10
32
(K10a
55
)
A knot diagram
1
Linearized knot diagam
7 5 8 9 10 1 6 4 3 2
Solving Sequence
3,8
4 9 5 10 6 2 7 1
c
3
c
8
c
4
c
9
c
5
c
2
c
7
c
1
c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
33
− 2u
32
+ ··· + u
2
+ 1i
I
u
2
= hu + 1i
* 2 irreducible components of dim
C
= 0, with total 34 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
= hu
33
− 2u
32
+ · · · + u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
9
=
u
−u
3
+ u
a
5
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
−u
3
+ 2u
−u
3
+ u
a
6
=
−u
10
+ 5u
8
− 8u
6
+ 3u
4
+ u
2
+ 1
−u
10
+ 4u
8
− 5u
6
+ 2u
4
− u
2
a
2
=
u
6
− 3u
4
+ 2u
2
+ 1
−u
8
+ 4u
6
− 4u
4
a
7
=
u
21
− 10u
19
+ ··· + 2u
3
+ u
u
21
− 9u
19
+ 33u
17
− 62u
15
+ 62u
13
− 33u
11
+ 13u
9
− 6u
7
+ u
5
− u
3
+ u
a
1
=
u
17
− 8u
15
+ 25u
13
− 36u
11
+ 19u
9
+ 4u
7
− 2u
5
− 4u
3
+ u
−u
19
+ 9u
17
− 32u
15
+ 55u
13
− 43u
11
+ 9u
9
+ 4u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
31
− 56u
29
− 4u
28
+ 344u
27
+ 52u
26
− 1204u
25
− 292u
24
+ 2600u
23
+ 912u
22
−
3476u
21
−1692u
20
+ 2672u
19
+ 1824u
18
−912u
17
−1016u
16
+ 40u
15
+ 276u
14
−156u
13
−
260u
12
+ 144u
11
+ 276u
10
+ 24u
9
−48u
8
−16u
7
−12u
6
+ 16u
5
−8u
4
−20u
3
−8u
2
−4u −6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
33
− 2u
32
+ ··· − 2u + 1
c
2
u
33
− 6u
32
+ ··· + 128u − 23
c
3
, c
4
, c
8
u
33
− 2u
32
+ ··· + u
2
+ 1
c
5
u
33
+ u
31
+ ··· − 8u + 1
c
7
, c
10
u
33
+ 10u
32
+ ··· − 2u + 1
c
9
u
33
+ 3u
32
+ ··· + 32u + 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
33
− 10y
32
+ ··· − 2y −1
c
2
y
33
+ 14y
32
+ ··· − 2062y −529
c
3
, c
4
, c
8
y
33
− 30y
32
+ ··· − 2y −1
c
5
y
33
+ 2y
32
+ ··· − 2y −1
c
7
, c
10
y
33
+ 26y
32
+ ··· + 6y −1
c
9
y
33
− 3y
32
+ ··· + 394y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.145930 + 0.199234I
2.02163 − 5.40417I −1.16809 + 6.21521I
u = −1.145930 − 0.199234I
2.02163 + 5.40417I −1.16809 − 6.21521I
u = 1.226990 + 0.119877I
2.93624 + 0.57729I − 61.088687 + 0.10I
u = 1.226990 − 0.119877I
2.93624 − 0.57729I − 61.088687 + 0.10I
u = −0.313132 + 0.699748I
1.61043 − 8.54919I −1.81653 + 8.15424I
u = −0.313132 − 0.699748I
1.61043 + 8.54919I −1.81653 − 8.15424I
u = 0.325114 + 0.672913I
2.49948 + 2.85888I 0.03469 − 3.31371I
u = 0.325114 − 0.672913I
2.49948 − 2.85888I 0.03469 + 3.31371I
u = −0.592603 + 0.413344I
2.74796 + 4.66940I 0.86326 − 2.61989I
u = −0.592603 − 0.413344I
2.74796 − 4.66940I 0.86326 + 2.61989I
u = −0.225806 + 0.667717I
−3.60419 − 3.13953I −8.34254 + 5.36114I
u = −0.225806 − 0.667717I
−3.60419 + 3.13953I −8.34254 − 5.36114I
u = 0.529781 + 0.441659I
3.40197 + 0.91195I 2.34870 − 3.13722I
u = 0.529781 − 0.441659I
3.40197 − 0.91195I 2.34870 + 3.13722I
u = 1.323560 + 0.186117I
3.02759 + 0.73587I −0.673126 + 0.769843I
u = 1.323560 − 0.186117I
3.02759 − 0.73587I −0.673126 − 0.769843I
u = −0.065742 + 0.645142I
−1.19428 + 2.21654I −6.16344 − 2.48417I
u = −0.065742 − 0.645142I
−1.19428 − 2.21654I −6.16344 + 2.48417I
u = −0.596679
−1.73897 −4.71290
u = 1.387740 + 0.260179I
1.53217 + 6.51294I −2.89383 − 5.98872I
u = 1.387740 − 0.260179I
1.53217 − 6.51294I −2.89383 + 5.98872I
u = −1.396540 + 0.216616I
5.26725 − 4.07711I 4.72201 + 3.88410I
u = −1.396540 − 0.216616I
5.26725 + 4.07711I 4.72201 − 3.88410I
u = 0.245019 + 0.527971I
0.007405 + 1.282000I −0.00329 − 5.16805I
u = 0.245019 − 0.527971I
0.007405 − 1.282000I −0.00329 + 5.16805I
u = −1.42908 + 0.26025I
8.11565 − 6.26770I 4.18982 + 3.24511I
u = −1.42908 − 0.26025I
8.11565 + 6.26770I 4.18982 − 3.24511I
u = 1.44655 + 0.13460I
9.14238 − 2.78863I 4.90822 + 2.57820I
u = 1.44655 − 0.13460I
9.14238 + 2.78863I 4.90822 − 2.57820I
u = 1.42746 + 0.27209I
7.18048 + 12.09090I 2.43573 − 8.11579I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.42746 − 0.27209I
7.18048 − 12.09090I 2.43573 + 8.11579I
u = −1.44503 + 0.15402I
9.63768 − 3.04389I 5.82618 + 2.90426I
u = −1.44503 − 0.15402I
9.63768 + 3.04389I 5.82618 − 2.90426I
6
II. I
u
2
= hu + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
−1
a
4
=
1
−1
a
9
=
−1
0
a
5
=
0
−1
a
10
=
−1
0
a
6
=
1
−1
a
2
=
1
−1
a
7
=
−1
0
a
1
=
0
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
8
, c
10
u + 1
c
2
u − 1
c
9
u
8
(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
10
y −1
c
9
y
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
−1.64493 −6.00000
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u + 1)(u
33
− 2u
32
+ ··· − 2u + 1)
c
2
(u − 1)(u
33
− 6u
32
+ ··· + 128u − 23)
c
3
, c
4
, c
8
(u + 1)(u
33
− 2u
32
+ ··· + u
2
+ 1)
c
5
(u + 1)(u
33
+ u
31
+ ··· − 8u + 1)
c
7
, c
10
(u + 1)(u
33
+ 10u
32
+ ··· − 2u + 1)
c
9
u(u
33
+ 3u
32
+ ··· + 32u + 7)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y −1)(y
33
− 10y
32
+ ··· − 2y −1)
c
2
(y −1)(y
33
+ 14y
32
+ ··· − 2062y −529)
c
3
, c
4
, c
8
(y −1)(y
33
− 30y
32
+ ··· − 2y −1)
c
5
(y −1)(y
33
+ 2y
32
+ ··· − 2y −1)
c
7
, c
10
(y −1)(y
33
+ 26y
32
+ ··· + 6y −1)
c
9
y(y
33
− 3y
32
+ ··· + 394y −49)
12
