8
14
(K8a
1
)
1
Arc Sequences
6 4 8 7 1 5 2 3
Solving Sequence
1,3
8 4 2 7 5 6
c
8
c
3
c
2
c
7
c
4
c
6
c
1
, c
5
Representation Ideals
I = I
u
1
I
u
1
= hu
15
+ u
14
+ 4u
13
+ 3u
12
+ 8u
11
+ 6u
10
+ 10u
9
+ 7u
8
+ 8u
7
+ 6u
6
+ 6u
5
+ 4u
4
+ 4u
3
+ 2u
2
+ 2u + 1i
There are 1 irreducible components with 15 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
15
+ u
14
+ 4u
13
+ 3u
12
+ 8u
11
+ 6u
10
+ 10u
9
+ 7u
8
+ 8u
7
+ 6u
6
+
6u
5
+ 4u
4
+ 4u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
3
=
0
u
a
8
=
1
u
2
a
4
=
u
u
3
+ u
a
2
=
u
3
u
5
+ u
3
+ u
a
7
=
u
6
u
4
+ 1
u
8
2u
6
2u
4
a
5
=
u
11
+ 2u
9
+ 2u
7
u
3
u
13
+ 3u
11
+ 5u
9
+ 4u
7
+ 2u
5
+ u
3
+ u
a
6
=
u
13
2u
11
3u
9
2u
7
2u
5
2u
3
u
u
13
+ 3u
11
+ 5u
9
+ 4u
7
+ 2u
5
+ u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
13
4u
12
12u
11
12u
10
20u
9
24u
8
20u
7
24u
6
16u
5
16u
4
16u
3
8u
2
8u10
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.758945 0.422629I
3.01689 4.09199I 0.95573 + 3.15094I
u = 0.758945 + 0.422629I
3.01689 + 4.09199I 0.95573 3.15094I
u = 0.594032 1.095617I
1.02630 + 9.21780I 4.14540 7.39135I
u = 0.594032 + 1.095617I
1.02630 9.21780I 4.14540 + 7.39135I
u = 0.538411
1.42428 6.56339
u = 0.426893 1.085665I
4.20816 + 3.60340I 10.16372 4.47672I
u = 0.426893 + 1.085665I
4.20816 3.60340I 10.16372 + 4.47672I
u = 0.146928 1.062740I
1.82075 2.07402I 7.82822 + 2.67122I
u = 0.146928 + 1.062740I
1.82075 + 2.07402I 7.82822 2.67122I
u = 0.385605 0.867795I
0.35117 1.66084I 2.48958 + 3.96405I
u = 0.385605 + 0.867795I
0.35117 + 1.66084I 2.48958 3.96405I
u = 0.594997 1.040825I
1.98305 3.51852I 2.28698 + 2.59027I
u = 0.594997 + 1.040825I
1.98305 + 3.51852I 2.28698 2.59027I
u = 0.715401 0.518352I
3.53338 1.50523I 0.15133 + 2.74048I
u = 0.715401 + 0.518352I
3.53338 + 1.50523I 0.15133 2.74048I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
15
+ u
14
+ ··· + 2u + 1)
c
2
(u
15
+ 7u
14
+ ··· + 4u
2
1)
c
3
, c
8
(u
15
+ u
14
+ ··· + 2u + 1)
c
4
, c
6
(u
15
+ 5u
14
+ ··· + 12u
3
+ 1)
c
7
(u
15
+ u
14
+ ··· 4u 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
15
5y
14
+ ··· + 12y
3
1)
c
2
(y
15
+ 3y
14
+ ··· + 8y 1)
c
3
, c
8
(y
15
+ 7y
14
+ ··· + 4y
2
1)
c
4
, c
6
(y
15
+ 11y
14
+ ··· 84y
2
1)
c
7
(y
15
y
14
+ ··· + 16y 1)
5