10
126
(K10n
17
)
A knot diagram
1
Linearized knot diagam
4 8 5 2 9 10 5 2 6 7
Solving Sequence
5,9
6
2,10
4 1 3 8 7
c
5
c
9
c
4
c
1
c
3
c
8
c
7
c
2
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
10
− u
9
+ 5u
8
+ 3u
7
− 9u
6
+ u
5
+ 8u
4
− 6u
3
− 3u
2
+ b + u,
u
10
+ u
9
− 4u
8
− 3u
7
+ 4u
6
− u
5
− u
4
+ 4u
3
+ u
2
+ a + 3u + 1,
u
11
+ 2u
10
− 4u
9
− 8u
8
+ 6u
7
+ 7u
6
− 10u
5
+ u
4
+ 11u
3
+ 1i
I
u
2
= hb + 1, a, u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 13 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−u
10
−u
9
+· · ·+b + u, u
10
+u
9
+· · ·+a + 1, u
11
+2u
10
+· · ·+11u
3
+1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
6
=
1
−u
2
a
2
=
−u
10
− u
9
+ 4u
8
+ 3u
7
− 4u
6
+ u
5
+ u
4
− 4u
3
− u
2
− 3u − 1
u
10
+ u
9
− 5u
8
− 3u
7
+ 9u
6
− u
5
− 8u
4
+ 6u
3
+ 3u
2
− u
a
10
=
u
−u
3
+ u
a
4
=
u
3
− 2u
u
10
+ u
9
− 4u
8
− 3u
7
+ 5u
6
− u
5
− 4u
4
+ 5u
3
+ 3u
2
a
1
=
u
3
− 2u
−u
5
+ 3u
3
− u
a
3
=
u
10
+ u
9
− 4u
8
− 3u
7
+ 5u
6
− u
5
− 4u
4
+ 6u
3
+ 3u
2
− 2u
u
10
+ u
9
− 4u
8
− 3u
7
+ 5u
6
− u
5
− 4u
4
+ 5u
3
+ 3u
2
a
8
=
u
4
− 3u
2
+ 1
u
4
− 2u
2
a
7
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
9
− u
8
− 6u
7
+ 7u
6
+ 11u
5
− 17u
4
+ 2u
3
+ 16u
2
− 15u + 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
11
− 3u
10
+ ··· − 7u + 1
c
2
, c
8
u
11
− u
10
+ ··· − 4u + 4
c
3
u
11
+ 15u
10
+ ··· + 51u + 1
c
5
, c
6
, c
9
c
10
u
11
− 2u
10
− 4u
9
+ 8u
8
+ 6u
7
− 7u
6
− 10u
5
− u
4
+ 11u
3
− 1
c
7
u
11
+ 12u
9
+ 2u
8
+ 32u
7
+ 17u
6
− 28u
5
+ 27u
4
− 15u
3
+ 2u
2
− 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
11
− 15y
10
+ ··· + 51y −1
c
2
, c
8
y
11
+ 15y
10
+ ··· + 88y −16
c
3
y
11
− 35y
10
+ ··· + 1959y −1
c
5
, c
6
, c
9
c
10
y
11
− 12y
10
+ ··· − 2y
2
− 1
c
7
y
11
+ 24y
10
+ ··· + 2y
2
− 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.555784 + 0.826080I
a = 1.70442 + 0.91227I
b = 1.75765 − 0.08981I
−11.41260 + 2.72618I 1.17921 − 2.48457I
u = 0.555784 − 0.826080I
a = 1.70442 − 0.91227I
b = 1.75765 + 0.08981I
−11.41260 − 2.72618I 1.17921 + 2.48457I
u = 1.30287
a = −0.964097
b = −1.44606
1.42853 5.86840
u = −1.395180 + 0.126727I
a = −0.158907 + 0.922695I
b = −0.665578 − 0.815452I
3.45898 − 2.75386I 6.03924 + 3.05522I
u = −1.395180 − 0.126727I
a = −0.158907 − 0.922695I
b = −0.665578 + 0.815452I
3.45898 + 2.75386I 6.03924 − 3.05522I
u = −0.509387
a = 0.753099
b = 0.150577
0.764590 13.1750
u = 0.205266 + 0.391152I
a = −1.19521 − 1.33382I
b = −0.887105 + 0.326749I
−1.67531 + 0.87131I −1.62556 − 2.85981I
u = 0.205266 − 0.391152I
a = −1.19521 + 1.33382I
b = −0.887105 − 0.326749I
−1.67531 − 0.87131I −1.62556 + 2.85981I
u = 1.58287
a = 0.388562
b = 0.514377
8.06663 13.5530
u = −1.55405 + 0.30396I
a = 0.560911 − 1.017150I
b = 1.68559 + 0.26432I
−4.54812 − 6.90426I 4.10911 + 3.24808I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.55405 − 0.30396I
a = 0.560911 + 1.017150I
b = 1.68559 − 0.26432I
−4.54812 + 6.90426I 4.10911 − 3.24808I
6
II. I
u
2
= hb + 1, a, u
2
− u − 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
6
=
1
−u − 1
a
2
=
0
−1
a
10
=
u
−u − 1
a
4
=
1
−1
a
1
=
−1
0
a
3
=
0
−1
a
8
=
0
u
a
7
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u − 1)
2
c
2
, c
8
u
2
c
4
(u + 1)
2
c
5
, c
6
u
2
− u − 1
c
7
, c
9
, c
10
u
2
+ u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
(y −1)
2
c
2
, c
8
y
2
c
5
, c
6
, c
7
c
9
, c
10
y
2
− 3y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0
b = −1.00000
−0.657974 3.00000
u = 1.61803
a = 0
b = −1.00000
7.23771 3.00000
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
2
)(u
11
− 3u
10
+ ··· − 7u + 1)
c
2
, c
8
u
2
(u
11
− u
10
+ ··· − 4u + 4)
c
3
((u − 1)
2
)(u
11
+ 15u
10
+ ··· + 51u + 1)
c
4
((u + 1)
2
)(u
11
− 3u
10
+ ··· − 7u + 1)
c
5
, c
6
(u
2
− u − 1)(u
11
− 2u
10
+ ··· + 11u
3
− 1)
c
7
(u
2
+ u − 1)
· (u
11
+ 12u
9
+ 2u
8
+ 32u
7
+ 17u
6
− 28u
5
+ 27u
4
− 15u
3
+ 2u
2
− 2u + 1)
c
9
, c
10
(u
2
+ u − 1)(u
11
− 2u
10
+ ··· + 11u
3
− 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
2
)(y
11
− 15y
10
+ ··· + 51y −1)
c
2
, c
8
y
2
(y
11
+ 15y
10
+ ··· + 88y −16)
c
3
((y −1)
2
)(y
11
− 35y
10
+ ··· + 1959y −1)
c
5
, c
6
, c
9
c
10
(y
2
− 3y + 1)(y
11
− 12y
10
+ ··· − 2y
2
− 1)
c
7
(y
2
− 3y + 1)(y
11
+ 24y
10
+ ··· + 2y
2
− 1)
12
