
16
− 9u
15
+ ··· − 34u + 14)
c
7
, c
9
((u − 1)
3
)(u
16
+ 7u
15
+ ··· + 10u + 1)(u
24
+ 13u
23
+ ··· + 4u + 1)
21
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y(y −2)
2
(y
12
− 11y
11
+ ··· + 2y + 1)
2
(y
16
− 15y
15
+ ··· − 20y + 4)
c
3
, c
4
, c
8
c
10
((y −1)
3
)(y
16
− 7y
15
+ ··· − 10y + 1)(y
24
− 13y
23
+ ··· − 4y + 1)
c
5
y(y −2)
2
(y
12
+ y
11
+ ··· − 2y + 1)
2
· (y
16
− 3y
15
+ ··· − 1156y + 196)
c
7
, c
9
((y −1)
3
)(y
16
+ 9y
15
+ ··· − 38y + 1)(y
24
− 5y
23
+ ··· + 48y + 1)
22
