10
85
(K10a
86
)
1
Arc Sequences
4 5 1 8 9 10 3 2 6 7
Solving Sequence
1,3
4
2,7
8 10 6 9 5
c
3
c
1
c
7
c
10
c
6
c
9
c
5
c
2
, c
4
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu + 1, a + 1, b 1i
I
u
2
= hu
29
+ 2u
28
+ ··· 9u + 1, 2.16918 × 10
16
u
28
3.79023 × 10
17
u
27
+ ··· + 3.02352 × 10
17
a 1.74512 × 10
16
,
2.34491 × 10
17
u
28
+ 1.73875 × 10
17
u
27
+ ··· + 3.02352 × 10
17
b + 6.18559 × 10
16
i
There are 2 irreducible components with 30 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 + 1, a + 1, b 1i
(i) Arc colorings
a
1
=
1
0
a
3
=
0
1
a
4
=
1
1
a
2
=
0
1
a
7
=
1
1
a
8
=
1
0
a
10
=
0
1
a
6
=
1
0
a
9
=
1
1
a
5
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.00000
3.28987 12.0000
3
II. I
u
2
=
hu
29
+2u
28
+· · ·9u+1, 2.17×10
16
u
28
3.79×10
17
u
27
+· · ·+3.02×10
17
a
1.75 × 10
16
, 2.34 × 10
17
u
28
+ 1.74 × 10
17
u
27
+ · · · + 3.02 × 10
17
b + 6.19 × 10
16
i
(i) Arc colorings
a
1
=
1
0
a
3
=
0
u
a
4
=
u
u
a
2
=
u
2
+ 1
u
2
a
7
=
0.0717437u
28
+ 1.25358u
27
+ ··· + 21.5376u + 0.0577183
0.775558u
28
0.575074u
27
+ ··· + 7.16758u 0.204583
a
8
=
0.0717437u
28
+ 1.25358u
27
+ ··· + 21.5376u + 0.0577183
1.41463u
28
0.710391u
27
+ ··· + 17.0867u 1.31468
a
10
=
0.0829529u
28
+ 1.03921u
27
+ ··· + 26.6779u 1.22187
0.731438u
28
0.325885u
27
+ ··· + 9.93497u 0.877342
a
6
=
0.261638u
28
0.178034u
27
+ ··· + 3.68602u + 1.51415
0.470019u
28
+ 0.323811u
27
+ ··· 2.84943u + 0.737809
a
9
=
0.578702u
28
+ 1.89506u
27
+ ··· + 22.6358u + 0.360961
0.966849u
28
0.796401u
27
+ ··· + 11.6636u 0.705536
a
5
=
0.171046u
28
1.43230u
27
+ ··· 29.2191u + 4.37130
2.88979u
28
+ 1.79958u
27
+ ··· 26.2371u + 3.06084
(ii) Obstruction class = 1
(iii) Cusp Shapes =
392058011584073088
302351788217056211
u
28
+
660165514284351834
302351788217056211
u
27
+ ···
1675183498912280626
302351788217056211
u +
90668892332919188
9753283490872781
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.48890 0.50890I
a = 0.257715 + 1.199624I
b = 1.69430 + 0.14926I
15.0062 + 11.7978I 10.91529 5.90770I
u = 1.48890 + 0.50890I
a = 0.257715 1.199624I
b = 1.69430 0.14926I
15.0062 11.7978I 10.91529 + 5.90770I
u = 1.37576 0.41947I
a = 0.229891 1.043788I
b = 0.945899 0.527377I
5.88176 + 9.11420I 9.68617 7.34304I
u = 1.37576 + 0.41947I
a = 0.229891 + 1.043788I
b = 0.945899 + 0.527377I
5.88176 9.11420I 9.68617 + 7.34304I
u = 1.356432 0.153459I
a = 0.404784 0.876034I
b = 1.71056 0.17143I
14.3721 + 3.4330I 13.24262 2.50700I
u = 1.356432 + 0.153459I
a = 0.404784 + 0.876034I
b = 1.71056 + 0.17143I
14.3721 3.4330I 13.24262 + 2 .50700I
u = 1.261298 0.271343I
a = 0.125060 + 0.842813I
b = 0.072018 + 0.813066I
2.76973 + 4.67347I 7.79912 5.64410I
u = 1.261298 + 0.271343I
a = 0.125060 0.842813I
b = 0.072018 0.813066I
2.76973 4.67347I 7.79912 + 5.64410I
u = 1.256043 0.054517I
a = 0.150079 0.757078I
b = 0.919400 0.642162I
5.23846 0.21078I 14.1588 0.1661I
u = 1.256043 + 0.054517I
a = 0.150079 + 0.757078I
b = 0.919400 + 0.642162I
5.23846 + 0.21078I 14.1588 + 0.1661I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.595128
a = 1.32278
b = 1.22499
2.40006 1.29978
u = 0.030189 0.602701I
a = 0.681585 1.151231I
b = 0.066518 0.465152I
1.04879 1.39671I 0.58532 + 2 .57538I
u = 0.030189 + 0.602701I
a = 0.681585 + 1.151231I
b = 0.066518 + 0 .465152I
1.04879 + 1 .39671I 0.58532 2.57538I
u = 0.158430 0.999897I
a = 0.592133 + 0.633857I
b = 0.778126 + 0 .317868I
1.06268 4.13663I 7.22942 + 7.89079I
u = 0.158430 + 0.999897I
a = 0.592133 0.633857I
b = 0.778126 0.317868I
1.06268 + 4.13663I 7.22942 7.89079I
u = 0.176438 0.081936I
a = 3.54961 1.59378I
b = 0.652302 0.186228I
1.260031 0.425153I 8.09157 + 0.86414I
u = 0.176438 + 0.081936I
a = 3.54961 + 1.59378I
b = 0.652302 + 0.186228I
1.260031 + 0.425153I 8.09157 0.86414I
u = 0.237411 1.299692I
a = 0.722991 0.420259I
b = 1.65082 0.07616I
9.53565 5.57255I 9.51799 + 5.65949I
u = 0.237411 + 1.299692I
a = 0.722991 + 0.420259I
b = 1.65082 + 0.07616I
9.53565 + 5.57255I 9.51799 5.65949I
u = 0.290265 0.446237I
a = 2.60984 + 0.41696I
b = 1.62115 + 0 .04828I
9.19873 1.27855I 8.61698 1.60098I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.290265 + 0.446237I
a = 2.60984 0.41696I
b = 1.62115 0.04828I
9.19873 + 1.27855I 8.61698 + 1 .60098I
u = 1.04582
a = 3.43855
b = 0.696417
2.85586 37.1855
u = 1.050707 0.171455I
a = 0.619193 0.870102I
b = 0.175263 0.221780I
1.92999 0.70792I 4.69463 1.16490I
u = 1.050707 + 0.171455I
a = 0.619193 + 0.870102I
b = 0.175263 + 0.221780I
1.92999 + 0.70792I 4.69463 + 1 .16490I
u = 1.17150
a = 3.14643
b = 1.64256
11.1512 4.65638
u = 1.40031 0.49154I
a = 0.001614 + 0.588755I
b = 0.852692 + 0.102964I
4.90102 1.77997I 15.0441 + 4.1634I
u = 1.40031 + 0.49154I
a = 0.001614 0.588755I
b = 0.852692 0.102964I
4.90102 + 1.77997I 15.0441 4.1634I
u = 1.64397 0.65512I
a = 0.164167 0.645212I
b = 1.67141 0.02331I
13.80519 2.24218I 14.8597 + 2.9793I
u = 1.64397 + 0.65512I
a = 0.164167 + 0.645212I
b = 1.67141 + 0.02331I
13.80519 + 2.24218I 14.8597 2.9793I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
(u + 1)(u
29
+ 2u
28
+ ··· 9u + 1)
c
2
(u)(u
29
+ 5u
28
+ ··· + 2u 2)
c
4
(u 1)(u
29
+ 2u
28
+ ··· + u + 1)
c
5
, c
6
, c
9
c
10
(u + 1)(u
29
+ 2u
28
+ ··· 5u 1)
c
7
(u + 1)(u
29
4u
27
+ ··· 277u 173)
c
8
(u 1)(u
29
+ 2u
28
+ ··· 51u 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
(y 1)(y
29
24y
28
+ ··· + 35y 1)
c
2
(y)(y
29
+ 9y
28
+ ··· 8y 4)
c
4
(y 1)(y
29
+ 4y
28
+ ··· y 1)
c
5
(y 1)(y
29
36y
28
+ ··· y 1)
c
6
, c
9
, c
10
(y 1)(y
29
36y
28
+ ··· y 1)
c
7
(y 1)(y
29
8y
28
+ ··· 224637y 29929)
c
8
(y 1)(y
29
36y
28
+ ··· + 2499y 1)
9