10
128
(K10n
22
)
A knot diagram
1
Linearized knot diagam
5 6 7 10 3 2 10 7 4 8
Solving Sequence
7,10
8
1,5
2 4 3 6 9
c
7
c
10
c
1
c
4
c
3
c
6
c
9
c
2
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
7
− 3u
6
+ 17u
5
+ 4u
4
− 31u
3
+ 12u
2
+ 4b − u + 1, −u
7
− 3u
6
+ u
5
+ 8u
4
+ 3u
3
− 6u
2
+ 2a − 9u − 1,
u
8
+ 4u
7
− 13u
5
− 3u
4
+ 15u
3
+ 3u
2
+ 2u − 1i
I
u
2
= hb
3
− b
2
+ 1, a, u − 1i
* 2 irreducible components of dim
C
= 0, with total 11 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−3u
7
− 3u
6
+ · · · + 4b + 1, −u
7
− 3u
6
+ · · · + 2a − 1, u
8
+ 4u
7
−
13u
5
− 3u
4
+ 15u
3
+ 3u
2
+ 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
8
=
1
u
2
a
1
=
−u
−u
3
+ u
a
5
=
1
2
u
7
+
3
2
u
6
+ ··· +
9
2
u +
1
2
3
4
u
7
+
3
4
u
6
+ ··· +
1
4
u −
1
4
a
2
=
−1
−
1
4
u
7
−
3
4
u
6
+ ··· +
5
4
u +
1
4
a
4
=
1
2
u
7
+
3
2
u
6
+ ··· +
9
2
u +
1
2
−
3
4
u
7
−
7
4
u
6
+ ··· −
5
4
u +
1
4
a
3
=
−
1
4
u
7
−
1
4
u
6
+ ··· +
13
4
u +
3
4
−
3
4
u
7
−
7
4
u
6
+ ··· −
5
4
u +
1
4
a
6
=
−
1
4
u
7
−
3
4
u
6
+ ··· +
5
4
u +
5
4
−
1
2
u
6
− u
5
+ ··· − 2u +
1
2
a
9
=
−u
2
+ 1
u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
7
−
11
2
u
6
− 3u
5
+
39
2
u
4
+
23
2
u
3
− 27u
2
− 6u −
23
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
8
+ 2u
7
− 7u
6
− 12u
5
+ 7u
4
+ 2u
3
− 2u
2
− 3u − 1
c
2
, c
5
, c
6
u
8
− 2u
7
+ 5u
6
− 6u
5
+ 7u
4
− 6u
3
+ 2u
2
− u − 1
c
4
, c
9
u
8
− u
7
− 10u
6
+ 7u
5
+ 19u
4
+ 23u
3
+ 12u + 8
c
7
, c
10
u
8
− 4u
7
+ 13u
5
− 3u
4
− 15u
3
+ 3u
2
− 2u − 1
c
8
u
8
+ 16u
7
+ 98u
6
+ 283u
5
+ 381u
4
+ 191u
3
− 45u
2
+ 10u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
y
8
− 18y
7
+ 111y
6
− 254y
5
+ 135y
4
− 90y
3
+ 2y
2
− 5y + 1
c
2
, c
5
, c
6
y
8
+ 6y
7
+ 15y
6
+ 14y
5
− 9y
4
− 30y
3
− 22y
2
− 5y + 1
c
4
, c
9
y
8
− 21y
7
+ 152y
6
− 383y
5
+ 79y
4
− 857y
3
− 248y
2
− 144y + 64
c
7
, c
10
y
8
− 16y
7
+ 98y
6
− 283y
5
+ 381y
4
− 191y
3
− 45y
2
− 10y + 1
c
8
y
8
− 60y
7
+ ··· − 190y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.251300 + 0.394571I
a = 0.381129 − 0.818334I
b = 1.066800 − 0.340674I
−1.14011 − 1.32248I −11.15537 + 1.48485I
u = 1.251300 − 0.394571I
a = 0.381129 + 0.818334I
b = 1.066800 + 0.340674I
−1.14011 + 1.32248I −11.15537 − 1.48485I
u = −0.202560 + 0.429200I
a = −0.93266 + 1.25163I
b = 1.031990 + 0.436432I
2.74105 − 2.12062I −5.41411 + 2.85603I
u = −0.202560 − 0.429200I
a = −0.93266 − 1.25163I
b = 1.031990 − 0.436432I
2.74105 + 2.12062I −5.41411 − 2.85603I
u = 0.266855
a = 1.86561
b = −0.260126
−0.675825 −14.7130
u = −2.08865 + 0.23775I
a = 1.276340 − 0.114214I
b = 2.96514 − 1.78943I
−14.2177 + 5.8605I −11.51154 − 2.72065I
u = −2.08865 − 0.23775I
a = 1.276340 + 0.114214I
b = 2.96514 + 1.78943I
−14.2177 − 5.8605I −11.51154 + 2.72065I
u = −2.18705
a = −1.31522
b = −3.86773
−18.5039 −14.1250
5
II. I
u
2
= hb
3
− b
2
+ 1, a, u − 1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
1
a
8
=
1
1
a
1
=
−1
0
a
5
=
0
b
a
2
=
−1
−b
2
a
4
=
0
b
a
3
=
b
b
a
6
=
−b
2
+ 1
−b
2
+ b + 1
a
9
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = b
2
+ 3b − 13
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
3
+ u
2
− 1
c
2
u
3
− u
2
+ 2u − 1
c
4
, c
9
u
3
c
5
, c
6
u
3
+ u
2
+ 2u + 1
c
7
(u − 1)
3
c
8
, c
10
(u + 1)
3
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
y
3
− y
2
+ 2y −1
c
2
, c
5
, c
6
y
3
+ 3y
2
+ 2y −1
c
4
, c
9
y
3
c
7
, c
8
, c
10
(y −1)
3
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = 0.877439 + 0.744862I
1.37919 − 2.82812I −10.15260 + 3.54173I
u = 1.00000
a = 0
b = 0.877439 − 0.744862I
1.37919 + 2.82812I −10.15260 − 3.54173I
u = 1.00000
a = 0
b = −0.754878
−2.75839 −14.6950
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
+ u
2
− 1)(u
8
+ 2u
7
− 7u
6
− 12u
5
+ 7u
4
+ 2u
3
− 2u
2
− 3u − 1)
c
2
(u
3
− u
2
+ 2u − 1)(u
8
− 2u
7
+ 5u
6
− 6u
5
+ 7u
4
− 6u
3
+ 2u
2
− u − 1)
c
4
, c
9
u
3
(u
8
− u
7
− 10u
6
+ 7u
5
+ 19u
4
+ 23u
3
+ 12u + 8)
c
5
, c
6
(u
3
+ u
2
+ 2u + 1)(u
8
− 2u
7
+ 5u
6
− 6u
5
+ 7u
4
− 6u
3
+ 2u
2
− u − 1)
c
7
(u − 1)
3
(u
8
− 4u
7
+ 13u
5
− 3u
4
− 15u
3
+ 3u
2
− 2u − 1)
c
8
(u + 1)
3
· (u
8
+ 16u
7
+ 98u
6
+ 283u
5
+ 381u
4
+ 191u
3
− 45u
2
+ 10u + 1)
c
10
(u + 1)
3
(u
8
− 4u
7
+ 13u
5
− 3u
4
− 15u
3
+ 3u
2
− 2u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
(y
3
− y
2
+ 2y −1)
· (y
8
− 18y
7
+ 111y
6
− 254y
5
+ 135y
4
− 90y
3
+ 2y
2
− 5y + 1)
c
2
, c
5
, c
6
(y
3
+ 3y
2
+ 2y −1)
· (y
8
+ 6y
7
+ 15y
6
+ 14y
5
− 9y
4
− 30y
3
− 22y
2
− 5y + 1)
c
4
, c
9
y
3
(y
8
− 21y
7
+ ··· − 144y + 64)
c
7
, c
10
(y −1)
3
· (y
8
− 16y
7
+ 98y
6
− 283y
5
+ 381y
4
− 191y
3
− 45y
2
− 10y + 1)
c
8
((y −1)
3
)(y
8
− 60y
7
+ ··· − 190y + 1)
11
