10
136
(K10n
3
)
A knot diagram
1
Linearized knot diagam
9 1 7 3 10 4 5 2 5 3
Solving Sequence
3,10
1
2,6
5 4 7 8 9
c
10
c
2
c
5
c
4
c
6
c
7
c
9
c
1
, c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
4
− u
3
+ 6u
2
+ 2b − 3u − 1, u
4
− u
3
+ 6u
2
+ 2a − 3u + 1, u
5
− u
4
+ 5u
3
− 3u
2
− u + 1i
I
u
2
= hb, a + 1, u
2
− u + 1i
I
u
3
= h−7u
5
+ 4u
4
− 41u
3
+ 14u
2
+ 23b − 57u − 18, −32u
5
+ 15u
4
− 194u
3
+ 110u
2
+ 23a − 254u − 10,
u
6
+ 6u
4
− u
3
+ 7u
2
+ 3u + 1i
I
u
4
= hb, a − u, u
2
− u + 1i
* 4 irreducible components of dim
C
= 0, with total 15 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
4
−u
3
+6u
2
+2b−3u−1, u
4
−u
3
+6u
2
+2a−3u+1, u
5
−u
4
+5u
3
−3u
2
−u+1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
1
=
1
u
2
a
2
=
u
u
3
+ u
a
6
=
−
1
2
u
4
+
1
2
u
3
+ ··· +
3
2
u −
1
2
−
1
2
u
4
+
1
2
u
3
+ ··· +
3
2
u +
1
2
a
5
=
−1
−
1
2
u
4
+
1
2
u
3
+ ··· +
3
2
u +
1
2
a
4
=
−1
−
1
2
u
4
+
1
2
u
3
+ ··· +
3
2
u +
1
2
a
7
=
−u
4
+
1
2
u
3
− 5u
2
+ u +
1
2
u
a
8
=
−u
4
+ u
3
− 5u
2
+ 2u
−
1
2
u
3
− u
2
+ 2u −
1
2
a
9
=
1
2
u
4
−
1
2
u
3
+ 3u
2
−
3
2
u +
1
2
−u
2
− u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 5u
4
− 4u
3
+ 24u
2
− 9u − 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
c
8
u
5
+ u
4
+ u
3
− u
2
+ u − 1
c
2
, c
4
, c
10
u
5
+ u
4
+ 5u
3
+ 3u
2
− u − 1
c
5
, c
9
u
5
+ 5u
4
+ 6u
3
− 4u
2
− 8u − 4
c
7
u
5
− 4u
4
+ 15u
3
− 10u
2
− u − 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
y
5
+ y
4
+ 5y
3
+ 3y
2
− y −1
c
2
, c
4
, c
10
y
5
+ 9y
4
+ 17y
3
− 17y
2
+ 7y −1
c
5
, c
9
y
5
− 13y
4
+ 60y
3
− 72y
2
+ 32y −16
c
7
y
5
+ 14y
4
+ 143y
3
− 146y
2
− 39y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.581386 + 0.247464I
a = −0.410284 − 0.453785I
b = 0.589716 − 0.453785I
−1.59034 − 1.66520I −2.97976 + 4.53195I
u = 0.581386 − 0.247464I
a = −0.410284 + 0.453785I
b = 0.589716 + 0.453785I
−1.59034 + 1.66520I −2.97976 − 4.53195I
u = −0.504717
a = −2.11802
b = −1.11802
1.42879 7.49490
u = 0.17097 + 2.22112I
a = 1.46930 − 0.60354I
b = 2.46930 − 0.60354I
14.8579 − 7.7463I 4.23230 + 4.04224I
u = 0.17097 − 2.22112I
a = 1.46930 + 0.60354I
b = 2.46930 + 0.60354I
14.8579 + 7.7463I 4.23230 − 4.04224I
5
II. I
u
2
= hb, a + 1, u
2
− u + 1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
1
=
1
u − 1
a
2
=
u
u − 1
a
6
=
−1
0
a
5
=
−1
0
a
4
=
−1
u − 1
a
7
=
−u
−u
a
8
=
0
−u
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u − 4
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
10
u
2
− u + 1
c
2
, c
3
, c
4
c
7
, c
8
u
2
+ u + 1
c
5
, c
9
u
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
y
2
+ y + 1
c
5
, c
9
y
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = −1.00000
b = 0
− 4.05977I 0. + 6.92820I
u = 0.500000 − 0.866025I
a = −1.00000
b = 0
4.05977I 0. − 6.92820I
9
III. I
u
3
= h−7u
5
+ 4u
4
+ · · · + 23b − 18, −32u
5
+ 15u
4
+ · · · + 23a − 10, u
6
+
6u
4
− u
3
+ 7u
2
+ 3u + 1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
1
=
1
u
2
a
2
=
u
u
3
+ u
a
6
=
1.39130u
5
− 0.652174u
4
+ ··· + 11.0435u + 0.434783
0.304348u
5
− 0.173913u
4
+ ··· + 2.47826u + 0.782609
a
5
=
1.08696u
5
− 0.478261u
4
+ ··· + 8.56522u − 0.347826
0.304348u
5
− 0.173913u
4
+ ··· + 2.47826u + 0.782609
a
4
=
1.08696u
5
− 0.478261u
4
+ ··· + 8.56522u − 0.347826
0.173913u
5
+ 0.0434783u
4
+ ··· + 2.13043u + 0.304348
a
7
=
−0.217391u
5
− 0.304348u
4
+ ··· + 0.0869565u − 3.13043
u
a
8
=
0.173913u
5
+ 0.0434783u
4
+ ··· + 1.13043u + 2.30435
−0.0869565u
5
+ 0.478261u
4
+ ··· − 1.56522u + 0.347826
a
9
=
−0.391304u
5
− 0.347826u
4
+ ··· − 2.04348u − 2.43478
0.0434783u
5
+ 0.260870u
4
+ ··· + 1.78261u − 0.173913
(ii) Obstruction class = −1
(iii) Cusp Shapes =
15
23
u
5
−
2
23
u
4
+
101
23
u
3
−
30
23
u
2
+
155
23
u +
101
23
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
c
8
u
6
+ 2u
5
+ 2u
4
+ u
3
+ 3u
2
+ 3u + 1
c
2
, c
4
, c
10
u
6
+ 6u
4
+ u
3
+ 7u
2
− 3u + 1
c
5
, c
9
(u
3
− 2u
2
− u − 2)
2
c
7
u
6
+ u
5
+ 14u
4
+ 33u
3
+ 58u
2
+ 45u + 17
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
y
6
+ 6y
4
+ y
3
+ 7y
2
− 3y + 1
c
2
, c
4
, c
10
y
6
+ 12y
5
+ 50y
4
+ 85y
3
+ 67y
2
+ 5y + 1
c
5
, c
9
(y
3
− 6y
2
− 7y −4)
2
c
7
y
6
+ 27y
5
+ 246y
4
+ 479y
3
+ 870y
2
− 53y + 289
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.447867 + 1.186990I
a = 0.183999 + 0.561035I
b = 0.329484 + 0.802255I
0.60803 − 2.56897I 3.12391 + 2.13317I
u = 0.447867 − 1.186990I
a = 0.183999 − 0.561035I
b = 0.329484 − 0.802255I
0.60803 + 2.56897I 3.12391 − 2.13317I
u = −0.221168 + 0.280722I
a = −1.50390 + 3.84210I
b = 0.329484 + 0.802255I
0.60803 − 2.56897I 3.12391 + 2.13317I
u = −0.221168 − 0.280722I
a = −1.50390 − 3.84210I
b = 0.329484 − 0.802255I
0.60803 + 2.56897I 3.12391 − 2.13317I
u = −0.22670 + 2.19389I
a = −1.68010 − 0.20448I
b = −2.65897
15.2333 4.75217 + 0.I
u = −0.22670 − 2.19389I
a = −1.68010 + 0.20448I
b = −2.65897
15.2333 4.75217 + 0.I
13
IV. I
u
4
= hb, a − u, u
2
− u + 1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
1
=
1
u − 1
a
2
=
u
u − 1
a
6
=
u
0
a
5
=
u
0
a
4
=
u
1
a
7
=
1
−u
a
8
=
0
−u
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
10
u
2
− u + 1
c
2
, c
3
, c
4
c
7
, c
8
u
2
+ u + 1
c
5
, c
9
u
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
y
2
+ y + 1
c
5
, c
9
y
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0
0 3.00000
u = 0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = 0
0 3.00000
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
2
− u + 1)
2
(u
5
+ u
4
+ u
3
− u
2
+ u − 1)
· (u
6
+ 2u
5
+ 2u
4
+ u
3
+ 3u
2
+ 3u + 1)
c
2
, c
4
((u
2
+ u + 1)
2
)(u
5
+ u
4
+ ··· − u − 1)(u
6
+ 6u
4
+ ··· − 3u + 1)
c
3
, c
8
(u
2
+ u + 1)
2
(u
5
+ u
4
+ u
3
− u
2
+ u − 1)
· (u
6
+ 2u
5
+ 2u
4
+ u
3
+ 3u
2
+ 3u + 1)
c
5
, c
9
u
4
(u
3
− 2u
2
− u − 2)
2
(u
5
+ 5u
4
+ 6u
3
− 4u
2
− 8u − 4)
c
7
(u
2
+ u + 1)
2
(u
5
− 4u
4
+ 15u
3
− 10u
2
− u − 2)
· (u
6
+ u
5
+ 14u
4
+ 33u
3
+ 58u
2
+ 45u + 17)
c
10
((u
2
− u + 1)
2
)(u
5
+ u
4
+ ··· − u − 1)(u
6
+ 6u
4
+ ··· − 3u + 1)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
((y
2
+ y + 1)
2
)(y
5
+ y
4
+ ··· − y −1)(y
6
+ 6y
4
+ ··· − 3y + 1)
c
2
, c
4
, c
10
(y
2
+ y + 1)
2
(y
5
+ 9y
4
+ 17y
3
− 17y
2
+ 7y −1)
· (y
6
+ 12y
5
+ 50y
4
+ 85y
3
+ 67y
2
+ 5y + 1)
c
5
, c
9
y
4
(y
3
− 6y
2
− 7y −4)
2
(y
5
− 13y
4
+ 60y
3
− 72y
2
+ 32y −16)
c
7
(y
2
+ y + 1)
2
(y
5
+ 14y
4
+ 143y
3
− 146y
2
− 39y −4)
· (y
6
+ 27y
5
+ 246y
4
+ 479y
3
+ 870y
2
− 53y + 289)
19
