9
17
(K9a
14
)
1
Arc Sequences
5 7 9 1 2 3 6 4 8
Solving Sequence
2,7
3 6 8 5 1 4 9
c
2
c
6
c
7
c
5
c
1
c
4
c
9
c
3
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
7
+ 2u
5
+ u
4
+ 2u
3
+ u
2
+ 1i
I
u
2
= hu
12
u
11
+ 4u
10
4u
9
+ 7u
8
7u
7
+ 5u
6
5u
5
+ u
4
u
3
+ 1i
There are 2 irreducible components with 19 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
7
+ 2u
5
+ u
4
+ 2u
3
+ u
2
+ 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
6
=
u
u
3
+ u
a
8
=
u
3
u
5
+ u
3
+ u
a
5
=
u
3
u
3
+ u
a
1
=
u
6
u
4
+ 1
u
6
2u
4
u
2
a
4
=
u
6
+ u
5
+ u
4
+ 2u
3
+ u
2
+ u
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ u + 1
a
9
=
u
6
+ u
5
u
4
+ 1
u
6
+ u
5
u
4
+ u
3
+ 1
a
9
=
u
6
+ u
5
u
4
+ 1
u
6
+ u
5
u
4
+ u
3
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
6
4u
5
4u
4
8u
3
8u
2
4u 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.824481
3.34763 1.23744
u = 0.468927 1.008505I
2.15041 + 6.00484I 4.26608 8.08638I
u = 0.468927 + 1.008505I
2.15041 6.00484I 4.26608 + 8.08638I
u = 0.391915 0.631080I
0.40799 1.46776I 1.41234 + 4.85424I
u = 0.391915 + 0.631080I
0.40799 + 1.46776I 1.41234 4.85424I
u = 0.489252 1.239918I
10.5657 9.4746I 7.52754 + 6.21855I
u = 0.489252 + 1.239918I
10.5657 + 9.4746I 7.52754 6.21855I
3
II. I
u
2
= hu
12
u
11
+ 4u
10
4u
9
+ 7u
8
7u
7
+ 5u
6
5u
5
+ u
4
u
3
+ 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
6
=
u
u
3
+ u
a
8
=
u
3
u
5
+ u
3
+ u
a
5
=
u
3
u
3
+ u
a
1
=
u
6
u
4
+ 1
u
6
2u
4
u
2
a
4
=
u
9
2u
7
u
5
+ 2u
3
+ u
u
9
3u
7
3u
5
+ u
a
9
=
u
10
3u
8
4u
6
u
4
+ u
2
+ u + 1
2u
11
u
10
+ 8u
9
4u
8
+ 14u
7
7u
6
+ 10u
5
6u
4
+ u
3
3u
2
u 2
a
9
=
u
10
3u
8
4u
6
u
4
+ u
2
+ u + 1
2u
11
u
10
+ 8u
9
4u
8
+ 14u
7
7u
6
+ 10u
5
6u
4
+ u
3
3u
2
u 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
9
+ 12u
7
+ 12u
5
4u
3
8u 6
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.525382 0.335320I
0.32962 1.97241I 0.57572 + 3.68478I
u = 0.525382 + 0.335320I
0.32962 + 1.97241I 0.57572 3.68478I
u = 0.460851 1.226446I
6.98545 + 4.59213I 4.58114 3.20482I
u = 0.460851 + 1.226446I
6.98545 4.59213I 4.58114 + 3.20482I
u = 0.206575 1.062083I
4.02872 9.41678
u = 0.206575 + 1.062083I
4.02872 9.41678
u = 0.386547 0.899125I
0.32962 1.97241I 0.57572 + 3.68478I
u = 0.386547 + 0.899125I
0.32962 + 1.97241I 0.57572 3.68478I
u = 0.436607 1.253745I
10.9500 8.26950
u = 0.436607 + 1.253745I
10.9500 8.26950
u = 0.869654 0.049931I
6.98545 + 4.59213I 4.58114 3.20482I
u = 0.869654 + 0.049931I
6.98545 4.59213I 4.58114 + 3.20482I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
, c
5
(u
6
u
5
3u
4
+ 2u
3
+ 2u
2
+ u 1)
2
(u
7
+ 3u
6
+ u
5
2u
4
+ 2u
3
+ 3u
2
+ u + 2)
c
2
, c
3
, c
6
c
8
(u
7
+ 2u
5
u
4
+ 2u
3
u
2
1)
(u
12
+ u
11
+ 4u
10
+ 4u
9
+ 7u
8
+ 7u
7
+ 5u
6
+ 5u
5
+ u
4
+ u
3
+ 1)
c
7
, c
9
(u
7
+ 4u
6
+ ··· 2u 1)(u
12
+ 7u
11
+ ··· + 2u
2
+ 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
5
(y
6
7y
5
+ 17y
4
16y
3
+ 6y
2
5y + 1)
2
(y
7
7y
6
+ 17y
5
16y
4
+ 6y
3
+ 3y
2
11y 4)
c
2
, c
3
, c
6
c
8
(y
7
+ 4y
6
+ ··· 2y 1)(y
12
+ 7y
11
+ ··· + 2y
2
+ 1)
c
7
, c
9
(y
7
+ 12y
5
+ ··· 2y 1)(y
12
5y
11
+ ··· + 4y + 1)
7