10
135
(K10n
5
)
A knot diagram
1
Linearized knot diagam
3 9 1 7 4 9 5 2 7 8
Solving Sequence
2,8
9 3
1,5
7 4 6 10
c
8
c
2
c
1
c
7
c
4
c
5
c
10
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
20
− u
19
+ ··· + b − 2u, u
18
+ u
17
+ ··· + a + 2, u
21
+ 2u
20
+ ··· + u − 1i
I
u
2
= hb + 1, a + u, u
3
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 24 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−u
20
−u
19
+· · · + b− 2u, u
18
+u
17
+· · · + a + 2, u
21
+2u
20
+· · · + u − 1i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
3
=
u
−u
3
+ u
a
1
=
−u
3
u
5
− u
3
+ u
a
5
=
−u
18
− u
17
+ ··· − u − 2
u
20
+ u
19
+ ··· + 4u
2
+ 2u
a
7
=
−u
20
− u
19
+ ··· + u + 3
−u
20
− u
19
+ ··· − 5u
2
− u
a
4
=
u
5
+ u
−u
7
+ u
5
− 2u
3
+ u
a
6
=
2u
20
+ 3u
19
+ ··· + 2u − 4
2u
20
+ u
19
+ ··· + 8u
3
+ 7u
2
a
10
=
−u
5
− u
u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −7u
20
− 10u
19
+ 19u
18
+ 42u
17
− 31u
16
− 99u
15
+ 18u
14
+ 172u
13
+ 44u
12
− 198u
11
−
125u
10
+ 168u
9
+ 183u
8
− 76u
7
− 166u
6
− 8u
5
+ 93u
4
+ 26u
3
− 41u
2
− 28u + 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
21
− 8u
20
+ ··· + 17u − 1
c
2
, c
8
u
21
− 2u
20
+ ··· + u + 1
c
4
, c
7
u
21
− 4u
20
+ ··· − 2u + 1
c
5
u
21
+ 6u
20
+ ··· − 2u + 1
c
6
, c
9
u
21
− u
20
+ ··· + 4u + 8
c
10
u
21
+ 2u
20
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
y
21
+ 12y
20
+ ··· + 137y − 1
c
2
, c
8
y
21
− 8y
20
+ ··· + 17y − 1
c
4
, c
7
y
21
− 6y
20
+ ··· − 2y − 1
c
5
y
21
+ 22y
20
+ ··· + 66y − 1
c
6
, c
9
y
21
+ 21y
20
+ ··· − 176y − 64
c
10
y
21
− 24y
20
+ ··· + 17y − 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.567882 + 0.851579I
a = −0.521076 − 0.321393I
b = −1.047460 + 0.802568I
2.42497 + 4.94435I −1.24866 − 2.70559I
u = −0.567882 − 0.851579I
a = −0.521076 + 0.321393I
b = −1.047460 − 0.802568I
2.42497 − 4.94435I −1.24866 + 2.70559I
u = −0.848992 + 0.598239I
a = 1.09099 − 1.32571I
b = 1.272850 + 0.072825I
−3.02655 − 2.36605I −0.59037 + 2.67274I
u = −0.848992 − 0.598239I
a = 1.09099 + 1.32571I
b = 1.272850 − 0.072825I
−3.02655 + 2.36605I −0.59037 − 2.67274I
u = −0.427156 + 0.796867I
a = −0.517814 + 0.424717I
b = −0.770704 − 0.886977I
3.29052 − 1.36266I −0.18856 + 2.27516I
u = −0.427156 − 0.796867I
a = −0.517814 − 0.424717I
b = −0.770704 + 0.886977I
3.29052 + 1.36266I −0.18856 − 2.27516I
u = 0.707761 + 0.560391I
a = −0.427154 − 0.668417I
b = 0.837997 + 0.449477I
−1.83472 + 0.21101I −3.18710 − 0.57244I
u = 0.707761 − 0.560391I
a = −0.427154 + 0.668417I
b = 0.837997 − 0.449477I
−1.83472 − 0.21101I −3.18710 + 0.57244I
u = 0.951460 + 0.595395I
a = 0.13569 + 1.78932I
b = 0.666759 − 0.637720I
−1.06863 + 4.45806I −0.43689 − 6.14529I
u = 0.951460 − 0.595395I
a = 0.13569 − 1.78932I
b = 0.666759 + 0.637720I
−1.06863 − 4.45806I −0.43689 + 6.14529I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.853051 + 0.160221I
a = −0.985793 + 0.858266I
b = 0.040042 − 0.421446I
1.46918 − 0.34630I 5.96536 + 0.53554I
u = −0.853051 − 0.160221I
a = −0.985793 − 0.858266I
b = 0.040042 + 0.421446I
1.46918 + 0.34630I 5.96536 − 0.53554I
u = 1.169830 + 0.051846I
a = 0.79206 − 1.88563I
b = −0.960607 + 0.961815I
8.83595 + 3.51416I 4.91512 − 2.66916I
u = 1.169830 − 0.051846I
a = 0.79206 + 1.88563I
b = −0.960607 − 0.961815I
8.83595 − 3.51416I 4.91512 + 2.66916I
u = 0.882737 + 0.780973I
a = −0.878224 − 0.429656I
b = −0.622642 + 0.052532I
−3.85955 + 2.93752I 2.97600 − 3.43881I
u = 0.882737 − 0.780973I
a = −0.878224 + 0.429656I
b = −0.622642 − 0.052532I
−3.85955 − 2.93752I 2.97600 + 3.43881I
u = −1.083580 + 0.616829I
a = 1.133680 − 0.321228I
b = −0.721179 + 1.021470I
5.21503 − 3.89686I 2.41425 + 2.65107I
u = −1.083580 − 0.616829I
a = 1.133680 + 0.321228I
b = −0.721179 − 1.021470I
5.21503 + 3.89686I 2.41425 − 2.65107I
u = −1.075840 + 0.689537I
a = −0.86208 + 1.97593I
b = −1.117050 − 0.836949I
3.96319 − 10.68720I 0.56681 + 6.96141I
u = −1.075840 − 0.689537I
a = −0.86208 − 1.97593I
b = −1.117050 + 0.836949I
3.96319 + 10.68720I 0.56681 − 6.96141I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.289436
a = −1.92057
b = 0.843987
−1.20998 −9.37190
7
II. I
u
2
= hb + 1, a + u, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
3
=
u
−u
2
+ u + 1
a
1
=
−u
2
+ 1
−u
2
a
5
=
−u
−1
a
7
=
−u + 1
−1
a
4
=
−1
0
a
6
=
−u + 1
−1
a
10
=
1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
2
− 7u − 2
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ u
2
+ 2u + 1
c
2
u
3
+ u
2
− 1
c
3
, c
10
u
3
− u
2
+ 2u − 1
c
4
(u − 1)
3
c
5
, c
7
(u + 1)
3
c
6
, c
9
u
3
c
8
u
3
− u
2
+ 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
10
y
3
+ 3y
2
+ 2y − 1
c
2
, c
8
y
3
− y
2
+ 2y − 1
c
4
, c
5
, c
7
(y − 1)
3
c
6
, c
9
y
3
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.877439 − 0.744862I
b = −1.00000
−4.66906 + 2.82812I −7.71191 − 2.59975I
u = 0.877439 − 0.744862I
a = −0.877439 + 0.744862I
b = −1.00000
−4.66906 − 2.82812I −7.71191 + 2.59975I
u = −0.754878
a = 0.754878
b = −1.00000
−0.531480 4.42380
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
+ u
2
+ 2u + 1)(u
21
− 8u
20
+ ··· + 17u − 1)
c
2
(u
3
+ u
2
− 1)(u
21
− 2u
20
+ ··· + u + 1)
c
3
(u
3
− u
2
+ 2u − 1)(u
21
− 8u
20
+ ··· + 17u − 1)
c
4
((u − 1)
3
)(u
21
− 4u
20
+ ··· − 2u + 1)
c
5
((u + 1)
3
)(u
21
+ 6u
20
+ ··· − 2u + 1)
c
6
, c
9
u
3
(u
21
− u
20
+ ··· + 4u + 8)
c
7
((u + 1)
3
)(u
21
− 4u
20
+ ··· − 2u + 1)
c
8
(u
3
− u
2
+ 1)(u
21
− 2u
20
+ ··· + u + 1)
c
10
(u
3
− u
2
+ 2u − 1)(u
21
+ 2u
20
+ ··· + 3u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
(y
3
+ 3y
2
+ 2y − 1)(y
21
+ 12y
20
+ ··· + 137y − 1)
c
2
, c
8
(y
3
− y
2
+ 2y − 1)(y
21
− 8y
20
+ ··· + 17y − 1)
c
4
, c
7
((y − 1)
3
)(y
21
− 6y
20
+ ··· − 2y − 1)
c
5
((y − 1)
3
)(y
21
+ 22y
20
+ ··· + 66y − 1)
c
6
, c
9
y
3
(y
21
+ 21y
20
+ ··· − 176y − 64)
c
10
(y
3
+ 3y
2
+ 2y − 1)(y
21
− 24y
20
+ ··· + 17y − 1)
13
