10
27
(K10a
58
)
A knot diagram
1
Linearized knot diagam
6 10 9 8 7 2 1 3 4 5
Solving Sequence
1,6
2 7 8 5 4 10 3 9
c
1
c
6
c
7
c
5
c
4
c
10
c
2
c
9
c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
35
− u
34
+ ··· + 2u − 1i
* 1 irreducible components of dim
C
= 0, with total 35 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
35
− u
34
+ · · · + 2u − 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
−u
−u
3
+ u
a
8
=
−u
3
−u
3
+ u
a
5
=
u
3
u
5
− u
3
+ u
a
4
=
−u
11
+ 2u
9
− 2u
7
+ u
3
−u
11
+ 3u
9
− 4u
7
+ 3u
5
− u
3
+ u
a
10
=
u
8
− u
6
+ u
4
+ 1
u
10
− 2u
8
+ 3u
6
− 2u
4
+ u
2
a
3
=
u
16
− 3u
14
+ 5u
12
− 4u
10
+ 3u
8
− 2u
6
+ 2u
4
+ 1
u
18
− 4u
16
+ 9u
14
− 12u
12
+ 11u
10
− 6u
8
+ 2u
6
+ u
2
a
9
=
−u
32
+ 7u
30
+ ··· + 2u
4
+ 1
−u
32
+ 8u
30
+ ··· − 4u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
33
+ 32u
31
−4u
30
−132u
29
+ 28u
28
+ 348u
27
−100u
26
−644u
25
+ 224u
24
+ 868u
23
−
344u
22
−880u
21
+376u
20
+700u
19
−312u
18
−488u
17
+228u
16
+336u
15
−180u
14
−232u
13
+
140u
12
+ 136u
11
− 88u
10
− 72u
9
+ 44u
8
+ 32u
7
− 24u
6
− 16u
5
+ 16u
4
+ 4u
3
− 8u
2
+ 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
35
− u
34
+ ··· + 2u − 1
c
2
, c
4
u
35
+ 3u
34
+ ··· + 14u + 5
c
3
, c
8
, c
9
u
35
− u
34
+ ··· + u
2
− 1
c
5
u
35
+ 17u
34
+ ··· + 2u + 1
c
7
u
35
− 3u
34
+ ··· + 58u − 7
c
10
u
35
+ u
34
+ ··· − 8u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
35
− 17y
34
+ ··· + 2y −1
c
2
, c
4
y
35
+ 23y
34
+ ··· + 166y −25
c
3
, c
8
, c
9
y
35
− 29y
34
+ ··· + 2y −1
c
5
y
35
+ 3y
34
+ ··· − 14y −1
c
7
y
35
+ 11y
34
+ ··· + 1446y −49
c
10
y
35
− y
34
+ ··· + 34y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.890522 + 0.542191I
3.03937 + 0.83862I 7.46140 + 0.32367I
u = 0.890522 − 0.542191I
3.03937 − 0.83862I 7.46140 − 0.32367I
u = −0.996188 + 0.423828I
−1.53766 + 1.71623I 0.733091 + 0.125972I
u = −0.996188 − 0.423828I
−1.53766 − 1.71623I 0.733091 − 0.125972I
u = 0.665614 + 0.623440I
3.70229 − 5.45820I 8.60996 + 5.96309I
u = 0.665614 − 0.623440I
3.70229 + 5.45820I 8.60996 − 5.96309I
u = 0.903342
2.34444 4.14110
u = −0.688085 + 0.531421I
−0.78083 + 2.01862I 2.90867 − 4.63726I
u = −0.688085 − 0.531421I
−0.78083 − 2.01862I 2.90867 + 4.63726I
u = 1.059800 + 0.502369I
−0.80902 − 4.67146I 3.48727 + 7.37463I
u = 1.059800 − 0.502369I
−0.80902 + 4.67146I 3.48727 − 7.37463I
u = −1.146120 + 0.254789I
−2.52028 − 4.45397I 0.84761 + 2.81525I
u = −1.146120 − 0.254789I
−2.52028 + 4.45397I 0.84761 − 2.81525I
u = 0.308085 + 0.766136I
1.96589 + 7.38977I 7.01566 − 5.00078I
u = 0.308085 − 0.766136I
1.96589 − 7.38977I 7.01566 + 5.00078I
u = 1.142990 + 0.287310I
−6.81373 + 0.30557I −3.68573 − 0.05854I
u = 1.142990 − 0.287310I
−6.81373 − 0.30557I −3.68573 + 0.05854I
u = −0.460984 + 0.678579I
6.79721 − 1.04155I 11.85373 + 0.57295I
u = −0.460984 − 0.678579I
6.79721 + 1.04155I 11.85373 − 0.57295I
u = −1.141570 + 0.325389I
−3.32477 + 3.85709I −0.01107 − 3.91391I
u = −1.141570 − 0.325389I
−3.32477 − 3.85709I −0.01107 + 3.91391I
u = −1.053770 + 0.564883I
5.05997 + 5.85664I 8.52563 − 5.76903I
u = −1.053770 − 0.564883I
5.05997 − 5.85664I 8.52563 + 5.76903I
u = −0.276974 + 0.740238I
−2.57455 − 3.36312I 2.16603 + 3.13288I
u = −0.276974 − 0.740238I
−2.57455 + 3.36312I 2.16603 − 3.13288I
u = 1.131430 + 0.520956I
−2.00084 − 4.02658I 1.98982 + 2.90516I
u = 1.131430 − 0.520956I
−2.00084 + 4.02658I 1.98982 − 2.90516I
u = −1.134810 + 0.545503I
−5.06633 + 8.22097I −0.85255 − 6.68822I
u = −1.134810 − 0.545503I
−5.06633 − 8.22097I −0.85255 + 6.68822I
u = 1.134940 + 0.561389I
−0.46048 − 12.38410I 3.84214 + 8.57579I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.134940 − 0.561389I
−0.46048 + 12.38410I 3.84214 − 8.57579I
u = 0.217277 + 0.699987I
0.592334 − 0.599446I 5.29885 + 0.74081I
u = 0.217277 − 0.699987I
0.592334 + 0.599446I 5.29885 − 0.74081I
u = 0.396163 + 0.521609I
1.091810 + 0.446317I 8.73891 − 2.08073I
u = 0.396163 − 0.521609I
1.091810 − 0.446317I 8.73891 + 2.08073I
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u
35
− u
34
+ ··· + 2u − 1
c
2
, c
4
u
35
+ 3u
34
+ ··· + 14u + 5
c
3
, c
8
, c
9
u
35
− u
34
+ ··· + u
2
− 1
c
5
u
35
+ 17u
34
+ ··· + 2u + 1
c
7
u
35
− 3u
34
+ ··· + 58u − 7
c
10
u
35
+ u
34
+ ··· − 8u − 1
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
35
− 17y
34
+ ··· + 2y −1
c
2
, c
4
y
35
+ 23y
34
+ ··· + 166y −25
c
3
, c
8
, c
9
y
35
− 29y
34
+ ··· + 2y −1
c
5
y
35
+ 3y
34
+ ··· − 14y −1
c
7
y
35
+ 11y
34
+ ··· + 1446y −49
c
10
y
35
− y
34
+ ··· + 34y −1
8
