10
123
(K10a
121
)
1
Arc Sequences
8 9 10 1 2 3 4 5 6 7
Solving Sequence
1,8 2,4
5 9 7 10 3 6
c
1
c
4
c
8
c
7
c
10
c
3
c
6
c
2
, c
5
, c
9
Representation Ideals
I =
8
\
i=1
I
u
i
1
The knot diagram image is adapter from “C. Livingston and A. H. Mo ore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I
u
1
= hu
2
a + bu u + 1, bau ba au b + 1, b
2
u + ba bu + au + 2b 2,
b
2
a + a
2
b + a
2
u + b
2
+ 2ba b 2ai
I
u
2
= h−u
3
+ a + 2, u
3
+ b 1, u
4
+ u
3
2u 1i
I
u
3
= hu
3
2u
2
+ a + 2u, u
4
3u
3
+ 4u
2
2u + 1, u
3
+ 2u
2
+ b 2u + 1i
I
u
4
= h3u
10
+ 6u
9
u
8
14u
7
8u
6
+ 10u
5
+ 11u
4
2u
3
5u
2
+ 1,
18u
8
+ 30u
7
9u
6
67u
5
26u
4
+ 41u
3
+ 35u
2
+ 2a 7u 8,
6u
9
+ 3u
8
17u
7
22u
6
+ 19u
5
+ 31u
4
5u
3
22u
2
+ 2b 2u + 6i
I
u
5
= h3u
10
+ 6u
9
u
8
14u
7
8u
6
+ 10u
5
+ 11u
4
2u
3
5u
2
+ 1,
18u
9
30u
8
+ 9u
7
+ 67u
6
+ 26u
5
41u
4
35u
3
+ 7u
2
+ 2b + 8u,
18u
9
+ 30u
8
9u
7
67u
6
26u
5
+ 41u
4
+ 35u
3
7u
2
+ 2a 8u + 2i
I
u
6
= hu
10
7u
9
+ 28u
8
77u
7
+ 159u
6
251u
5
+ 308u
4
288u
3
+ 200u
2
96u + 32,
31u
9
+ 183u
8
622u
7
+ 1411u
6
2327u
5
+ 2703u
4
2174u
3
+ 880u
2
+ 288a + 56u 256,
17u
9
123u
8
+ 488u
7
1301u
6
+ 2539u
5
3687u
4
+ 4024u
3
3128u
2
+ 144b + 1616u 496i
I
u
7
= hu
10
+ 4u
9
+ 9u
8
+ 14u
7
+ 20u
6
+ 26u
5
+ 31u
4
+ 30u
3
+ 23u
2
+ 12u + 3,
10u
9
35u
8
70u
7
98u
6
139u
5
175u
4
201u
3
175u
2
+ 6a 116u 41,
2u
9
+ 7u
8
+ 14u
7
+ 19u
6
+ 27u
5
+ 34u
4
+ 40u
3
+ 33u
2
+ 2b + 22u + 6i
I
u
8
= hu
10
+ 4u
9
+ 9u
8
+ 14u
7
+ 20u
6
+ 26u
5
+ 31u
4
+ 30u
3
+ 23u
2
+ 12u + 3,
2u
9
+ 6u
8
+ 11u
7
+ 14u
6
+ 21u
5
+ 25u
4
+ 28u
3
+ 20u
2
+ 2a + 13u + 2,
5u
9
20u
8
42u
7
61u
6
85u
5
109u
4
125u
3
114u
2
+ 6b 79u 30i
There are 8 irreducible components with 58 representations.
There are 1 irreducible components of dim
C
= 1 for 10
123
2
I. I
u
1
= hu
2
a + bu u + 1, bau ba au b + 1, b
2
u + ba bu + au + 2b
2, b
2
a + a
2
u + · · · b 2ai
(i) Arc colorings
a
1
=
1
0
a
8
=
a
b
a
2
=
ba + 1
b
2
a
4
=
0
u
a
5
=
u
u
a
9
=
u
2
b + bu + a u + 1
u
2
b bu + b + u 1
a
7
=
a
bu + b + u 1
a
10
=
b + a
b
2
u + u
2
b b
2
2bu u
2
+ b + u 1
a
3
=
b
2
a + b
2
u + a
2
b + b
2
+ 2ba 2bu + b 2a 2
b
2
ba + bu b + 2
a
6
=
b
2
+ ba bu 1
b
3
b
2
a + b
2
u b
2
bu + 3b + u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
3
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
4
II. I
u
2
= h−u
3
+ a + 2, u
3
+ b 1, u
4
+ u
3
2u 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
u
3
2
u
3
+ 1
a
2
=
2u
3
u
2
+ u + 4
u
3
+ u
2
u 2
a
4
=
0
u
a
5
=
u
u
a
9
=
u
3
+ u
2
2
u
3
u
2
+ 1
a
7
=
u
3
2
u
a
10
=
u
3
+ 2
u
2
a
3
=
2u
3
u
2
+ u + 3
u
3
+ u
2
1
a
6
=
2u
3
+ u
2
u 4
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
3
5u + 5
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.809017 0.981593I
a = 0.190983 0.981593I
b = 0.809017 + 0.981593I
9.37207I 9.81593I
u = 0.809017 + 0.981593I
a = 0.190983 + 0.981593I
b = 0.809017 0.981593I
9.37207I 9.81593I
u = 0.535687
a = 2.15372
b = 1.15372
4.48216 8.44704
u = 1.15372
a = 0.464313
b = 0.535687
4.48216 8.44704
6
III.
I
u
3
= hu
3
2u
2
+ a + 2u, u
4
3u
3
+ 4u
2
2u + 1, u
3
+ 2u
2
+ b 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
u
3
+ 2u
2
2u
u
3
2u
2
+ 2u 1
a
2
=
u
2
+ u
u
3
u
2
+ u
a
4
=
0
u
a
5
=
u
u
a
9
=
u
3
+ 3u
2
2u
u
3
3u
2
+ 2u 1
a
7
=
u
3
+ 2u
2
2u
u
a
10
=
u
3
2u
2
+ 2u
u
2
a
3
=
u
2
u + 1
u
3
3u
2
+ 2u 1
a
6
=
u
2
u
2u
2
+ 2u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
3
10u
2
+ 15u 5
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.190983 0.587785I
a = 0.809017 + 0.587785I
b = 0.190983 0.587785I
1.38939I 5.87785I
u = 0.190983 + 0.587785I
a = 0.809017 0.587785I
b = 0.190983 + 0.587785I
1.38939I 5.87785I
u = 1.30902 0.95106I
a = 0.309017 + 0.951057I
b = 1.30902 0.95106I
17.0857I 9.51057I
u = 1.30902 + 0.95106I
a = 0.309017 0.951057I
b = 1.30902 + 0.95106I
17.0857I 9.51057I
8
IV.
I
u
4
= h3u
10
+6u
9
+· · ·5u
2
+1, 18u
8
+30u
7
+· · ·+2a8, 6u
9
+3u
8
+· · ·+2b+6 i
(i) Arc colorings
a
1
=
1
0
a
8
=
9u
8
15u
7
+ ··· +
7
2
u + 4
3u
9
3
2
u
8
+ ··· + u 3
a
2
=
3
2
u
9
3
2
u
8
+ ··· u 3
9
2
u
9
6u
8
+ ··· +
5
2
u
1
2
a
4
=
0
u
a
5
=
u
u
a
9
=
3
2
u
9
9u
8
+ ··· +
5
2
u +
5
2
3
2
u
9
3
2
u
8
+ ··· + 2u
3
2
a
7
=
9u
8
15u
7
+ ··· +
7
2
u + 4
u
a
10
=
9u
9
+ 15u
8
+ ··· 4u + 1
u
2
a
3
=
3
2
u
9
3
2
u
8
+ ··· + 3u +
5
2
3
2
u
9
3
2
u
8
+ ··· + 2u
3
2
a
6
=
6u
9
21
2
u
8
+ ··· +
11
2
u
3
2
3
2
u
9
+
7
2
u
7
+ ···
3
2
u
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 12u
9
+ 6u
8
+ 64u
7
+ 52u
6
90u
5
110u
4
+ 30u
3
+ 86u
2
+ 4u 24
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.081495 0.798609I
a = 0.308366 + 1.226789I
b = 1.20165 0.91842I
2.49243 8.64801I 4.04126 + 7.50135I
u = 1.081495 + 0.798609I
a = 0.308366 1.226789I
b = 1.20165 + 0.91842I
2.49243 + 8.64801I 4.04126 7.50135I
u = 0.744942 0.201707I
a = 1.09410 1.78861I
b = 0.719811 + 1.046886I
2.49243 8.64801I 4.04126 + 7.50135I
u = 0.744942 + 0.201707I
a = 1.09410 + 1.78861I
b = 0.719811 1.046886I
2.49243 + 8.64801I 4.04126 7.50135I
u = 0.707358 0.648629I
a = 0.57894 + 1.46671I
b = 0.684636 0.234182I
3.61397 2.21654I 5.38699 + 4.72022I
u = 0.707358 + 0.648629I
a = 0.57894 1.46671I
b = 0.684636 + 0.234182I
3.61397 + 2.21654I 5.38699 4.72022I
u = 0.550514 0.187402I
a = 0.50999 + 2.80284I
b = 0.108840 1.043638I
0.806279I 8.22652I
u = 0.550514 + 0.187402I
a = 0.50999 2.80284I
b = 0.108840 + 1.043638I
0.806279I 8.22652I
u = 0.983280 0.164908I
a = 0.008608 + 0.822327I
b = 0.724687 0.940396I
3.61397 + 2.21654I 5.38699 4.72022I
u = 0.983280 + 0.164908I
a = 0.008608 0.822327I
b = 0.724687 + 0.940396I
3.61397 2.21654I 5.38699 + 4.72022I
10
V. I
u
5
=
h3u
10
+6u
9
+· · ·5u
2
+1, 18u
9
30u
8
+· · ·+2b+8u, 18u
9
+30u
8
+· · ·+2a+2i
(i) Arc colorings
a
1
=
1
0
a
8
=
9u
9
15u
8
+ ··· + 4u 1
9u
9
+ 15u
8
+ ···
7
2
u
2
4u
a
2
=
3
2
u
9
3
2
u
8
+ ··· u 3
21
2
u
9
+
33
2
u
8
+ ··· 3u + 4
a
4
=
0
u
a
5
=
u
u
a
9
=
9u
9
15u
8
+ ··· + 4u 1
9u
9
+ 15u
8
+ ···
9
2
u
2
4u
a
7
=
9u
9
15u
8
+ ··· + 4u 1
9
2
u
9
+
15
2
u
8
+ ··· u 1
a
10
=
6u
9
15
2
u
8
+ ··· +
3
2
u
7
2
15
2
u
9
+ 9u
8
+ ···
1
2
u + 2
a
3
=
15
2
u
9
+
21
2
u
8
+ ··· 4u + 2
3u
9
3u
8
+ 4u
7
+ 10u
6
2u
5
8u
4
3u
3
+ 5u
2
+ u
a
6
=
3u
9
23
2
u
7
+ ···
7
2
u +
9
2
9
2
u
9
+
3
2
u
8
+ ··· +
1
2
u 7
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 12u
9
+ 6u
8
+ 64u
7
+ 52u
6
90u
5
110u
4
+ 30u
3
+ 86u
2
+ 4u 24
11
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.081495 0.798609I
a = 0.313222 1.080503I
b = 1.31322 + 1.08050I
2.49243 8.64801I 4.04126 + 7.50135I
u = 1.081495 + 0.798609I
a = 0.313222 + 1.080503I
b = 1.31322 1.08050I
2.49243 + 8.64801I 4.04126 7.50135I
u = 0.744942 0.201707I
a = 0.54573 + 1.55310I
b = 0.45427 1.55310I
2.49243 8.64801I 4.04126 + 7.50135I
u = 0.744942 + 0.201707I
a = 0.54573 1.55310I
b = 0.45427 + 1.55310I
2.49243 + 8.64801I 4.04126 7.50135I
u = 0.707358 0.648629I
a = 0.360867 0.661972I
b = 1.36087 + 0.66197I
3.61397 2.21654I 5.38699 + 4.72022I
u = 0.707358 + 0.648629I
a = 0.360867 + 0.661972I
b = 1.36087 0.66197I
3.61397 + 2.21654I 5.38699 4.72022I
u = 0.550514 0.187402I
a = 0.75550 + 1.63857I
b = 0.24450 1.63857I
0.806279I 8.22652I
u = 0.550514 + 0.187402I
a = 0.75550 1.63857I
b = 0.24450 + 1.63857I
0.806279I 8.22652I
u = 0.983280 0.164908I
a = 0.872856 + 0.809997I
b = 0.127144 0.809997I
3.61397 + 2.21654I 5.38699 4.72022I
u = 0.983280 + 0.164908I
a = 0.872856 0.809997I
b = 0.127144 + 0.809997I
3.61397 2.21654I 5.38699 + 4.72022I
12
VI. I
u
6
= hu
10
7u
9
+ · · · 96u + 32, 31u
9
+ 183u
8
+ · · · + 288a
256, 17u
9
123u
8
+ · · · + 144b 496i
(i) Arc colorings
a
1
=
1
0
a
8
=
31
288
u
9
61
96
u
8
+ ···
7
36
u +
8
9
0.118056u
9
+ 0.854167u
8
+ ··· 11.2222u + 3.44444
a
2
=
1
48
u
9
+
7
48
u
8
+ ···
23
4
u +
10
3
1
24
u
8
5
24
u
7
+ ···
1
3
u
2
3
a
4
=
0
u
a
5
=
u
u
a
9
=
0.0243056u
9
+ 0.114583u
8
+ ··· 14.5278u + 5.55556
0.0347222u
9
+ 0.104167u
8
+ ··· + 3.11111u 1.22222
a
7
=
31
288
u
9
61
96
u
8
+ ···
7
36
u +
8
9
7
48
u
9
+
15
16
u
8
+ ···
10
3
u
1
3
a
10
=
1
48
u
9
+
3
16
u
8
+ ···
73
12
u +
8
3
1
8
u
8
+
5
8
u
7
+ ··· + 5u 2
a
3
=
1
16
u
9
1
2
u
8
+ ··· + 9u
7
2
3
16
u
9
+
19
16
u
8
+ ···
23
2
u + 4
a
6
=
0.0555556u
9
0.229167u
8
+ ··· 6.27778u + 3.05556
0.118056u
9
+ 0.729167u
8
+ ··· 2.72222u + 0.444444
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
4
27
u
9
11
18
u
8
+
173
54
u
7
331
27
u
6
+
1765
54
u
5
1103
18
u
4
+
4645
54
u
3
2239
27
u
2
+
1420
27
u
554
27
13
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 0.127144 0.809997I
a = 0.012729 + 1.215928I
b = 0.983280 0.164908I
3.61397 + 2.21654I 5.38699 4.72022I
u = 0.127144 + 0.809997I
a = 0.012729 1.215928I
b = 0.983280 + 0.164908I
3.61397 2.21654I 5.38699 + 4.72022I
u = 0.24450 1.63857I
a = 0.062837 + 0.345348I
b = 0.550514 0.187402I
0.806279I 8.22652I
u = 0.24450 + 1.63857I
a = 0.062837 0.345348I
b = 0.550514 + 0.187402I
0.806279I 8.22652I
u = 0.45427 1.55310I
a = 0.248876 0.406856I
b = 0.744942 0.201707I
2.49243 8.64801I 4.04126 + 7.50135I
u = 0.45427 + 1.55310I
a = 0.248876 + 0.406856I
b = 0.744942 + 0.201707I
2.49243 + 8.64801I 4.04126 7.50135I
u = 1.31322 1.08050I
a = 0.192716 0.766695I
b = 1.081495 + 0.798609I
2.49243 + 8.64801I 4.04126 7.50135I
u = 1.31322 + 1.08050I
a = 0.192716 + 0.766695I
b = 1.081495 0.798609I
2.49243 8.64801I 4.04126 + 7.50135I
u = 1.36087 0.66197I
a = 0.232842 0.589891I
b = 0.707358 + 0.648629I
3.61397 + 2.21654I 5.38699 4.72022I
u = 1.36087 + 0.66197I
a = 0.232842 + 0.589891I
b = 0.707358 0.648629I
3.61397 2.21654I 5.38699 + 4.72022I
14
VII. I
u
7
=
hu
10
+4u
9
+· · ·+12u+3, 10u
9
35u
8
+· · ·+6a41, 2u
9
+7u
8
+· · ·+2b+6i
(i) Arc colorings
a
1
=
1
0
a
8
=
5
3
u
9
+
35
6
u
8
+ ··· +
58
3
u +
41
6
u
9
7
2
u
8
+ ··· 11u 3
a
2
=
2
3
u
9
8
3
u
8
+ ···
59
6
u
7
2
3
2
u
9
+ 5u
8
+ ··· +
33
2
u + 6
a
4
=
0
u
a
5
=
u
u
a
9
=
5
3
u
9
+
16
3
u
8
+ ··· +
52
3
u +
35
6
u
9
3u
8
6u
7
8u
6
12u
5
14u
4
17u
3
13u
2
9u 2
a
7
=
5
3
u
9
+
35
6
u
8
+ ··· +
58
3
u +
41
6
u
9
3u
8
+ ··· 6u
1
2
a
10
=
5
3
u
9
35
6
u
8
+ ···
58
3
u
41
6
2u
9
+ 6u
8
+ ··· +
27
2
u +
7
2
a
3
=
10
9
u
9
+
31
9
u
8
+ ··· +
205
18
u +
23
6
1
6
u
9
1
6
u
8
+ ··· +
5
3
u + 2
a
6
=
2
3
u
9
+
8
3
u
8
+ ··· +
59
6
u +
7
2
1
2
u
9
3
2
u
8
+ ··· 5u
2
2u
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
80
9
u
9
296
9
u
8
66u
7
838
9
u
6
1174
9
u
5
1504
9
u
4
1706
9
u
3
496
3
u
2
946
9
u
112
3
15
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
1(vol +
1CS) Cusp shape
u = 1.20165 0.91842I
a = 0.247490 0.853751I
b = 0.719811 + 1.046886I
2.49243 8.64801I 4.04126 + 7.50135I
u = 1.20165 + 0.91842I
a = 0.247490 + 0.853751I
b = 0.719811 1.046886I
2.49243 + 8.64801I 4.04126 7.50135I
u = 0.724687 0.940396I
a = 0.615566 + 0.571236I
b = 0.684636 + 0.234182I
3.61397 + 2.21654I 5.38699 4.72022I
u = 0.724687 + 0.940396I
a = 0.615566 0.571236I
b = 0.684636 0.234182I
3.61397 2.21654I 5.38699 + 4.72022I
u = 0.684636 0.234182I
a = 0.634847 1.164559I
b = 0.724687 + 0.940396I
3.61397 2.21654I 5.38699 + 4.72022I
u = 0.684636 + 0.234182I
a = 0.634847 + 1.164559I
b = 0.724687 0.940396I
3.61397 + 2.21654I 5.38699 4.72022I
u = 0.108840 1.043638I
a = 0.232054 + 0.503294I
b = 0.108840 1.043638I
0.806279I 8.22652I
u = 0.108840 + 1.043638I
a = 0.232054 0.503294I
b = 0.108840 + 1.043638I
0.806279I 8.22652I
u = 0.719811 1.046886I
a = 0.201383 0.573113I
b = 1.20165 + 0.91842I
2.49243 + 8.64801I 4.04126 7.50135I
u = 0.719811 + 1.046886I
a = 0.201383 + 0.573113I
b = 1.20165 0.91842I
2.49243 8.64801I 4.04126 + 7.50135I
16
VIII.
I
u
8
= hu
10
+4u
9
+· · ·+12u+3, 2u
9
+6u
8
+· · ·+2a+2, 5u
9
20u
8
+· · ·+6b30i
(i) Arc colorings
a
1
=
1
0
a
8
=
u
9
3u
8
+ ···
13
2
u 1
5
6
u
9
+
10
3
u
8
+ ··· +
79
6
u + 5
a
2
=
2
3
u
9
8
3
u
8
+ ···
59
6
u
7
2
u
9
19
6
u
8
+ ···
19
2
u
10
3
a
4
=
0
u
a
5
=
u
u
a
9
=
1
2
u
8
+
3
2
u
7
+ ··· + 5u + 2
1
6
u
9
1
6
u
8
+ ··· +
5
3
u + 2
a
7
=
u
9
3u
8
+ ···
13
2
u 1
1
3
u
9
+
4
3
u
8
+ ··· +
25
6
u + 2
a
10
=
5
6
u
9
+
7
3
u
8
+ ··· +
20
3
u +
5
2
1
2
u
9
+
4
3
u
8
+ ··· + 2u
1
3
a
3
=
1
6
u
9
7
6
u
8
+ ···
47
6
u
7
2
1
2
u
9
5
3
u
8
+ ···
11
2
u
17
6
a
6
=
1
3
u
9
2
3
u
8
+ ···
19
6
u
7
6
7
9
u
9
25
9
u
8
+ ···
74
9
u
8
3
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
80
9
u
9
296
9
u
8
66u
7
838
9
u
6
1174
9
u
5
1504
9
u
4
1706
9
u
3
496
3
u
2
946
9
u
112
3
17
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
1(vol +
1CS) Cusp shape
u = 1.20165 0.91842I
a = 0.042195 + 0.838960I
b = 1.081495 0.798609I
2.49243 8.64801I 4.04126 + 7.50135I
u = 1.20165 + 0.91842I
a = 0.042195 0.838960I
b = 1.081495 + 0.798609I
2.49243 + 8.64801I 4.04126 7.50135I
u = 0.724687 0.940396I
a = 0.508239 0.336371I
b = 0.983280 0.164908I
3.61397 + 2.21654I 5.38699 4.72022I
u = 0.724687 + 0.940396I
a = 0.508239 + 0.336371I
b = 0.983280 + 0.164908I
3.61397 2.21654I 5.38699 + 4.72022I
u = 0.684636 0.234182I
a = 1.36825 + 0.90556I
b = 0.707358 0.648629I
3.61397 2.21654I 5.38699 + 4.72022I
u = 0.684636 + 0.234182I
a = 1.36825 0.90556I
b = 0.707358 + 0.648629I
3.61397 + 2.21654I 5.38699 4.72022I
u = 0.108840 1.043638I
a = 0.978481 0.206335I
b = 0.550514 0.187402I
0.806279I 8.22652I
u = 0.108840 + 1.043638I
a = 0.978481 + 0.206335I
b = 0.550514 + 0.187402I
0.806279I 8.22652I
u = 0.719811 1.046886I
a = 0.059798 1.188945I
b = 0.744942 + 0.201707I
2.49243 + 8.64801I 4.04126 7.50135I
u = 0.719811 + 1.046886I
a = 0.059798 + 1.188945I
b = 0.744942 0.201707I
2.49243 8.64801I 4.04126 + 7.50135I
18
IX. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
(u
2
3u + 1)
2
(u
4
4u
2
+ 5u 1)(u
4
3u
3
+ 4u
2
2u + 1)
(u
4
+ u
3
2u 1)(u
8
2u
7
+ ··· 3u + 1)
(u
10
7u
9
+ ··· 96u + 32)
(3 + 12u + 23u
2
+ 30u
3
+ 31u
4
+ 26u
5
+ 20u
6
+ 14u
7
+ 9u
8
+ 4u
9
+ u
10
)
2
(3u
10
+ 6u
9
u
8
14u
7
8u
6
+ 10u
5
+ 11u
4
2u
3
5u
2
+ 1)
2
c
2
, c
6
(u
2
+ 3u + 1)
2
(u
4
4u
2
5u 1)(u
4
3u
3
+ 4u
2
2u + 1)
(u
4
u
3
+ 2u 1)(u
8
+ 2u
7
+ ··· + 3u + 1)
(u
10
7u
9
+ ··· 96u + 32)
(3 + 12u + 23u
2
+ 30u
3
+ 31u
4
+ 26u
5
+ 20u
6
+ 14u
7
+ 9u
8
+ 4u
9
+ u
10
)
2
(3u
10
+ 6u
9
u
8
14u
7
8u
6
+ 10u
5
+ 11u
4
2u
3
5u
2
+ 1)
2
c
3
, c
5
(u
4
4u
2
+ 5u 1)(u
4
3u
3
+ 4u
2
2u + 1)(u
4
+ u
3
2u 1)
2
(u
8
+ u
7
+ 4u
6
3u
5
u
4
11u
3
+ 16u
2
7u + 1)
(u
10
7u
9
+ ··· 96u + 32)
(3 + 12u + 23u
2
+ 30u
3
+ 31u
4
+ 26u
5
+ 20u
6
+ 14u
7
+ 9u
8
+ 4u
9
+ u
10
)
2
(3u
10
+ 6u
9
u
8
14u
7
8u
6
+ 10u
5
+ 11u
4
2u
3
5u
2
+ 1)
2
c
4
(u
4
3u
3
+ 4u
2
2u + 1)(u
4
u
3
+ 2u 1)
3
(1 + 2u + 4u
2
+ 3u
3
+ u
4
)
2
(u
10
7u
9
+ ··· 96u + 32)
(3 + 12u + 23u
2
+ 30u
3
+ 31u
4
+ 26u
5
+ 20u
6
+ 14u
7
+ 9u
8
+ 4u
9
+ u
10
)
2
(3u
10
+ 6u
9
u
8
14u
7
8u
6
+ 10u
5
+ 11u
4
2u
3
5u
2
+ 1)
2
c
8
, c
10
(u
4
4u
2
5u 1)(u
4
3u
3
+ 4u
2
2u + 1)(u
4
u
3
+ 2u 1)
2
(u
8
u
7
+ 4u
6
+ 3u
5
u
4
+ 11u
3
+ 16u
2
+ 7u + 1)
(u
10
7u
9
+ ··· 96u + 32)
(3 + 12u + 23u
2
+ 30u
3
+ 31u
4
+ 26u
5
+ 20u
6
+ 14u
7
+ 9u
8
+ 4u
9
+ u
10
)
2
(3u
10
+ 6u
9
u
8
14u
7
8u
6
+ 10u
5
+ 11u
4
2u
3
5u
2
+ 1)
2
c
9
(u
4
3u
3
+ 4u
2
2u + 1)
3
(u
4
+ u
3
2u 1)
3
(u
10
7u
9
+ ··· 96u + 32)
(3 + 12u + 23u
2
+ 30u
3
+ 31u
4
+ 26u
5
+ 20u
6
+ 14u
7
+ 9u
8
+ 4u
9
+ u
10
)
2
(3u
10
+ 6u
9
u
8
14u
7
8u
6
+ 10u
5
+ 11u
4
2u
3
5u
2
+ 1)
2
19
X. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
6
c
7
(y
2
7y + 1)
2
(y
4
8y
3
+ 14y
2
17y + 1)(y
4
y
3
+ 2y
2
4y + 1)
(y
4
y
3
+ 6y
2
+ 4y + 1)
(y
8
+ 6y
7
+ 5y
6
11y
5
+ 14y
4
y
3
+ 15y
2
+ y + 1)
(9 6y 5y
2
+ 22y
3
+ 39y
4
+ 48y
5
+ 36y
6
+ 18y
7
+ 9y
8
+ 2y
9
+ y
10
)
2
(y
10
+ 7y
9
+ ··· + 3584y + 1024)
(1 10y + 47y
2
130y
3
+ 239y
4
316y
5
+ 316y
6
234y
7
+ 121y
8
42y
9
+ 9y
10
)
2
c
3
, c
5
, c
8
c
10
(y
4
8y
3
+ 14y
2
17y + 1)(y
4
y
3
+ 2y
2
4y + 1)
2
(y
4
y
3
+ 6y
2
+ 4y + 1)
(y
8
+ 7y
7
+ 20y
6
+ 37y
5
+ 79y
4
187y
3
+ 100y
2
17y + 1)
(9 6y 5y
2
+ 22y
3
+ 39y
4
+ 48y
5
+ 36y
6
+ 18y
7
+ 9y
8
+ 2y
9
+ y
10
)
2
(y
10
+ 7y
9
+ ··· + 3584y + 1024)
(1 10y + 47y
2
130y
3
+ 239y
4
316y
5
+ 316y
6
234y
7
+ 121y
8
42y
9
+ 9y
10
)
2
c
4
, c
9
(y
4
y
3
+ 2y
2
4y + 1)
3
(y
4
y
3
+ 6y
2
+ 4y + 1)
3
(9 6y 5y
2
+ 22y
3
+ 39y
4
+ 48y
5
+ 36y
6
+ 18y
7
+ 9y
8
+ 2y
9
+ y
10
)
2
(y
10
+ 7y
9
+ ··· + 3584y + 1024)
(1 10y + 47y
2
130y
3
+ 239y
4
316y
5
+ 316y
6
234y
7
+ 121y
8
42y
9
+ 9y
10
)
2
20