10
15
(K10a
68
)
A knot diagram
1
Linearized knot diagam
3 8 1 9 10 4 2 7 5 6
Solving Sequence
3,8
2 1 4 7 9 5 6 10
c
2
c
1
c
3
c
7
c
8
c
4
c
6
c
10
c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
21
− u
20
+ ··· + u − 1i
* 1 irreducible components of dim
C
= 0, with total 21 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
21
− u
20
+ · · · + u − 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
4
=
u
4
− u
2
+ 1
u
4
a
7
=
u
−u
3
+ u
a
9
=
−u
3
u
5
− u
3
+ u
a
5
=
u
12
− u
10
+ 3u
8
− 2u
6
+ 2u
4
− u
2
+ 1
−u
14
+ 2u
12
− 5u
10
+ 6u
8
− 6u
6
+ 4u
4
− u
2
a
6
=
u
11
− 2u
9
+ 4u
7
− 4u
5
+ 3u
3
u
11
− u
9
+ 2u
7
− u
5
− u
3
+ u
a
10
=
u
20
− 3u
18
+ 9u
16
− 16u
14
+ 24u
12
− 25u
10
+ 21u
8
− 10u
6
+ 3u
4
− u
2
+ 1
u
20
− 2u
18
+ 6u
16
− 8u
14
+ 9u
12
− 6u
10
+ 4u
6
− 3u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
19
+ 4u
18
+ 8u
17
− 12u
16
− 28u
15
+ 32u
14
+ 40u
13
− 56u
12
−
64u
11
+ 72u
10
+ 64u
9
− 68u
8
− 56u
7
+ 44u
6
+ 36u
5
− 8u
4
− 16u
3
− 4u
2
+ 8u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
8
u
21
+ 5u
20
+ ··· + 3u + 1
c
2
, c
7
u
21
− u
20
+ ··· + u − 1
c
4
, c
5
, c
9
c
10
u
21
− u
20
+ ··· − u − 1
c
6
u
21
+ 7u
20
+ ··· + 57u + 23
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
8
y
21
+ 23y
20
+ ··· − 21y −1
c
2
, c
7
y
21
− 5y
20
+ ··· + 3y −1
c
4
, c
5
, c
9
c
10
y
21
− 25y
20
+ ··· + 3y −1
c
6
y
21
− 13y
20
+ ··· + 903y −529
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.953485
4.41569 −0.452350
u = 0.874819 + 0.364250I
−0.42770 − 3.55745I 0.30280 + 8.52474I
u = 0.874819 − 0.364250I
−0.42770 + 3.55745I 0.30280 − 8.52474I
u = −0.953468 + 0.447109I
6.90304 + 5.27729I 3.50266 − 5.86843I
u = −0.953468 − 0.447109I
6.90304 − 5.27729I 3.50266 + 5.86843I
u = −0.797642 + 0.208550I
−1.36175 + 0.64933I −4.62516 − 0.62543I
u = −0.797642 − 0.208550I
−1.36175 − 0.64933I −4.62516 + 0.62543I
u = −0.863139 + 0.856542I
7.11051 − 0.45995I 6.17329 + 1.45528I
u = −0.863139 − 0.856542I
7.11051 + 0.45995I 6.17329 − 1.45528I
u = 0.900058 + 0.818905I
4.54603 − 3.06102I 1.66624 + 2.52883I
u = 0.900058 − 0.818905I
4.54603 + 3.06102I 1.66624 − 2.52883I
u = 0.853497 + 0.897241I
15.5974 + 2.5355I 7.87177 − 0.33713I
u = 0.853497 − 0.897241I
15.5974 − 2.5355I 7.87177 + 0.33713I
u = −0.352374 + 0.669848I
8.81523 − 1.18870I 8.06950 + 0.14927I
u = −0.352374 − 0.669848I
8.81523 + 1.18870I 8.06950 − 0.14927I
u = −0.945375 + 0.826771I
6.85351 + 6.71941I 5.45682 − 6.57422I
u = −0.945375 − 0.826771I
6.85351 − 6.71941I 5.45682 + 6.57422I
u = 0.974286 + 0.843873I
15.2130 − 8.9734I 7.18924 + 5.14301I
u = 0.974286 − 0.843873I
15.2130 + 8.9734I 7.18924 − 5.14301I
u = 0.332595 + 0.443596I
1.162670 + 0.391903I 7.61900 − 1.22999I
u = 0.332595 − 0.443596I
1.162670 − 0.391903I 7.61900 + 1.22999I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
8
u
21
+ 5u
20
+ ··· + 3u + 1
c
2
, c
7
u
21
− u
20
+ ··· + u − 1
c
4
, c
5
, c
9
c
10
u
21
− u
20
+ ··· − u − 1
c
6
u
21
+ 7u
20
+ ··· + 57u + 23
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
8
y
21
+ 23y
20
+ ··· − 21y −1
c
2
, c
7
y
21
− 5y
20
+ ··· + 3y −1
c
4
, c
5
, c
9
c
10
y
21
− 25y
20
+ ··· + 3y −1
c
6
y
21
− 13y
20
+ ··· + 903y −529
7
