10
133
(K10n
4
)
A knot diagram
1
Linearized knot diagam
8 1 6 3 10 4 10 2 5 3
Solving Sequence
3,6
4
7,10
8 1 2 5 9
c
3
c
6
c
7
c
10
c
2
c
5
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
+ 5u
10
+ 9u
9
+ 2u
8
− 15u
7
− 18u
6
+ u
5
+ 13u
4
+ 5u
3
− u
2
+ 4b − 7u + 1,
− u
11
− 5u
10
− 11u
9
− 8u
8
+ 9u
7
+ 24u
6
+ 13u
5
− 7u
4
− 13u
3
− 3u
2
+ 2a + 5u + 5,
u
12
+ 4u
11
+ 8u
10
+ 5u
9
− 5u
8
− 15u
7
− 9u
6
+ 8u
4
+ 2u
3
− 2u
2
− 4u − 1i
I
u
2
= hb
3
+ b
2
+ 2b + 1, a, u − 1i
* 2 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
11
+5u
10
+· · ·+4b+1, −u
11
−5u
10
+· · ·+2a+5, u
12
+4u
11
+· · ·−4u−1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
4
=
1
u
2
a
7
=
−u
−u
3
+ u
a
10
=
1
2
u
11
+
5
2
u
10
+ ··· −
5
2
u −
5
2
−
1
4
u
11
−
5
4
u
10
+ ··· +
7
4
u −
1
4
a
8
=
u
4
− u
2
− 2u + 1
−
1
4
u
11
−
3
4
u
10
+ ··· +
3
4
u +
1
4
a
1
=
3
4
u
11
+
15
4
u
10
+ ··· −
17
4
u −
9
4
−
1
4
u
11
−
5
4
u
10
+ ··· +
7
4
u −
1
4
a
2
=
−
1
4
u
11
−
3
4
u
10
+ ··· +
3
4
u +
1
4
1
4
u
11
+
3
4
u
10
+ ··· +
1
4
u −
1
4
a
5
=
−u
2
+ 1
u
2
a
9
=
3
2
u
11
+
9
2
u
10
+ ··· −
3
2
u −
3
2
−
3
4
u
11
−
7
4
u
10
+ ··· +
1
4
u −
3
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 2u
11
+
17
2
u
10
+ 16u
9
+
13
2
u
8
−
39
2
u
7
− 34u
6
− 9u
5
+
35
2
u
4
+ 19u
3
+
3
2
u
2
− 12u −
19
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
12
+ 2u
11
+ u
10
− 2u
9
+ u
8
+ 6u
7
+ 4u
6
− 3u
5
+ 6u
3
+ 3u
2
− u − 1
c
2
, c
10
u
12
+ 2u
11
+ ··· + 7u + 1
c
3
, c
6
u
12
− 4u
11
+ 8u
10
− 5u
9
− 5u
8
+ 15u
7
− 9u
6
+ 8u
4
− 2u
3
− 2u
2
+ 4u − 1
c
4
u
12
+ 14u
10
+ ··· + 12u + 1
c
5
, c
9
u
12
+ u
11
+ ··· + 36u + 8
c
7
u
12
− 2u
11
+ ··· − 175u − 49
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
12
− 2y
11
+ ··· − 7y + 1
c
2
, c
10
y
12
+ 18y
11
+ ··· − 7y + 1
c
3
, c
6
y
12
+ 14y
10
+ ··· − 12y + 1
c
4
y
12
+ 28y
11
+ ··· − 136y + 1
c
5
, c
9
y
12
− 21y
11
+ ··· − 464y + 64
c
7
y
12
+ 54y
11
+ ··· − 39739y + 2401
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.267707 + 0.884422I
a = 0.991606 + 0.968229I
b = 0.208639 − 1.095630I
3.72986 − 1.03019I −1.27943 + 1.44119I
u = −0.267707 − 0.884422I
a = 0.991606 − 0.968229I
b = 0.208639 + 1.095630I
3.72986 + 1.03019I −1.27943 − 1.44119I
u = −0.561933 + 0.696285I
a = −0.925264 − 0.846250I
b = −0.544421 + 1.250460I
2.66318 + 4.39533I −2.94428 − 5.22312I
u = −0.561933 − 0.696285I
a = −0.925264 + 0.846250I
b = −0.544421 − 1.250460I
2.66318 − 4.39533I −2.94428 + 5.22312I
u = 1.11609
a = 0.469158
b = −0.247448
−2.23241 0.00782210
u = 0.703419 + 0.354505I
a = 0.543453 + 0.851824I
b = −0.137910 − 0.436156I
−0.87372 − 1.32529I −6.28742 + 4.78445I
u = 0.703419 − 0.354505I
a = 0.543453 − 0.851824I
b = −0.137910 + 0.436156I
−0.87372 + 1.32529I −6.28742 − 4.78445I
u = −1.18067 + 1.13803I
a = −0.702429 − 1.111310I
b = −0.15451 + 1.86459I
14.0447 + 7.7983I −3.16952 − 4.22102I
u = −1.18067 − 1.13803I
a = −0.702429 + 1.111310I
b = −0.15451 − 1.86459I
14.0447 − 7.7983I −3.16952 + 4.22102I
u = −1.10559 + 1.21488I
a = 0.744589 + 1.118150I
b = 0.11602 − 1.80584I
14.3370 + 0.8045I −2.71291 + 0.16086I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.10559 − 1.21488I
a = 0.744589 − 1.118150I
b = 0.11602 + 1.80584I
14.3370 − 0.8045I −2.71291 − 0.16086I
u = −0.291129
a = −1.77307
b = −0.728189
−1.41716 −6.22070
6
II. I
u
2
= hb
3
+ b
2
+ 2b + 1, a, u − 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
1
a
4
=
1
1
a
7
=
−1
0
a
10
=
0
b
a
8
=
−1
−b
2
a
1
=
−b
b
a
2
=
b
2
+ 1
−b
2
a
5
=
0
1
a
9
=
0
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7b
2
− 5b − 17
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
− u
2
+ 1
c
2
u
3
+ u
2
+ 2u + 1
c
3
(u − 1)
3
c
4
, c
6
(u + 1)
3
c
5
, c
9
u
3
c
7
, c
10
u
3
− u
2
+ 2u − 1
c
8
u
3
+ u
2
− 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
3
− y
2
+ 2y −1
c
2
, c
7
, c
10
y
3
+ 3y
2
+ 2y −1
c
3
, c
4
, c
6
(y −1)
3
c
5
, c
9
y
3
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = −0.215080 + 1.307140I
1.37919 + 2.82812I −4.28809 − 2.59975I
u = 1.00000
a = 0
b = −0.215080 − 1.307140I
1.37919 − 2.82812I −4.28809 + 2.59975I
u = 1.00000
a = 0
b = −0.569840
−2.75839 −16.4240
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 1)
· (u
12
+ 2u
11
+ u
10
− 2u
9
+ u
8
+ 6u
7
+ 4u
6
− 3u
5
+ 6u
3
+ 3u
2
− u − 1)
c
2
(u
3
+ u
2
+ 2u + 1)(u
12
+ 2u
11
+ ··· + 7u + 1)
c
3
(u − 1)
3
· (u
12
− 4u
11
+ 8u
10
− 5u
9
− 5u
8
+ 15u
7
− 9u
6
+ 8u
4
− 2u
3
− 2u
2
+ 4u − 1)
c
4
((u + 1)
3
)(u
12
+ 14u
10
+ ··· + 12u + 1)
c
5
, c
9
u
3
(u
12
+ u
11
+ ··· + 36u + 8)
c
6
(u + 1)
3
· (u
12
− 4u
11
+ 8u
10
− 5u
9
− 5u
8
+ 15u
7
− 9u
6
+ 8u
4
− 2u
3
− 2u
2
+ 4u − 1)
c
7
(u
3
− u
2
+ 2u − 1)(u
12
− 2u
11
+ ··· − 175u − 49)
c
8
(u
3
+ u
2
− 1)
· (u
12
+ 2u
11
+ u
10
− 2u
9
+ u
8
+ 6u
7
+ 4u
6
− 3u
5
+ 6u
3
+ 3u
2
− u − 1)
c
10
(u
3
− u
2
+ 2u − 1)(u
12
+ 2u
11
+ ··· + 7u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y
3
− y
2
+ 2y −1)(y
12
− 2y
11
+ ··· − 7y + 1)
c
2
, c
10
(y
3
+ 3y
2
+ 2y −1)(y
12
+ 18y
11
+ ··· − 7y + 1)
c
3
, c
6
((y −1)
3
)(y
12
+ 14y
10
+ ··· − 12y + 1)
c
4
((y −1)
3
)(y
12
+ 28y
11
+ ··· − 136y + 1)
c
5
, c
9
y
3
(y
12
− 21y
11
+ ··· − 464y + 64)
c
7
(y
3
+ 3y
2
+ 2y −1)(y
12
+ 54y
11
+ ··· − 39739y + 2401)
12
