10
145
(K10n
14
)
A knot diagram
1
Linearized knot diagam
8 10 8 3 9 2 4 1 3 6
Solving Sequence
3,8
4
5,10
2 1 7 6 9
c
3
c
4
c
2
c
1
c
7
c
6
c
9
c
5
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h3u
4
− 9u
3
+ 31u
2
+ 118b − 54u + 26, 17u
4
+ 8u
3
+ 215u
2
+ 236a − 70u + 167,
u
5
+ 2u
4
+ 15u
3
+ 14u
2
+ 17u + 4i
I
u
2
= hb − a + u + 1, a
2
− 2au − 2a + 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
= h3u
4
− 9u
3
+ 31u
2
+ 118b − 54u + 26, 17u
4
+ 8u
3
+ 215u
2
+ 236a −
70u + 167, u
5
+ 2u
4
+ 15u
3
+ 14u
2
+ 17u + 4i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
5
=
u
2
+ 1
−u
2
a
10
=
−0.0720339u
4
− 0.0338983u
3
+ ··· + 0.296610u − 0.707627
−0.0254237u
4
+ 0.0762712u
3
+ ··· + 0.457627u − 0.220339
a
2
=
0.0550847u
4
+ 0.0847458u
3
+ ··· + 0.508475u + 1.39407
−0.0169492u
4
+ 0.0508475u
3
+ ··· + 0.305085u + 0.186441
a
1
=
0.0550847u
4
+ 0.0847458u
3
+ ··· + 0.508475u + 1.39407
0.110169u
4
+ 0.169492u
3
+ ··· + 0.516949u + 0.288136
a
7
=
−u
u
3
+ u
a
6
=
0.0720339u
4
+ 0.0338983u
3
+ ··· − 0.296610u + 0.707627
−0.110169u
4
− 0.169492u
3
+ ··· − 0.516949u − 0.288136
a
9
=
−0.0466102u
4
− 0.110169u
3
+ ··· − 0.161017u − 0.487288
−0.0254237u
4
+ 0.0762712u
3
+ ··· + 0.457627u − 0.220339
(ii) Obstruction class = −1
(iii) Cusp Shapes =
33
59
u
4
+
78
59
u
3
+
518
59
u
2
+
645
59
u +
994
59
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
8
c
9
u
5
+ u
4
+ 8u
3
− 4u
2
+ 3u − 1
c
3
, c
7
u
5
− 2u
4
+ 15u
3
− 14u
2
+ 17u − 4
c
4
u
5
+ 26u
4
+ 203u
3
+ 298u
2
+ 177u − 16
c
5
, c
6
u
5
+ u
4
+ 26u
3
+ 18u
2
− 4u − 4
c
10
u
5
+ 4u
4
+ 7u
3
+ 4u
2
− u − 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
8
c
9
y
5
+ 15y
4
+ 78y
3
+ 34y
2
+ y −1
c
3
, c
7
y
5
+ 26y
4
+ 203y
3
+ 298y
2
+ 177y −16
c
4
y
5
− 270y
4
+ 26067y
3
− 16110y
2
+ 40865y −256
c
5
, c
6
y
5
+ 51y
4
+ 632y
3
− 524y
2
+ 160y −16
c
10
y
5
− 2y
4
+ 15y
3
− 14y
2
+ 17y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.345349 + 1.000390I
a = −0.062320 + 0.860264I
b = −0.078472 + 0.559300I
−1.59932 − 2.36167I 6.81651 + 4.70099I
u = −0.345349 − 1.000390I
a = −0.062320 − 0.860264I
b = −0.078472 − 0.559300I
−1.59932 + 2.36167I 6.81651 − 4.70099I
u = −0.281507
a = −0.863015
b = −0.371844
0.702837 14.4400
u = −0.51390 + 3.52451I
a = −0.131172 − 0.610069I
b = 0.76439 − 2.80121I
15.2298 − 5.0449I 4.96361 + 1.80446I
u = −0.51390 − 3.52451I
a = −0.131172 + 0.610069I
b = 0.76439 + 2.80121I
15.2298 + 5.0449I 4.96361 − 1.80446I
5
II. I
u
2
= hb − a + u + 1, a
2
− 2au − 2a + u + 1, u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u + 1
a
5
=
−u
u + 1
a
10
=
a
a − u − 1
a
2
=
au + a − u
−1
a
1
=
au + a − u
−au − 2
a
7
=
−u
u + 1
a
6
=
−a
au + 2
a
9
=
u + 1
a − u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 4
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
8
c
9
(u
2
+ 1)
2
c
3
, c
4
(u
2
+ u + 1)
2
c
5
u
4
− 2u
3
+ 2u
2
− 4u + 4
c
6
u
4
+ 2u
3
+ 2u
2
+ 4u + 4
c
7
(u
2
− u + 1)
2
c
10
u
4
− u
2
+ 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
8
c
9
(y + 1)
4
c
3
, c
4
, c
7
(y
2
+ y + 1)
2
c
5
, c
6
y
4
− 4y
2
+ 16
c
10
(y
2
− y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.500000 − 0.133975I
b = − 1.000000I
−3.28987 − 2.02988I 2.00000 + 3.46410I
u = −0.500000 + 0.866025I
a = 0.50000 + 1.86603I
b = 1.000000I
−3.28987 − 2.02988I 2.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 0.500000 + 0.133975I
b = 1.000000I
−3.28987 + 2.02988I 2.00000 − 3.46410I
u = −0.500000 − 0.866025I
a = 0.50000 − 1.86603I
b = − 1.000000I
−3.28987 + 2.02988I 2.00000 − 3.46410I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
8
c
9
(u
2
+ 1)
2
(u
5
+ u
4
+ 8u
3
− 4u
2
+ 3u − 1)
c
3
(u
2
+ u + 1)
2
(u
5
− 2u
4
+ 15u
3
− 14u
2
+ 17u − 4)
c
4
(u
2
+ u + 1)
2
(u
5
+ 26u
4
+ 203u
3
+ 298u
2
+ 177u − 16)
c
5
(u
4
− 2u
3
+ 2u
2
− 4u + 4)(u
5
+ u
4
+ 26u
3
+ 18u
2
− 4u − 4)
c
6
(u
4
+ 2u
3
+ 2u
2
+ 4u + 4)(u
5
+ u
4
+ 26u
3
+ 18u
2
− 4u − 4)
c
7
(u
2
− u + 1)
2
(u
5
− 2u
4
+ 15u
3
− 14u
2
+ 17u − 4)
c
10
(u
4
− u
2
+ 1)(u
5
+ 4u
4
+ 7u
3
+ 4u
2
− u − 2)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
8
c
9
(y + 1)
4
(y
5
+ 15y
4
+ 78y
3
+ 34y
2
+ y −1)
c
3
, c
7
(y
2
+ y + 1)
2
(y
5
+ 26y
4
+ 203y
3
+ 298y
2
+ 177y −16)
c
4
(y
2
+ y + 1)
2
(y
5
− 270y
4
+ 26067y
3
− 16110y
2
+ 40865y −256)
c
5
, c
6
(y
4
− 4y
2
+ 16)(y
5
+ 51y
4
+ 632y
3
− 524y
2
+ 160y −16)
c
10
(y
2
− y + 1)
2
(y
5
− 2y
4
+ 15y
3
− 14y
2
+ 17y −4)
11
