10
99
(K10a
103
)
1
Arc Sequences
5 9 1 8 2 10 4 6 3 7
Solving Sequence
4,7
8
5,10
1 2 3 6 9
c
7
c
4
c
10
c
1
c
3
c
6
c
9
c
2
, c
5
, c
8
Representation Ideals
I =
8
\
i=1
I
u
i
2
\
j=1
I
v
j
I
u
1
= ha + 1, u + 1i
I
u
2
= hu + 1, ba a
2
b + a 1i
I
u
3
= ha
2
b ba + a
2
a + 1, ba b + a + u 1i
I
u
4
= ha, b u, u
3
u + 1i
I
u
5
= ha + 1, u
8
3u
6
+ 3u
4
2u
3
+ 2u
2
+ 2u 1, u
7
u
6
+ 3u
5
+ 2u
4
3u
3
+ 2u
2
+ 2b 2i
I
u
6
= hu + 1, a
3
a 1, a
2
+ b ai
I
u
7
= hu
3
u 1, u
2
+ a + 1, u
2
+ b ui
I
u
8
= h3u
12
12u
11
+ 14u
10
+ 4u
9
20u
8
+ 10u
7
+ 32u
6
108u
5
+ 163u
4
142u
3
+ 96u
2
62u + 23,
849u
11
+ 2262u
10
+ ··· + 1656a + 17759, 75u
11
132u
10
+ ··· + 36b 100i
I
v
1
= ha, v 1, b + 1i
I
v
2
= ha, v 1, b
3
b + 1i
There are 10 irreducible components with 33 representations.
There are 3 irreducible components of dim
C
= 1 for 10
99
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= ha + 1, u + 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
1
a
8
=
1
1
a
5
=
0
1
a
10
=
1
b
a
1
=
1
b + 1
a
2
=
1
b
a
3
=
b + 2
b
2
2b 1
a
6
=
1
b 1
a
9
=
b 3
b
2
+ 3b + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
3
II. I
u
2
= hu + 1, ba a
2
b + a 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
1
a
8
=
1
1
a
5
=
0
1
a
10
=
a
b
a
1
=
a
b a
a
2
=
a
b
a
3
=
b + a
b
2
+ ba + b a + 1
a
6
=
ba b + a 1
ba 1
a
9
=
b
b
2
ba + a 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
5
III. I
u
3
= ha
2
b ba + a
2
a + 1, ba b + a + u 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
ba + b a + 1
a
8
=
ba + b a + 1
ba + b a + 1
a
5
=
b
2
a b
2
+ 2ba b + a + 1
b
2
a b
2
+ 2ba b + a
a
10
=
a
b
a
1
=
a
ba + a 1
a
2
=
b
2
a b
2
+ 2ba b + 2a + 1
b
2
a b
2
+ 3ba b + 2a 1
a
3
=
ba + 2
b
2
a + b + a
a
6
=
a
ba a + 1
a
9
=
2ba + b a + 2
b
2
a ba + 2b + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
7
IV. I
u
4
= ha, b u, u
3
u + 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
8
=
u
u
a
5
=
u
2
+ 1
u
2
a
10
=
0
u
a
1
=
0
u
a
2
=
u
2
1
u
2
+ u
a
3
=
1
u
2
a
6
=
0
u
a
9
=
u
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.32472
a = 0
b = 1.32472
1.64493 6.00000
u = 0.662359 0.562280I
a = 0
b = 0.662359 0.562280I
1.64493 6.00000
u = 0.662359 + 0.562280I
a = 0
b = 0.662359 + 0.562280I
1.64493 6.00000
9
V. I
u
5
=
ha+1, u
8
3u
6
+3u
4
2u
3
+2u
2
+2u1, u
7
u
6
+3u
5
+2u
4
3u
3
+2u
2
+2b2i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
8
=
u
u
a
5
=
u
2
+ 1
u
2
a
10
=
1
1
2
u
7
+
1
2
u
6
+ ··· u
2
+ 1
a
1
=
1
1
2
u
7
+
1
2
u
6
+ ··· +
3
2
u
3
+ 1
a
2
=
1
2
u
7
1
2
u
6
+ ···
1
2
u
2
3
2
1
2
u
7
3
2
u
5
+ ··· +
1
2
u + 1
a
3
=
1
2
u
7
+
1
2
u
6
+ ··· +
3
2
u
3
+ 2
1
2
u
4
+
1
2
u
2
1
2
u 1
a
6
=
u
1
2
u
7
u
5
u
2
+ u +
1
2
a
9
=
1
2
u
7
3
2
u
5
+ u
3
1
2
u
2
+
3
2
u
1
2
u
4
u
3
1
2
u
2
+
1
2
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
7
+ 10u
5
2u
4
6u
3
+ 12u
2
10u 6
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.38933 0.55684I
a = 1.00000
b = 2.84334 0.09252I
14.3343I 7.84155I
u = 1.38933 + 0.55684I
a = 1.00000
b = 2.84334 + 0.09252I
14.3343I 7.84155I
u = 0.752536
a = 1.00000
b = 0.141782
1.28346 8.36994
u = 0.221678 0.868597I
a = 1.00000
b = 0.451465 + 0.606992I
7.42191 + 3.34562I 7.11001 1.68383I
u = 0.221678 + 0.868597I
a = 1.00000
b = 0.451465 0.606992I
7.42191 3.34562I 7.11001 + 1.68383I
u = 0.382196
a = 1.00000
b = 0.906253
1.28346 8.36994
u = 1.352817 0.318023I
a = 1.00000
b = 2.58744 0.16401I
7.42191 3.34562I 7.11001 + 1.68383I
u = 1.352817 + 0.318023I
a = 1.00000
b = 2.58744 + 0.16401I
7.42191 + 3.34562I 7.11001 1.68383I
11
VI. I
u
6
= hu + 1, a
3
a 1, a
2
+ b ai
(i) Arc colorings
a
4
=
1
0
a
7
=
0
1
a
8
=
1
1
a
5
=
0
1
a
10
=
a
a
2
+ a
a
1
=
a
a
2
a
2
=
a
a
2
+ a
a
3
=
a
a
2
a
a
6
=
a
2
a
2
+ a
a
9
=
a
a
2
+ a
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.662359 0.562280I
b = 0.539798 + 0.182582I
1.64493 6.00000
u = 1.00000
a = 0.662359 + 0.562280I
b = 0.539798 0.182582I
1.64493 6.00000
u = 1.00000
a = 1.32472
b = 3.07960
1.64493 6.00000
13
VII. I
u
7
= hu
3
u 1, u
2
+ a + 1, u
2
+ b ui
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
8
=
u
u
a
5
=
u
2
+ 1
u
2
a
10
=
u
2
1
u
2
+ u
a
1
=
u
2
1
u
2
a
2
=
u
2
1
u
2
a
3
=
u + 1
u
2
u
a
6
=
u
2
+ 1
u
2
a
9
=
u 1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
14
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
1(vol +
1CS) Cusp shape
u = 0.662359 0.562280I
a = 0.877439 + 0.744862I
b = 0.539798 + 0.182582I
1.64493 6.00000
u = 0.662359 + 0.562280I
a = 0.877439 0.744862I
b = 0.539798 0.182582I
1.64493 6.00000
u = 1.32472
a = 0.754878
b = 3.07960
1.64493 6.00000
15
VIII. I
u
8
= h3u
12
12u
11
+ · · · 62u + 23, 849u
11
+ 2262u
10
+ · · · +
1656a + 17759, 75u
11
132u
10
+ · · · + 36b 100i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
8
=
u
u
a
5
=
u
2
+ 1
u
2
a
10
=
0.512681u
11
1.36594u
10
+ ··· + 12.3303u 10.7240
2.08333u
11
+ 3.66667u
10
+ ··· 11.9722u + 2.77778
a
1
=
0.512681u
11
1.36594u
10
+ ··· + 12.3303u 10.7240
5u
11
107
8
u
10
+ ··· +
1313
24
u
281
12
a
2
=
0.887681u
11
+ 4.63406u
10
+ ··· 23.4197u + 16.5260
4.70833u
11
+ 19.4167u
10
+ ··· 91.5972u + 57.4028
a
3
=
0.918478u
11
+ 2.04891u
10
+ ··· 8.93297u + 4.14855
15
8
u
11
45
8
u
10
+ ··· +
93
4
u
87
8
a
6
=
0.0434783u
11
+ 1.07609u
10
+ ··· 4.40036u + 4.55978
7
4
u
11
23
8
u
10
+ ··· +
32
3
u
61
24
a
9
=
3.06884u
11
10.5670u
10
+ ··· + 48.8973u 27.8255
2.41667u
11
9.83333u
10
+ ··· + 46.5278u 28.2222
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
1
18
u
11
38
9
u
10
+
295
54
u
9
+
187
54
u
8
275
54
u
7
+
65
27
u
6
+2u
5
71
2
u
4
+
2645
54
u
3
1855
54
u
2
+
785
27
u
484
27
16
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
1(vol +
1CS) Cusp shape
u = 1.41250 0.63054I
a = 0.859432 0.131349I
b = 2.49933 + 0.06072I
4.49149 + 8.24229I 3.01193 6.51979I
u = 1.41250 + 0.63054I
a = 0.859432 + 0.131349I
b = 2.49933 0.06072I
4.49149 8.24229I 3.01193 + 6.51979I
u = 0.187861 0.726416I
a = 0.94142 1.24258I
b = 0.267214 0.128490I
4.33667I 5.70400I
u = 0.187861 + 0.726416I
a = 0.94142 + 1.24258I
b = 0.267214 + 0.128490I
4.33667I 5.70400I
u = 0.052828 1.195262I
a = 0.447926 0.960140I
b = 0.266546 + 0.731832I
4.49149 8.24229I 3.01193 + 6.51979I
u = 0.052828 + 1.195262I
a = 0.447926 + 0.960140I
b = 0.266546 0.731832I
4.49149 + 8.24229I 3.01193 6.51979I
u = 1.079483 0.450431I
a = 0.387373 0.511292I
b = 0.267214 + 0.128490I
4.33667I 5.70400I
u = 1.079483 + 0.450431I
a = 0.387373 + 0.511292I
b = 0.267214 0.128490I
4.33667I 5.70400I
u = 1.171282 0.484667I
a = 0.399040 + 0.855353I
b = 0.266546 + 0.731832I
4.49149 8.24229I 3.01193 + 6.51979I
u = 1.171282 + 0.484667I
a = 0.399040 0.855353I
b = 0.266546 0.731832I
4.49149 + 8.24229I 3.01193 6.51979I
17
Solution to I
u
8
1(vol +
1CS) Cusp shape
u = 1.296771 0.356378I
a = 1.137001 0.173771I
b = 2.49933 0.06072I
4.49149 8.24229I 3.01193 + 6.51979I
u = 1.296771 + 0.356378I
a = 1.137001 + 0.173771I
b = 2.49933 + 0.06072I
4.49149 + 8.24229I 3.01193 6.51979I
18
IX. I
v
1
= ha, v 1, b + 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
1
0
a
8
=
1
0
a
5
=
1
0
a
10
=
0
1
a
1
=
1
1
a
2
=
0
1
a
3
=
0
1
a
6
=
1
1
a
9
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
19
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
20
X. I
v
2
= ha, v 1, b
3
b + 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
1
0
a
8
=
1
0
a
5
=
1
0
a
10
=
0
b
a
1
=
b
b
a
2
=
0
b
a
3
=
b
2
+ 1
b
2
a
6
=
1
b
2
a
9
=
b
2
+ 1
b
2
+ b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
21
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
2
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.32472
1.64493 6.00000
v = 1.00000
a = 0
b = 0.662359 0.562280I
1.64493 6.00000
v = 1.00000
a = 0
b = 0.662359 + 0.562280I
1.64493 6.00000
22
XI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
u
3
(u 1)
10
(u + 1)(u
2
+ u 1)(u
3
u 1)(u
3
u + 1)
(u
8
3u
6
+ ··· 2u 1)(3u
12
+ 12u
11
+ ··· + 62u + 23)
c
2
, c
4
u
4
(u 1)
3
(u + 1)
3
(u
3
u 1)
2
(u
3
u + 1)(u
3
u
2
1)
(u
8
3u
6
+ ··· 2u 1)(3u
12
+ 12u
11
+ ··· + 62u + 23)
c
3
(u 1)(2u + 1)(u
2
u 1)(u
3
u 1)
2
(u
3
2u
2
3u 1)
(u
3
u
2
2u 1)(u
3
+ 2u
2
+ u + 1)
2
(u
6
+ u
5
+ 2u
4
u
3
+ 2u
2
+ 3)
2
(2u
8
10u
7
+ 23u
6
22u
5
7u
4
+ 37u
3
30u
2
+ 8)
c
5
u
3
(u 1)
3
(u + 1)
8
(u
2
u 1)(u
3
u 1)(u
3
u + 1)
(u
8
3u
6
+ ··· 2u 1)(3u
12
+ 12u
11
+ ··· + 62u + 23)
c
6
u
3
(u 1)
4
(u + 1)
7
(u
2
u 1)(u
3
u 1)(u
3
u + 1)
(u
8
3u
6
+ ··· 2u 1)(3u
12
+ 12u
11
+ ··· + 62u + 23)
c
7
, c
9
u
4
(u 1)
6
(u
3
u 1)(u
3
u + 1)
2
(u
3
+ u
2
+ 1)
(u
8
3u
6
+ ··· 2u 1)(3u
12
+ 12u
11
+ ··· + 62u + 23)
c
8
(u + 1)(2u 1)(u
2
+ u 1)(u
3
u 1)
2
(u
3
+ u
2
2u + 1)
(u
3
+ 2u
2
3u + 1)(1 + u + 2u
2
+ u
3
)
2
(3 + 2u
2
u
3
+ 2u
4
+ u
5
+ u
6
)
2
(2u
8
10u
7
+ 23u
6
22u
5
7u
4
+ 37u
3
30u
2
+ 8)
c
10
u
3
(u 1)
11
(u
2
+ u 1)(u
3
u 1)(u
3
u + 1)
(u
8
3u
6
+ ··· 2u 1)(3u
12
+ 12u
11
+ ··· + 62u + 23)
23
XII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
c
10
y
3
(y 1)
11
(y
2
3y + 1)(y
3
2y
2
+ y 1)
2
(y
8
6y
7
+ 15y
6
14y
5
5y
4
+ 14y
3
+ 6y
2
8y + 1)
(9y
12
60y
11
+ ··· + 572y + 529)
c
2
, c
4
, c
7
c
9
y
4
(y 1)
6
(y
3
2y
2
+ y 1)
3
(y
3
y
2
2y 1)
(y
8
6y
7
+ 15y
6
14y
5
5y
4
+ 14y
3
+ 6y
2
8y + 1)
(9y
12
60y
11
+ ··· + 572y + 529)
c
3
, c
8
(y 1)(4y 1)(y
2
3y + 1)(y
3
10y
2
+ 5y 1)(y
3
5y
2
+ 2y 1)
(y
3
2y
2
3y 1)
2
(y
3
2y
2
+ y 1)
2
(y
6
+ 3y
5
+ 10y
4
+ 13y
3
+ 16y
2
+ 12y + 9)
2
(4y
8
8y
7
+ 61y
6
186y
5
+ 329y
4
581y
3
+ 788y
2
480y + 64)
24