10
125
(K10n
15
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 9 10 4 5 6 8
Solving Sequence
5,8
9 6
3,10
2 1 4 7
c
8
c
5
c
9
c
2
c
1
c
4
c
7
c
3
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
6
− 5u
4
+ 6u
2
+ b + u − 1, −u
5
− u
4
+ 4u
3
+ 3u
2
+ a − 4u − 3, u
7
+ 2u
6
− 4u
5
− 8u
4
+ 4u
3
+ 9u
2
+ 2u − 1i
I
u
2
= hb, a − u − 1, u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 9 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
6
− 5u
4
+ 6u
2
+ b + u − 1, −u
5
− u
4
+ 4u
3
+ 3u
2
+ a − 4u −
3, u
7
+ 2u
6
− 4u
5
− 8u
4
+ 4u
3
+ 9u
2
+ 2u − 1i
(i) Arc colorings
a
5
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
6
=
u
−u
3
+ u
a
3
=
u
5
+ u
4
− 4u
3
− 3u
2
+ 4u + 3
−u
6
+ 5u
4
− 6u
2
− u + 1
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
u
5
+ u
4
− 4u
3
− 3u
2
+ 4u + 3
u
a
1
=
u
4
− 3u
2
+ 1
u
4
− 2u
2
a
4
=
u
6
+ u
5
− 4u
4
− 4u
3
+ 3u
2
+ 5u + 2
−u
6
+ 5u
4
− 6u
2
− u + 1
a
7
=
u
3
− 2u
−u
5
+ 3u
3
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
6
− 7u
5
+ 15u
4
+ 26u
3
− 13u
2
− 27u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
7
− u
6
+ 11u
5
− 8u
4
+ 13u
3
+ 10u
2
− 7u + 1
c
2
, c
4
u
7
− 3u
6
+ 5u
5
− 2u
4
− u
3
+ 4u
2
− u + 1
c
3
, c
7
u
7
+ u
6
+ 8u
5
+ u
4
+ 13u
3
− 5u
2
+ 4u + 4
c
5
, c
6
, c
8
c
9
u
7
− 2u
6
− 4u
5
+ 8u
4
+ 4u
3
− 9u
2
+ 2u + 1
c
10
u
7
+ 8u
6
+ 8u
5
− 30u
4
+ 102u
3
− 135u
2
+ 78u − 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
7
+ 21y
6
+ 131y
5
+ 228y
4
+ 177y
3
− 266y
2
+ 29y −1
c
2
, c
4
y
7
+ y
6
+ 11y
5
+ 8y
4
+ 13y
3
− 10y
2
− 7y −1
c
3
, c
7
y
7
+ 15y
6
+ 88y
5
+ 225y
4
+ 235y
3
+ 71y
2
+ 56y −16
c
5
, c
6
, c
8
c
9
y
7
− 12y
6
+ 56y
5
− 128y
4
+ 148y
3
− 81y
2
+ 22y −1
c
10
y
7
− 48y
6
+ 748y
5
+ 3048y
4
+ 3664y
3
− 2733y
2
+ 4194y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.689874 + 0.272602I
a = −0.177708 + 0.654657I
b = −0.515013 + 0.602362I
1.33573 − 0.48421I 6.10711 + 1.60895I
u = −0.689874 − 0.272602I
a = −0.177708 − 0.654657I
b = −0.515013 − 0.602362I
1.33573 + 0.48421I 6.10711 − 1.60895I
u = 1.45176 + 0.25511I
a = −0.314310 + 0.755649I
b = 0.25005 + 1.56572I
8.55355 + 2.69234I 5.72785 − 2.29938I
u = 1.45176 − 0.25511I
a = −0.314310 − 0.755649I
b = 0.25005 − 1.56572I
8.55355 − 2.69234I 5.72785 + 2.29938I
u = 0.236235
a = 3.72864
b = 0.444320
−1.26901 −9.72020
u = −1.88000 + 0.08028I
a = 0.627700 + 0.690043I
b = 0.54280 + 2.32525I
−18.3019 − 4.6120I 5.02514 + 1.92936I
u = −1.88000 − 0.08028I
a = 0.627700 − 0.690043I
b = 0.54280 − 2.32525I
−18.3019 + 4.6120I 5.02514 − 1.92936I
5
II. I
u
2
= hb, a − u − 1, u
2
+ u − 1i
(i) Arc colorings
a
5
=
0
u
a
8
=
1
0
a
9
=
1
u − 1
a
6
=
u
−u + 1
a
3
=
u + 1
0
a
10
=
u
−u
a
2
=
u + 1
−u
a
1
=
0
−u
a
4
=
u + 1
0
a
7
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
7
u
2
c
4
(u + 1)
2
c
5
, c
6
u
2
− u − 1
c
8
, c
9
, c
10
u
2
+ u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
2
c
3
, c
7
y
2
c
5
, c
6
, c
8
c
9
, c
10
y
2
− 3y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.61803
b = 0
−0.657974 5.00000
u = −1.61803
a = −0.618034
b = 0
7.23771 5.00000
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
(u
7
− u
6
+ 11u
5
− 8u
4
+ 13u
3
+ 10u
2
− 7u + 1)
c
2
(u − 1)
2
(u
7
− 3u
6
+ 5u
5
− 2u
4
− u
3
+ 4u
2
− u + 1)
c
3
, c
7
u
2
(u
7
+ u
6
+ 8u
5
+ u
4
+ 13u
3
− 5u
2
+ 4u + 4)
c
4
(u + 1)
2
(u
7
− 3u
6
+ 5u
5
− 2u
4
− u
3
+ 4u
2
− u + 1)
c
5
, c
6
(u
2
− u − 1)(u
7
− 2u
6
− 4u
5
+ 8u
4
+ 4u
3
− 9u
2
+ 2u + 1)
c
8
, c
9
(u
2
+ u − 1)(u
7
− 2u
6
− 4u
5
+ 8u
4
+ 4u
3
− 9u
2
+ 2u + 1)
c
10
(u
2
+ u − 1)(u
7
+ 8u
6
+ 8u
5
− 30u
4
+ 102u
3
− 135u
2
+ 78u − 7)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
2
(y
7
+ 21y
6
+ 131y
5
+ 228y
4
+ 177y
3
− 266y
2
+ 29y −1)
c
2
, c
4
(y −1)
2
(y
7
+ y
6
+ 11y
5
+ 8y
4
+ 13y
3
− 10y
2
− 7y −1)
c
3
, c
7
y
2
(y
7
+ 15y
6
+ 88y
5
+ 225y
4
+ 235y
3
+ 71y
2
+ 56y −16)
c
5
, c
6
, c
8
c
9
(y
2
− 3y + 1)(y
7
− 12y
6
+ ··· + 22y −1)
c
10
(y
2
− 3y + 1)
· (y
7
− 48y
6
+ 748y
5
+ 3048y
4
+ 3664y
3
− 2733y
2
+ 4194y −49)
11
