10
38
(K10a
29
)
1
Arc Sequences
8 6 5 9 3 10 1 7 4 2
Solving Sequence
1,7
8 2 9 10 6 3 5 4
c
7
c
1
c
8
c
10
c
6
c
2
c
5
c
3
c
4
, c
9
Representation Ideals
I = I
u
1
I
u
1
= hu
29
u
28
+ ··· + 3u 1i
There are 1 irreducible components with 29 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
29
u
28
+ · · · + 3u 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
8
=
u
u
a
2
=
u
2
+ 1
u
2
a
9
=
u
u
3
+ u
a
10
=
u
4
u
2
+ 1
u
4
a
6
=
u
9
2u
7
+ 3u
5
2u
3
+ u
u
9
+ u
7
u
5
+ u
a
3
=
u
16
+ 3u
14
7u
12
+ 10u
10
11u
8
+ 8u
6
4u
4
+ 1
u
16
2u
14
+ 4u
12
4u
10
+ 2u
8
2u
4
+ 2u
2
a
5
=
u
23
4u
21
+ ··· 4u
3
+ 2u
u
23
+ 3u
21
+ ··· + 2u
3
+ u
a
4
=
u
27
4u
25
+ ··· 7u
3
+ 2u
u
28
u
27
+ ··· 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
28
+ 20u
26
4u
25
68u
24
+ 16u
23
+ 164u
22
56u
21
308u
20
+ 128u
19
+ 464u
18
240u
17
564u
16
+ 356u
15
+ 540u
14
424u
13
392u
12
+
404u
11
+ 172u
10
292u
9
+ 8u
8
+ 132u
7
80u
6
16u
5
+ 68u
4
36u
3
20u
2
+ 28u 14
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.026367 0.233938I
2.56729 6.08103I 6.75508 + 6.19570I
u = 1.026367 + 0.233938I
2.56729 + 6.08103I 6.75508 6.19570I
u = 0.992356 0.761925I
9.22437 4.97924I 1.18288 + 2.83205I
u = 0.992356 + 0.761925I
9.22437 + 4.97924I 1.18288 2.83205I
u = 0.977674 0.081172I
3.81512 2.50065I 13.4942 + 5.2130I
u = 0.977674 + 0.081172I
3.81512 + 2.50065I 13.4942 5.2130I
u = 0.920683 0.717468I
2.89789 3.74340I 0.78236 + 3.16701I
u = 0.920683 + 0.717468I
2.89789 + 3.74340I 0.78236 3.16701I
u = 0.811061 0.735142I
3.23356 1.79478I 0.02040 + 2.96423I
u = 0.811061 + 0.735142I
3.23356 + 1.79478I 0.02040 2.96423I
u = 0.753827 0.841147I
9.96021 1.00685I 0.05949 + 2.19242I
u = 0.753827 + 0.841147I
9.96021 + 1.00685I 0.05949 2.19242I
u = 0.023100 0.676599I
5.95691 + 3.09358I 0.04639 2.70964I
u = 0.023100 + 0.676599I
5.95691 3.09358I 0.04639 + 2.70964I
u = 0.218842 0.390599I
0.332830 + 1.166296I 4.21359 5.75923I
u = 0.218842 + 0.390599I
0.332830 1.166296I 4.21359 + 5.75923I
u = 0.720110 0.729907I
1.63523 2.09123I 4.28547 + 3.54352I
u = 0.720110 + 0.729907I
1.63523 + 2.09123I 4.28547 3.54352I
u = 0.736471 0.843921I
9.64156 5.37662I 0.52039 + 2.73445I
u = 0.736471 + 0.843921I
9.64156 + 5.37662I 0.52039 2.73445I
u = 0.834085
1.36635 6.67119
u = 0.891442 0.617290I
0.99960 + 2.39368I 10.11411 2.65936I
u = 0.891442 + 0.617290I
0.99960 2.39368I 10.11411 + 2.65936I
u = 0.971124 0.697980I
0.88657 + 7.55674I 6.27529 8.69605I
u = 0.971124 + 0.697980I
0.88657 7.55674I 6.27529 + 8.69605I
u = 1.000892 0.266873I
2.82194 + 0.04233I 6.03677 1.08568I
u = 1.000892 + 0.266873I
2.82194 0.04233I 6.03677 + 1.08568I
u = 1.002945 0.755712I
8.8206 + 11.3493I 1.99701 7.67243I
u = 1.002945 + 0.755712I
8.8206 11.3493I 1.99701 + 7.67243I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
(u
29
+ u
28
+ ··· + 3u + 1)
c
2
, c
3
, c
5
(u
29
+ 7u
28
+ ··· u 1)
c
4
, c
9
(u
29
+ u
28
+ ··· + u 1)
c
6
(u
29
+ u
28
+ ··· + 15u 25)
c
8
, c
10
(u
29
+ 9u
28
+ ··· u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
(y
29
9y
28
+ ··· y 1)
c
2
, c
3
, c
5
(y
29
+ 31y
28
+ ··· + 15y 1)
c
4
, c
9
(y
29
+ 7y
28
+ ··· y 1)
c
6
(y
29
+ 11y
28
+ ··· 2925y 625)
c
8
, c
10
(y
29
+ 23y
28
+ ··· 17y 1)
5