10
62
(K10a
41
)
1
Arc Sequences
6 9 7 10 1 3 4 2 8 5
Solving Sequence
2,9 3,6
7 1 5 8 10 4
c
2
c
6
c
1
c
5
c
8
c
10
c
4
c
3
, c
7
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
2
2, b 1, 2a u + 2i
I
u
2
= hb
6
2b
4
b
3
+ b
2
+ b 1, b
3
+ b + u, b
5
b
3
b
2
+ ai
I
u
3
= hu
19
+ 2u
18
+ ··· + 2u
2
2, u
17
10u
15
+ ··· + 4a 4, 2u
18
2u
17
+ ··· + 4b + 2i
I
v
1
= hb + 1, v 1, ai
There are 4 irreducible components with 28 representations.
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
= hu
2
2, b 1, 2a u + 2i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
2
u 1
1
a
3
=
1
2
u
1
a
6
=
0
u
a
7
=
1
2
u
u + 1
a
1
=
1
2
a
5
=
u
u
a
8
=
1
2
u
1
a
10
=
1
0
a
4
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.41421
a = 1.70711
b = 1.00000
4.93480 8.00000
u = 1.41421
a = 0.292893
b = 1.00000
4.93480 8.00000
3
II. I
u
2
= hb
6
2b
4
b
3
+ b
2
+ b 1, b
3
+ b + u, b
5
b
3
b
2
+ ai
(i) Arc colorings
a
2
=
1
0
a
9
=
b
5
+ b
3
+ b
2
b
a
3
=
b
4
+ b
2
+ b
b
2
a
6
=
0
b
3
b
a
7
=
b
5
b
3
b
2
b
a
1
=
1
b
3
b + 1
a
5
=
b
3
+ b
b
3
+ b 1
a
8
=
b
5
+ b
3
+ b
2
+ b
b
a
10
=
b
3
+ b
b
3
+ b
a
4
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 10
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.618034
a = 2.43346
b = 1.22636
0.986960 10.0000
u = 1.61803
a = 0.034946 0.687037I
b = 0.726823 0.764732I
8.88264 10.0000
u = 1.61803
a = 0.034946 + 0.687037I
b = 0.726823 + 0.764732I
8.88264 10.0000
u = 0.618034
a = 0.401306 0.709835I
b = 0.613180 0.357727I
0.986960 10.0000
u = 0.618034
a = 0.401306 + 0.709835I
b = 0.613180 + 0.357727I
0.986960 10.0000
u = 1.61803
a = 1.30596
b = 1.45365
8.88264 10.0000
5
III. I
u
3
=
hu
19
+2u
18
+· · ·+2u
2
2, u
17
10u
15
+· · ·+4a4, 2u
18
2u
17
+· · ·+4b+2i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
4
u
17
+
5
2
u
15
+ ··· u
2
+ 1
1
2
u
18
+
1
2
u
17
+ ···
1
2
u
1
2
a
3
=
1
4
u
18
+
11
4
u
16
+ ···
5
2
u +
1
2
1
4
u
18
+
5
2
u
16
+ ··· +
1
2
u + 1
a
6
=
0
u
a
7
=
1
4
u
14
2u
12
+ ··· +
1
2
u
1
2
1
2
u
9
5
2
u
7
+ ··· u
2
+ u
a
1
=
1
u
2
a
5
=
u
u
3
+ u
a
8
=
1
2
u
18
+
1
4
u
17
+ ···
1
2
u +
1
2
1
2
u
18
+
1
2
u
17
+ ···
1
2
u
1
2
a
10
=
u
2
+ 1
u
4
+ 2u
2
a
4
=
u
3
2u
u
5
3u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
18
+ 22u
16
96u
14
2u
13
+ 210u
12
+ 16u
11
240u
10
46u
9
+ 128u
8
+ 56u
7
+ 12u
6
32u
5
66u
4
+ 24u
3
+ 20u
2
10u + 8
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.69053 0.10897I
a = 0.0255644 0.0608617I
b = 0.544790 1.075047I
16.6648 + 3.4892I 12.44780 0.95664I
u = 1.69053 + 0.10897I
a = 0.0255644 + 0.0608617I
b = 0.544790 + 1.075047I
16.6648 3.4892I 12.44780 + 0.95664I
u = 1.62272 0.09591I
a = 0.623688 + 0.994954I
b = 0.987432 + 0.715363I
8.08934 + 5.62533I 8.31274 4.90801I
u = 1.62272 + 0.09591I
a = 0.623688 0.994954I
b = 0.987432 0.715363I
8.08934 5.62533I 8.31274 + 4.90801I
u = 1.43916
a = 1.03559
b = 1.06686
3.34099 2.02408
u = 0.833626 0.586392I
a = 1.79371 0.61393I
b = 1.045523 0.721346I
6.16103 + 7.19649I 9.03544 6.33971I
u = 0.833626 + 0.586392I
a = 1.79371 + 0.61393I
b = 1.045523 + 0.721346I
6.16103 7.19649I 9.03544 + 6.33971I
u = 0.449480
a = 1.13904
b = 0.461609
0.876243 12.3181
u = 0.109594 0.768897I
a = 1.65341 + 0.80786I
b = 0.865727 0.693353I
3.98301 2.66673I 7.07144 + 2.45976I
u = 0.109594 + 0.768897I
a = 1.65341 0.80786I
b = 0.865727 + 0.693353I
3.98301 + 2.66673I 7.07144 2.45976I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.169186 0.450873I
a = 1.82059 0.34046I
b = 0.900628 + 0.262698I
1.52268 + 0.97340I 1.44998 1.44252I
u = 0.169186 + 0.450873I
a = 1.82059 + 0.34046I
b = 0.900628 0.262698I
1.52268 0.97340I 1.44998 + 1.44252I
u = 0.706968 0.375087I
a = 1.26691 + 1.26495I
b = 0.927980 + 0.516153I
0.05288 3.91264I 5.51817 + 7.54928I
u = 0.706968 + 0.375087I
a = 1.26691 1.26495I
b = 0.927980 0.516153I
0.05288 + 3.91264I 5.51817 7.54928I
u = 0.976743 0.434841I
a = 0.214159 + 0.505267I
b = 0.631148 0.824317I
7.39549 1.39372I 11.32275 + 1.16010I
u = 0.976743 + 0.434841I
a = 0.214159 0.505267I
b = 0.631148 + 0.824317I
7.39549 + 1.39372I 11.32275 1.16010I
u = 1.37410
a = 0.922720
b = 0.222173
6.50526 14.0762
u = 1.66085 0.17438I
a = 1.112291 0.812172I
b = 1.181397 0.773041I
14.6774 10.1415I 10.53245 + 5.16770I
u = 1.66085 + 0.17438I
a = 1.112291 + 0.812172I
b = 1.181397 + 0.773041I
14.6774 + 10.1415I 10.53245 5.16770I
8
IV. I
v
1
= hb + 1, v 1, ai
(i) Arc colorings
a
2
=
1
0
a
9
=
0
1
a
3
=
1
1
a
6
=
1
0
a
7
=
0
1
a
1
=
1
0
a
5
=
1
0
a
8
=
1
1
a
10
=
1
0
a
4
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
9
(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
10
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
, c
5
c
10
(u)(u
2
2)(u
2
+ u 1)
3
(u
19
2u
18
+ ··· 2u
2
+ 2)
c
2
(u 1)
2
(u + 1)(u
6
2u
4
+ ··· + u 1)(u
19
+ 2u
18
+ ··· + 5u 1)
c
3
(u 1)
2
(u + 1)(u
6
2u
4
+ ··· u 1)(u
19
+ 2u
18
+ ··· 7u + 1)
c
6
, c
7
(u 1)(u + 1)
2
(u
6
2u
4
+ ··· u 1)(u
19
+ 2u
18
+ ··· 7u + 1)
c
8
(u 1)(u + 1)
2
(u
6
2u
4
+ ··· + u 1)(u
19
+ 2u
18
+ ··· + 5u 1)
c
9
(u + 1)
3
(u
6
+ 4u
5
+ 6u
4
+ 7u
3
+ 7u
2
+ 3u + 1)
(u
19
+ 6u
18
+ ··· + 29u + 1)
11
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
5
c
10
(y)(y 2)
2
(1 3y + y
2
)
3
(y
19
22y
18
+ ··· + 8y 4)
c
2
, c
8
(y 1)
3
(y
6
4y
5
+ 6y
4
7y
3
+ 7y
2
3y + 1)
(y
19
6y
18
+ ··· + 29y 1)
c
3
, c
6
, c
7
(y 1)
3
(y
6
4y
5
+ 6y
4
7y
3
+ 7y
2
3y + 1)
(y
19
22y
18
+ ··· + 45y 1)
c
9
(y 1)
3
(y
6
4y
5
6y
4
+ 13y
3
+ 19y
2
+ 5y + 1)
(y
19
+ 18y
18
+ ··· + 429y 1)
12