10
143
(K10n
26
)
A knot diagram
1
Linearized knot diagam
6 5 10 8 1 2 9 5 3 9
Solving Sequence
1,5
6 2
3,9
8 4 7 10
c
5
c
1
c
2
c
8
c
4
c
7
c
10
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
12
− 2u
11
− 9u
10
+ 10u
9
+ 12u
8
− 18u
7
+ 7u
6
+ 4u
5
− 25u
4
+ 18u
3
+ 4u
2
+ 4b − 2u + 2,
− u
11
+ 4u
9
− 5u
7
− u
5
+ 2u
4
+ 6u
3
− 2u
2
+ 4a + 2u − 4,
u
13
− 2u
12
− 3u
11
+ 9u
10
− u
9
− 12u
8
+ 14u
7
− 7u
6
− 11u
5
+ 23u
4
− 12u
3
+ 2u
2
− 2i
I
u
2
= hb + 1, 2a + u, u
2
− 2i
I
u
3
= h−a
2
+ b + a, a
3
− 2a
2
+ a − 1, u + 1i
I
v
1
= ha, b − 1, v −1i
* 4 irreducible components of dim
C
= 0, with total 19 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
= h2u
12
−2u
11
+· · ·+4b+ 2, −u
11
+4u
9
+· · ·+4a−4, u
13
−2u
12
+· · ·+2u
2
−2i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
2
=
u
−u
3
+ u
a
3
=
−u
3
+ 2u
−u
3
+ u
a
9
=
1
4
u
11
− u
9
+ ··· −
1
2
u + 1
−
1
2
u
12
+
1
2
u
11
+ ··· +
1
2
u −
1
2
a
8
=
−
1
2
u
12
+
3
4
u
11
+ ··· −
1
2
u
2
+
1
2
−
1
2
u
12
+
1
2
u
11
+ ··· +
1
2
u −
1
2
a
4
=
−
1
4
u
11
+ u
9
+ ··· −
3
2
u + 1
−
1
4
u
11
+ u
9
+ ··· −
1
2
u
2
−
1
2
u
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
−
1
4
u
10
+ u
8
+ ··· −
1
2
u +
1
2
1
2
u
7
−
3
2
u
5
+
1
2
u
4
+ u
3
− u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 2u
12
− 10u
10
+ 2u
9
+ 18u
8
− 8u
7
− 4u
6
+ 10u
5
− 26u
4
+ 20u
2
− 2u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
u
13
+ 2u
12
+ ··· − 2u
2
+ 2
c
2
u
13
+ 3u
12
+ ··· − 92u + 46
c
3
, c
9
u
13
− 2u
12
+ ··· − 3u − 1
c
4
, c
8
u
13
+ 2u
12
+ ··· + 9u − 1
c
7
u
13
+ 18u
12
+ ··· + 65u + 1
c
10
u
13
− 2u
12
+ ··· + 17u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
y
13
− 10y
12
+ ··· + 8y −4
c
2
y
13
+ 23y
12
+ ··· + 7728y −2116
c
3
, c
9
y
13
− 2y
12
+ ··· + 17y −1
c
4
, c
8
y
13
− 18y
12
+ ··· + 65y −1
c
7
y
13
− 42y
12
+ ··· + 2989y −1
c
10
y
13
+ 22y
12
+ ··· + 205y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.116060 + 1.025320I
a = −1.94905 − 0.25674I
b = −1.69551 + 0.12749I
−10.21610 + 3.70097I 0.67358 − 2.50956I
u = 0.116060 − 1.025320I
a = −1.94905 + 0.25674I
b = −1.69551 − 0.12749I
−10.21610 − 3.70097I 0.67358 + 2.50956I
u = 1.197110 + 0.332616I
a = −0.447636 − 0.899887I
b = −0.583119 + 0.809161I
1.92578 + 4.88678I 6.41460 − 5.91732I
u = 1.197110 − 0.332616I
a = −0.447636 + 0.899887I
b = −0.583119 − 0.809161I
1.92578 − 4.88678I 6.41460 + 5.91732I
u = 1.236960 + 0.573659I
a = 0.918969 + 0.882216I
b = 1.67219 − 0.07727I
−6.78115 + 1.92961I 2.66803 − 0.98070I
u = 1.236960 − 0.573659I
a = 0.918969 − 0.882216I
b = 1.67219 + 0.07727I
−6.78115 − 1.92961I 2.66803 + 0.98070I
u = 1.38959
a = −0.810069
b = −0.135830
6.53354 13.9760
u = 0.094132 + 0.586012I
a = 0.854196 + 0.075054I
b = 0.787240 + 0.445864I
−1.38205 − 1.36942I −0.56235 + 3.09698I
u = 0.094132 − 0.586012I
a = 0.854196 − 0.075054I
b = 0.787240 − 0.445864I
−1.38205 + 1.36942I −0.56235 − 3.09698I
u = −1.45446
a = 0.0472843
b = −1.10499
3.37738 1.87580
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.40252 + 0.47847I
a = 0.81193 − 1.16730I
b = 1.62497 + 0.28976I
−5.44762 − 9.07090I 4.16718 + 5.02365I
u = −1.40252 − 0.47847I
a = 0.81193 + 1.16730I
b = 1.62497 − 0.28976I
−5.44762 + 9.07090I 4.16718 − 5.02365I
u = −0.418617
a = 1.38596
b = −0.370722
0.992576 11.4260
6
II. I
u
2
= hb + 1, 2a + u, u
2
− 2i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
6
=
1
−2
a
2
=
u
−u
a
3
=
0
−u
a
9
=
−
1
2
u
−1
a
8
=
−
1
2
u − 1
−1
a
4
=
−
1
2
u
−1
a
7
=
−1
0
a
10
=
−
1
2
u
u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
2
− 2
c
3
, c
4
(u + 1)
2
c
7
, c
8
, c
9
c
10
(u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
(y −2)
2
c
3
, c
4
, c
7
c
8
, c
9
, c
10
(y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = −0.707107
b = −1.00000
4.93480 8.00000
u = −1.41421
a = 0.707107
b = −1.00000
4.93480 8.00000
10
III. I
u
3
= h−a
2
+ b + a, a
3
− 2a
2
+ a − 1, u + 1i
(i) Arc colorings
a
1
=
0
−1
a
5
=
1
0
a
6
=
1
−1
a
2
=
−1
0
a
3
=
−1
0
a
9
=
a
a
2
− a
a
8
=
a
2
a
2
− a
a
4
=
−a
2
−a
a
7
=
0
−1
a
10
=
a
2
a
2
− a
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
(u − 1)
3
c
2
u
3
c
3
, c
4
, c
8
c
9
u
3
− u + 1
c
7
u
3
+ 2u
2
+ u + 1
c
10
u
3
− 2u
2
+ u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
(y −1)
3
c
2
y
3
c
3
, c
4
, c
8
c
9
y
3
− 2y
2
+ y −1
c
7
, c
10
y
3
− 2y
2
− 3y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.122561 + 0.744862I
b = −0.662359 − 0.562280I
1.64493 6.00000
u = −1.00000
a = 0.122561 − 0.744862I
b = −0.662359 + 0.562280I
1.64493 6.00000
u = −1.00000
a = 1.75488
b = 1.32472
1.64493 6.00000
14
IV. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
1
0
a
6
=
1
0
a
2
=
1
0
a
3
=
1
0
a
9
=
0
1
a
8
=
1
1
a
4
=
0
−1
a
7
=
1
0
a
10
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
c
3
, c
4
, c
7
c
10
u − 1
c
8
, c
9
u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
y
c
3
, c
4
, c
7
c
8
, c
9
, c
10
y −1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
u(u − 1)
3
(u
2
− 2)(u
13
+ 2u
12
+ ··· − 2u
2
+ 2)
c
2
u
4
(u
2
− 2)(u
13
+ 3u
12
+ ··· − 92u + 46)
c
3
(u − 1)(u + 1)
2
(u
3
− u + 1)(u
13
− 2u
12
+ ··· − 3u − 1)
c
4
(u − 1)(u + 1)
2
(u
3
− u + 1)(u
13
+ 2u
12
+ ··· + 9u − 1)
c
7
((u − 1)
3
)(u
3
+ 2u
2
+ u + 1)(u
13
+ 18u
12
+ ··· + 65u + 1)
c
8
((u − 1)
2
)(u + 1)(u
3
− u + 1)(u
13
+ 2u
12
+ ··· + 9u − 1)
c
9
((u − 1)
2
)(u + 1)(u
3
− u + 1)(u
13
− 2u
12
+ ··· − 3u − 1)
c
10
((u − 1)
3
)(u
3
− 2u
2
+ u − 1)(u
13
− 2u
12
+ ··· + 17u − 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
y(y −2)
2
(y −1)
3
(y
13
− 10y
12
+ ··· + 8y −4)
c
2
y
4
(y −2)
2
(y
13
+ 23y
12
+ ··· + 7728y −2116)
c
3
, c
9
((y −1)
3
)(y
3
− 2y
2
+ y −1)(y
13
− 2y
12
+ ··· + 17y −1)
c
4
, c
8
((y −1)
3
)(y
3
− 2y
2
+ y −1)(y
13
− 18y
12
+ ··· + 65y −1)
c
7
((y −1)
3
)(y
3
− 2y
2
− 3y −1)(y
13
− 42y
12
+ ··· + 2989y −1)
c
10
((y −1)
3
)(y
3
− 2y
2
− 3y −1)(y
13
+ 22y
12
+ ··· + 205y −1)
20
