8
16
(K8a
15
)
1
Arc Sequences
4 5 1 7 8 3 2 6
Solving Sequence
1,4 2,7
5 3 6 8
c
1
c
4
c
3
c
6
c
8
c
2
, c
5
, c
7
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
5
2u
4
+ 2u
2
u + 1, u
4
u
3
+ b, u
3
+ u
2
+ a + u 1i
I
u
2
= hu
12
+ u
11
4u
10
5u
9
+ 4u
8
+ 7u
7
+ 7u
6
u
5
16u
4
6u
3
+ 8u
2
+ 4u + 1,
90u
11
+ 30u
10
+ ··· + 749a 1363, 64u
11
478u
10
+ ··· + 749b + 96i
There are 2 irreducible components with 17 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
5
2u
4
+ 2u
2
u + 1, u
4
u
3
+ b, u
3
+ u
2
+ a + u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
7
=
u
3
u
2
u + 1
u
4
+ u
3
a
5
=
u
2
1
u
3
+ u
a
3
=
u
u
a
6
=
u
4
+ 2u
3
u
2
+ 1
u
a
8
=
u
3
2u
2
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
+ 4u
3
+ 4u 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.09851
a = 0.433797
b = 2.78175
3.68417 17.5210
u = 0.131705 0.621876I
a = 1.08716 + 0.99382I
b = 0.260129 + 0.087120I
1.32133 + 1.30034I 2.51370 2.13902I
u = 0.131705 + 0.621876I
a = 1.08716 0.99382I
b = 0.260129 0.087120I
1.32133 1.30034I 2.51370 + 2.13902I
u = 1.41755 0.49337I
a = 0.370261 0.961987I
b = 0.65101 + 2.08638I
6.88145 10.57902I 6.27422 + 6.37200I
u = 1.41755 + 0.49337I
a = 0.370261 + 0.961987I
b = 0.65101 2.08638I
6.88145 + 10.57902I 6.27422 6.37200I
3
II. I
u
2
= hu
12
+ u
11
+ · · · + 4u + 1, 90u
11
+ 30u
10
+ · · · + 749a
1363, 64u
11
478u
10
+ · · · + 749b + 96i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
7
=
0.120160u
11
0.0400534u
10
+ ··· 0.950601u + 1.81976
0.0854473u
11
+ 0.638184u
10
+ ··· 4.32043u 0.128171
a
5
=
1.24433u
11
2.08144u
10
+ ··· 2.66622u + 0.133511
0.837116u
11
+ 2.27904u
10
+ ··· 5.11081u 1.24433
a
3
=
u
u
a
6
=
0.0106809u
11
+ 0.670227u
10
+ ··· 3.95995u + 1.01602
0.216288u
11
0.0720961u
10
+ ··· 1.31108u + 0.675567
a
8
=
0.259012u
11
+ 0.246996u
10
+ ··· 5.47130u + 1.61148
0.134846u
11
+ 0.288385u
10
+ ··· 2.75567u + 0.297730
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1992
749
u
11
3660
749
u
10
+
4416
749
u
9
+
13008
749
u
8
+
4332
749
u
7
7844
749
u
6
19700
749
u
5
16832
749
u
4
+
11136
749
u
3
+
19508
749
u
2
+
6012
749
u +
4502
749
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.52069 0.58643I
a = 0.062603 + 0.510527I
b = 0.032442 1.116676I
6.04826 + 2.02988I 11.01951 3.46410I
u = 1.52069 + 0.58643I
a = 0.062603 0.510527I
b = 0.032442 + 1.116676I
6.04826 2.02988I 11.01951 + 3.46410I
u = 1.029010 0.216402I
a = 0.097176 0.542147I
b = 0.537643 + 1.250254I
1.91067 + 0.79824I 4.49024 + 0.48465I
u = 1.029010 + 0.216402I
a = 0.097176 + 0.542147I
b = 0.537643 1.250254I
1.91067 0.79824I 4.49024 0.48465I
u = 0.210547 0.250904I
a = 2.08966 + 0.00606I
b = 0.916994 + 0.524193I
1.91067 + 0.79824I 4.49024 + 0.48465I
u = 0.210547 + 0.250904I
a = 2.08966 0.00606I
b = 0.916994 0.524193I
1.91067 0.79824I 4.49024 0.48465I
u = 0.167732 1.153853I
a = 0.564253 0.648859I
b = 0.109077 + 0.117649I
1.91067 + 4.85801I 4.49024 6.44355I
u = 0.167732 + 1.153853I
a = 0.564253 + 0.648859I
b = 0.109077 0.117649I
1.91067 4.85801I 4.49024 + 6.44355I
u = 1.192209 0.314018I
a = 0.487055 + 1.246751I
b = 0.70032 2.02933I
1.91067 4.85801I 4.49024 + 6.44355I
u = 1.192209 + 0.314018I
a = 0.487055 1.246751I
b = 0.70032 + 2.02933I
1.91067 + 4.85801I 4.49024 6.44355I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.235769 0.092938I
a = 1.17224 + 0.88306I
b = 0.82232 1.61017I
6.04826 2.02988I 11.01951 + 3.46410I
u = 1.235769 + 0.092938I
a = 1.17224 0.88306I
b = 0.82232 + 1.61017I
6.04826 + 2.02988I 11.01951 3.46410I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
5
c
8
(u
5
+ 2u
4
2u
2
u 1)(u
12
u
11
+ ··· 4u + 1)
c
2
, c
4
(u
5
2u
2
+ 3u 1)(u
12
+ 3u
11
+ ··· + 4u + 1)
c
6
(u
2
u + 1)
6
(u
5
+ 7u
4
+ 19u
3
+ 30u
2
+ 24u + 8)
c
7
(u
3
u
2
+ 1)
4
(u
5
+ 7u
4
+ 18u
3
+ 23u
2
+ 14u + 4)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
5
c
8
(y
5
4y
4
+ ··· 3y 1)(y
12
9y
11
+ ··· + 80y
2
+ 1)
c
2
, c
4
(y
5
+ 6y
3
+ ··· + 5y 1)(y
12
+ 3y
11
+ ··· + 8y + 1)
c
6
(y
2
+ y + 1)
6
(y
5
11y
4
11y
3
100y
2
+ 96y 64)
c
7
(y
3
y
2
+ 2y 1)
4
(y
5
13y
4
+ 30y
3
81y
2
+ 12y 16)
8