10
44
(K10a
32
)
1
Arc Sequences
8 9 1 10 4 3 2 7 6 5
Solving Sequence
5,10
1 4 6 3 7 9 2 8
c
10
c
4
c
5
c
3
c
6
c
9
c
2
c
8
c
1
, c
7
Representation Ideals
I = I
u
1
I
u
1
= hu
39
u
38
+ ··· + 2u 1i
There are 1 irreducible components with 39 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
39
u
38
+ · · · + 2u 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
1
=
u
u
a
4
=
1
u
2
a
6
=
u
2
+ 1
u
4
a
3
=
u
4
u
2
+ 1
u
4
a
7
=
u
12
+ 3u
10
5u
8
+ 4u
6
2u
4
u
2
+ 1
u
12
2u
10
+ 2u
8
+ u
4
a
9
=
u
5
+ 2u
3
u
u
7
u
5
+ u
a
2
=
u
16
5u
14
+ 11u
12
12u
10
+ 5u
8
+ 2u
6
2u
4
+ 1
u
18
+ 4u
16
7u
14
+ 4u
12
+ 3u
10
6u
8
+ 2u
6
u
2
a
8
=
u
31
8u
29
+ ··· + 12u
7
4u
5
u
31
+ 7u
29
+ ··· 2u
13
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
38
+ 44u
36
4u
35
232u
34
+ 40u
33
+ 752u
32
192u
31
1620u
30
+ 564u
29
+
2316u
28
1092u
27
1948u
26
+ 1380u
25
+ 284u
24
980u
23
+ 1508u
22
+ 16u
21
1892u
20
+ 728u
19
+ 776u
18
660u
17
+ 444u
16
+ 64u
15
692u
14
+ 332u
13
+ 236u
12
252u
11
+ 128u
10
4u
9
132u
8
+ 96u
7
+ 20u
6
40u
5
+ 20u
4
8u
3
8u
2
+ 12u 10
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.213029 0.378072I
7.52915 + 4.12434I 8.59821 2.83806I
u = 1.213029 + 0.378072I
7.52915 4.12434I 8.59821 + 2.83806I
u = 1.210580 0.415258I
9.04143 3.95701I 10.59268 + 3.75109I
u = 1.210580 + 0.415258I
9.04143 + 3.95701I 10.59268 3.75109I
u = 1.185453 0.504016I
4.07758 7.98510I 3.85690 + 5.54137I
u = 1.185453 + 0.504016I
4.07758 + 7.98510I 3.85690 5.54137I
u = 1.142372 0.483180I
2.09760 6.17588I 2.65093 + 6.87938I
u = 1.142372 + 0.483180I
2.09760 + 6.17588I 2.65093 6.87938I
u = 1.027709 0.074094I
4.16770 3.61917I 10.06501 + 4.33455I
u = 1.027709 + 0.074094I
4.16770 + 3.61917I 10.06501 4.33455I
u = 0.824609 0.517095I
1.91036 2.98443I 1.85991 + 4.48194I
u = 0.824609 + 0.517095I
1.91036 + 2.98443I 1.85991 4.48194I
u = 0.704254 0.512490I
2.25704 1.23434I 3.23691 + 3.43750I
u = 0.704254 + 0.512490I
2.25704 + 1.23434I 3.23691 3.43750I
u = 0.180542 0.637095I
0.64272 + 1.83013I 1.22482 3.69155I
u = 0.180542 + 0.637095I
0.64272 1.83013I 1.22482 + 3.69155I
u = 0.132738 0.775160I
1.00162 + 3.25758I 0.69216 2.50620I
u = 0.132738 + 0.775160I
1.00162 3.25758I 0.69216 + 2.50620I
u = 0.076025 0.793162I
5.24072 0.25023I 6.76221 0.26522I
u = 0.076025 + 0.793162I
5.24072 + 0.25023I 6.76221 + 0.26522I
u = 0.139221 0.807285I
3.46412 8.12134I 3.90397 + 6.02892I
u = 0.139221 + 0.807285I
3.46412 + 8.12134I 3.90397 6.02892I
u = 0.339086 0.540694I
0.25067 + 2.27932I 0.43670 3.34383I
u = 0.339086 + 0.540694I
0.25067 2.27932I 0.43670 + 3.34383I
u = 0.632327 0.547010I
0.63380 3.33294I 0.02170 + 2.50936I
u = 0.632327 + 0.547010I
0.63380 + 3.33294I 0.02170 2.50936I
u = 0.867921 0.539600I
0.02772 + 7.71489I 1.96279 8.94046I
u = 0.867921 + 0.539600I
0.02772 7.71489I 1.96279 + 8.94046I
u = 0.898181
1.46294 6.39811
u = 0.913577 0.379498I
2.06381 + 1.25772I 6.67108 2.89583I
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.913577 + 0.379498I
2.06381 1.25772I 6.67108 + 2.89583I
u = 1.114964 0.441427I
2.49096 + 1.59434I 3.82288 0.43137I
u = 1.114964 + 0.441427I
2.49096 1.59434I 3.82288 + 0.43137I
u = 1.194180 0.388571I
4.89443 + 0.66747I 5.40097 0.84813I
u = 1.194180 + 0.388571I
4.89443 0.66747I 5.40097 + 0.84813I
u = 1.194899 0.512673I
6.5788 + 12.9690I 6.91871 9.04784I
u = 1.194899 + 0.512673I
6.5788 12.9690I 6.91871 + 9.04784I
u = 1.199996 0.486246I
8.53633 + 4.91106I 9.80910 3.06121I
u = 1.199996 + 0.486246I
8.53633 4.91106I 9.80910 + 3.06121I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
(u
39
+ u
38
+ ··· + 2u
3
1)
c
2
(u
39
+ u
38
+ ··· 18u + 17)
c
3
, c
9
(u
39
+ 3u
38
+ ··· + 12u + 1)
c
4
, c
10
(u
39
+ u
38
+ ··· + 2u + 1)
c
5
(u
39
+ 21u
38
+ ··· 2u
2
+ 1)
c
6
(u
39
+ 5u
38
+ ··· 12u 1)
c
8
(u
39
+ 19u
38
+ ··· + 2u
2
1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
(y
39
+ 19y
38
+ ··· + 2y
2
1)
c
2
(y
39
13y
38
+ ··· + 3588y 289)
c
3
, c
9
(y
39
+ 31y
38
+ ··· 36y 1)
c
4
, c
10
(y
39
21y
38
+ ··· + 2y
2
1)
c
5
(y
39
5y
38
+ ··· + 4y 1)
c
6
(y
39
y
38
+ ··· + 28y 1)
c
8
(y
39
+ 3y
38
+ ··· + 4y 1)
5