10
22
(K10a
112
)
1
Arc Sequences
6 9 8 7 1 10 3 4 2 5
Solving Sequence
3,7
8 4 5 9 2 10 1 6
c
7
c
3
c
4
c
8
c
2
c
9
c
10
c
6
c
1
, c
5
Representation Ideals
I = I
u
1
I
u
1
= hu
24
u
23
+ ··· + 2u
2
+ 1i
There are 1 irreducible components with 24 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
24
u
23
+ · · · + 2u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
8
=
u
u
a
4
=
u
2
+ 1
u
2
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
9
=
u
3
2u
u
3
+ u
a
2
=
u
6
3u
4
+ 2u
2
+ 1
u
6
+ 2u
4
u
2
a
10
=
u
9
4u
7
+ 5u
5
3u
u
9
+ 3u
7
3u
5
+ u
a
1
=
u
15
+ 6u
13
14u
11
+ 14u
9
2u
7
6u
5
+ 4u
3
2u
u
17
+ 7u
15
19u
13
+ 22u
11
3u
9
14u
7
+ 6u
5
+ 2u
3
+ u
a
6
=
u
19
+ 8u
17
26u
15
+ 40u
13
19u
11
24u
9
+ 30u
7
9u
3
u
19
7u
17
+ 20u
15
27u
13
+ 11u
11
+ 13u
9
14u
7
+ 3u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
21
+ 32u
19
+ 4u
18
108u
17
28u
16
+ 180u
15
+ 80u
14
104u
13
104u
12
120u
11
+ 24u
10
+ 216u
9
+ 88u
8
56u
7
76u
6
80u
5
12u
4
+ 36u
3
+ 24u
2
+ 8u + 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.293385 0.128068I
4.64383 2.66216I 8.07524 + 4.83074I
u = 1.293385 + 0.128068I
4.64383 + 2.66216I 8.07524 4.83074I
u = 1.291331 0.388939I
1.81113 6.59660I 1.74384 + 6.15928I
u = 1.291331 + 0.388939I
1.81113 + 6.59660I 1.74384 6.15928I
u = 1.234197 0.427679I
8.55472 + 0.67393I 2.54072 + 0.18139I
u = 1.234197 + 0.427679I
8.55472 0.67393I 2.54072 0.18139I
u = 0.832524
3.20914 1.52537
u = 0.691969
3.21354 0.806221
u = 0.240904 0.566295I
4.81497 3.00632I 4.21158 + 5.20782I
u = 0.240904 + 0.566295I
4.81497 + 3.00632I 4.21158 5.20782I
u = 0.047552 0.882738I
12.21816 5.35992I 5.68286 + 3.17670I
u = 0.047552 + 0.882738I
12.21816 + 5.35992I 5.68286 3.17670I
u = 0.023946 0.850260I
5.90820 + 2.14805I 2.49248 3.24690I
u = 0.023946 + 0.850260I
5.90820 2.14805I 2.49248 + 3.24690I
u = 0.208545 0.356460I
0.079333 + 0.910145I 1.70410 7.59691I
u = 0.208545 + 0.356460I
0.079333 0.910145I 1.70410 + 7.59691I
u = 1.20293
2.53343 1.89057
u = 1.252441 0.391136I
2.10558 + 2.30642I 0.925091 0.098908I
u = 1.252441 + 0.391136I
2.10558 2.30642I 0.925091 + 0.098908I
u = 1.30821
2.22926 4.75395
u = 1.311949 0.407404I
7.97363 + 9.98187I 1.73153 5.91019I
u = 1.311949 + 0.407404I
7.97363 9.98187I 1.73153 + 5.91019I
u = 1.317162 0.196052I
0.01480 + 5.67994I 2.05445 5.89837I
u = 1.317162 + 0.196052I
0.01480 5.67994I 2.05445 + 5.89837I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
10
(u
24
+ u
23
+ ··· + 2u
2
+ 1)
c
2
, c
4
, c
9
(u
24
+ 3u
23
+ ··· + 8u + 1)
c
3
, c
7
, c
8
(u
24
+ u
23
+ ··· + 2u
2
+ 1)
c
6
(u
24
+ 3u
23
+ ··· 20u 7)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
10
(y
24
23y
23
+ ··· + 4y + 1)
c
2
, c
4
, c
9
(y
24
+ 25y
23
+ ··· 20y + 1)
c
3
, c
7
, c
8
(y
24
19y
23
+ ··· + 4y + 1)
c
6
(y
24
11y
23
+ ··· 904y + 49)
5