8
13
(K8a
7
)
1
Arc Sequences
5 8 7 6 2 1 3 4
Solving Sequence
2,5
6 1 7 4 3 8
c
5
c
1
c
6
c
4
c
3
c
8
c
2
, c
7
Representation Ideals
I = I
u
1
I
u
1
= hu
14
u
13
3u
12
+ 4u
11
+ 4u
10
7u
9
u
8
+ 6u
7
2u
6
2u
5
+ 2u
4
u + 1i
There are 1 irreducible components with 14 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
14
u
13
3u
12
+4u
11
+4u
10
7u
9
u
8
+6u
7
2u
6
2u
5
+2u
4
u +1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
1
u
2
a
7
=
u
3
u
5
u
3
+ u
a
4
=
u
3
u
3
+ u
a
3
=
u
11
+ 2u
9
2u
7
+ u
3
u
13
3u
11
+ 5u
9
4u
7
+ 2u
5
u
3
+ u
a
8
=
u
8
u
6
+ u
4
+ 1
u
8
+ 2u
6
2u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
13
16u
11
+ 4 u
10
+ 28 u
9
12u
8
20u
7
+ 16 u
6
8u
4
+ 8 u
3
2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.157218 0.286866I
7.82627 + 0.47055I 5.32829 + 0.18349I
u = 1.157218 + 0.286866I
7.82627 0.47055I 5.32829 0.18349I
u = 1.068409 0.522447I
0.56380 5.07185I 1.67153 + 6.33126I
u = 1.068409 + 0.522447I
0.56380 + 5.07185I 1.67153 6.33126I
u = 0.403136 0.584808I
1.36265 + 0.62859I 6.31651 1.42251I
u = 0.403136 + 0.584808I
1.36265 0.62859I 6.31651 + 1.42251I
u = 0.268039 0.757899I
3.51248 3.62879I 0.33383 + 2.63226I
u = 0.268039 + 0.757899I
3.51248 + 3.62879I 0.33383 2.63226I
u = 0.728347 0.560551I
1.44038 + 2.19128I 1.23919 3.85718I
u = 0.728347 + 0.560551I
1.44038 2.19128I 1.23919 + 3.85718I
u = 0.989783 0.381937I
1.69471 + 1.40484I 1.50927 0.52948I
u = 0.989783 + 0.381937I
1.69471 1.40484I 1.50927 + 0.52948I
u = 1.142594 0.546762I
6.06421 + 8.53123I 2.72348 6.18031I
u = 1.142594 + 0.546762I
6.06421 8.53123I 2.72348 + 6.18031I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
14
+ u
13
+ ··· + u + 1)
c
2
, c
3
, c
7
(u
14
+ u
13
+ ··· + u + 1)
c
4
(u
14
+ 7u
13
+ ··· + u + 1)
c
6
(u
14
+ 3u
13
+ ··· + 7u + 3)
c
8
(u
14
+ u
13
+ ··· 3u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
14
7y
13
+ ··· y + 1)
c
2
, c
3
, c
7
(y
14
+ 13y
13
+ ··· y + 1)
c
4
(y
14
+ y
13
+ ··· + 7y + 1)
c
6
(y
14
+ 5y
13
+ ··· + 23y + 9)
c
8
(y
14
+ y
13
+ ··· y + 1)
5