9
30
(K9a
1
)
A knot diagram
1
Linearized knot diagam
5 9 1 7 2 4 6 3 8
Solving Sequence
4,7 1,5
2 3 6 8 9
c
4
c
1
c
3
c
6
c
7
c
9
c
2
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
27
− 4u
26
+ ··· + 2b − 3, −5u
27
+ 14u
26
+ ··· + 2a + 9, u
28
− 3u
27
+ ··· − u + 1i
I
u
2
= hb + a, a
2
− a + 1, u + 1i
* 2 irreducible components of dim
C
= 0, with total 30 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
27
−4u
26
+· · ·+2b−3, −5u
27
+14u
26
+· · ·+2a+9, u
28
−3u
27
+· · ·−u+1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
1
=
5
2
u
27
− 7u
26
+ ··· + 2u −
9
2
−
1
2
u
27
+ 2u
26
+ ··· + 2u +
3
2
a
5
=
1
−u
2
a
2
=
9
2
u
27
− 10u
26
+ ··· + 2u −
11
2
−
7
2
u
27
+ 9u
26
+ ··· + u +
9
2
a
3
=
1
2
u
27
− u
26
+ ··· + u +
3
2
−
1
2
u
27
+ u
26
+ ··· − u +
1
2
a
6
=
−u
u
a
8
=
u
3
−u
3
+ u
a
9
=
4u
27
− 9u
26
+ ··· + 3u − 5
−
5
2
u
27
+ 6u
26
+ ··· + 2u +
7
2
a
9
=
4u
27
− 9u
26
+ ··· + 3u − 5
−
5
2
u
27
+ 6u
26
+ ··· + 2u +
7
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
u
27
−u
26
−3u
25
+6u
24
−5u
22
+13u
21
−24u
20
−3u
19
+81u
18
−69u
17
−92u
16
+190u
15
+6u
14
−
242u
13
+142u
12
+176u
11
−212u
10
−40u
9
+182u
8
−38u
7
−93u
6
+53u
5
+30u
4
−29u
3
−u
2
+9u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
28
+ u
27
+ ··· + 8u + 4
c
2
, c
8
u
28
+ 2u
27
+ ··· + 2u + 1
c
3
u
28
− 2u
27
+ ··· − 22u + 17
c
4
, c
6
u
28
+ 3u
27
+ ··· + u + 1
c
7
u
28
− 13u
27
+ ··· + 7u + 1
c
9
u
28
+ 14u
27
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
28
− 15y
27
+ ··· − 88y + 16
c
2
, c
8
y
28
+ 14y
27
+ ··· + 2y + 1
c
3
y
28
− 10y
27
+ ··· − 246y + 289
c
4
, c
6
y
28
− 13y
27
+ ··· + 7y + 1
c
7
y
28
+ 7y
27
+ ··· − 61y + 1
c
9
y
28
+ 2y
27
+ ··· + 14y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.421904 + 0.904838I
a = −0.038492 − 0.189970I
b = −1.43260 − 0.55257I
−5.32101 − 6.23266I −4.14975 + 4.30079I
u = 0.421904 − 0.904838I
a = −0.038492 + 0.189970I
b = −1.43260 + 0.55257I
−5.32101 + 6.23266I −4.14975 − 4.30079I
u = −0.959758 + 0.402988I
a = −0.766770 + 1.057520I
b = −0.623667 − 0.562813I
1.85217 − 1.40144I 4.69947 + 1.74630I
u = −0.959758 − 0.402988I
a = −0.766770 − 1.057520I
b = −0.623667 + 0.562813I
1.85217 + 1.40144I 4.69947 − 1.74630I
u = 0.619172 + 0.839658I
a = −0.016226 + 0.286921I
b = −1.019470 − 0.068324I
−6.61232 + 2.08114I −5.79595 − 2.78862I
u = 0.619172 − 0.839658I
a = −0.016226 − 0.286921I
b = −1.019470 + 0.068324I
−6.61232 − 2.08114I −5.79595 + 2.78862I
u = 0.963620 + 0.456689I
a = 0.72093 + 1.60659I
b = −0.015157 − 1.395580I
1.56772 + 4.24816I 1.88645 − 6.97904I
u = 0.963620 − 0.456689I
a = 0.72093 − 1.60659I
b = −0.015157 + 1.395580I
1.56772 − 4.24816I 1.88645 + 6.97904I
u = 0.855481 + 0.371946I
a = −1.18245 − 1.31391I
b = 0.66840 + 1.28739I
0.967687 − 0.906276I −0.59768 − 1.67094I
u = 0.855481 − 0.371946I
a = −1.18245 + 1.31391I
b = 0.66840 − 1.28739I
0.967687 + 0.906276I −0.59768 + 1.67094I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.454354 + 0.784849I
a = −0.204179 + 0.058390I
b = 1.105360 + 0.510425I
−2.52313 − 1.47542I −1.29345 + 0.59666I
u = 0.454354 − 0.784849I
a = −0.204179 − 0.058390I
b = 1.105360 − 0.510425I
−2.52313 + 1.47542I −1.29345 − 0.59666I
u = −0.962167 + 0.550809I
a = 0.83827 − 1.25040I
b = 1.191130 + 0.619206I
−0.41268 − 5.75423I 0.10698 + 5.96655I
u = −0.962167 − 0.550809I
a = 0.83827 + 1.25040I
b = 1.191130 − 0.619206I
−0.41268 + 5.75423I 0.10698 − 5.96655I
u = −1.126550 + 0.202617I
a = −0.903208 + 0.571058I
b = 0.236722 − 0.655524I
2.40233 − 0.64414I 4.35398 − 1.30683I
u = −1.126550 − 0.202617I
a = −0.903208 − 0.571058I
b = 0.236722 + 0.655524I
2.40233 + 0.64414I 4.35398 + 1.30683I
u = −0.668097 + 0.525777I
a = 0.55770 − 1.31624I
b = 0.847077 − 0.345927I
−1.32210 + 1.34593I −1.91932 − 0.66126I
u = −0.668097 − 0.525777I
a = 0.55770 + 1.31624I
b = 0.847077 + 0.345927I
−1.32210 − 1.34593I −1.91932 + 0.66126I
u = 1.021030 + 0.695890I
a = 0.04209 − 1.42194I
b = −0.763781 + 0.287418I
−5.39487 + 3.62399I −4.20871 − 2.76186I
u = 1.021030 − 0.695890I
a = 0.04209 + 1.42194I
b = −0.763781 − 0.287418I
−5.39487 − 3.62399I −4.20871 + 2.76186I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.099170 + 0.618751I
a = 0.03996 + 1.72641I
b = 1.13985 − 0.88919I
−0.59978 + 6.77427I 1.77406 − 4.95962I
u = 1.099170 − 0.618751I
a = 0.03996 − 1.72641I
b = 1.13985 + 0.88919I
−0.59978 − 6.77427I 1.77406 + 4.95962I
u = −1.278740 + 0.117832I
a = 1.225830 − 0.293847I
b = −0.991759 + 0.593054I
0.65193 + 3.28147I −1.23266 − 4.99392I
u = −1.278740 − 0.117832I
a = 1.225830 + 0.293847I
b = −0.991759 − 0.593054I
0.65193 − 3.28147I −1.23266 + 4.99392I
u = 1.146350 + 0.652255I
a = 0.11235 − 1.78840I
b = −1.53314 + 0.75996I
−3.12706 + 11.95450I −1.04116 − 8.32221I
u = 1.146350 − 0.652255I
a = 0.11235 + 1.78840I
b = −1.53314 − 0.75996I
−3.12706 − 11.95450I −1.04116 + 8.32221I
u = −0.085781 + 0.348606I
a = −0.92580 + 1.34078I
b = 0.191038 + 0.606129I
−0.22315 − 1.43304I −1.58225 + 4.97603I
u = −0.085781 − 0.348606I
a = −0.92580 − 1.34078I
b = 0.191038 − 0.606129I
−0.22315 + 1.43304I −1.58225 − 4.97603I
7
II. I
u
2
= hb + a, a
2
− a + 1, u + 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
−1
a
1
=
a
−a
a
5
=
1
−1
a
2
=
a
−a
a
3
=
a
−a + 1
a
6
=
1
−1
a
8
=
−1
0
a
9
=
0
−a
a
9
=
0
−a
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a + 1
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
2
c
2
u
2
− u + 1
c
3
, c
8
, c
9
u
2
+ u + 1
c
4
(u + 1)
2
c
6
, c
7
(u − 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
2
c
2
, c
3
, c
8
c
9
y
2
+ y + 1
c
4
, c
6
, c
7
(y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.500000 + 0.866025I
b = −0.500000 − 0.866025I
1.64493 − 2.02988I 3.00000 + 3.46410I
u = −1.00000
a = 0.500000 − 0.866025I
b = −0.500000 + 0.866025I
1.64493 + 2.02988I 3.00000 − 3.46410I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
2
(u
28
+ u
27
+ ··· + 8u + 4)
c
2
(u
2
− u + 1)(u
28
+ 2u
27
+ ··· + 2u + 1)
c
3
(u
2
+ u + 1)(u
28
− 2u
27
+ ··· − 22u + 17)
c
4
((u + 1)
2
)(u
28
+ 3u
27
+ ··· + u + 1)
c
6
((u − 1)
2
)(u
28
+ 3u
27
+ ··· + u + 1)
c
7
((u − 1)
2
)(u
28
− 13u
27
+ ··· + 7u + 1)
c
8
(u
2
+ u + 1)(u
28
+ 2u
27
+ ··· + 2u + 1)
c
9
(u
2
+ u + 1)(u
28
+ 14u
27
+ ··· + 2u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
2
(y
28
− 15y
27
+ ··· − 88y + 16)
c
2
, c
8
(y
2
+ y + 1)(y
28
+ 14y
27
+ ··· + 2y + 1)
c
3
(y
2
+ y + 1)(y
28
− 10y
27
+ ··· − 246y + 289)
c
4
, c
6
((y −1)
2
)(y
28
− 13y
27
+ ··· + 7y + 1)
c
7
((y −1)
2
)(y
28
+ 7y
27
+ ··· − 61y + 1)
c
9
(y
2
+ y + 1)(y
28
+ 2y
27
+ ··· + 14y + 1)
13
