10
77
(K10a
18
)
1
Arc Sequences
4 8 5 2 10 1 9 3 7 6
Solving Sequence
5,10
6
1,4
2 7 3 9 8
c
5
c
10
c
1
c
6
c
3
c
9
c
8
c
2
, c
4
, c
7
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu 1, b + 1, a 1i
I
u
2
= hu
9
6u
8
+ 15u
7
17u
6
+ 3u
5
+ 12u
4
9u
3
u
2
+ 2u 1, u
7
5u
6
+ 10u
5
8u
4
u
3
+ 5u
2
+ a u 1,
u
8
+ 5u
7
10u
6
+ 8u
5
+ u
4
5u
3
+ u
2
+ b + ui
I
u
3
= hu
23
12u
22
+ ··· + 7u 1, 9375u
22
108217u
21
+ ··· + 5969b + 14071,
12427u
22
+ 145111u
21
+ ··· + 11938a 55091i
There are 3 irreducible components with 33 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, b + 1, a 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
1
1
a
6
=
1
0
a
1
=
0
1
a
4
=
1
0
a
2
=
1
1
a
7
=
1
1
a
3
=
1
1
a
9
=
1
1
a
8
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
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
0 0
3
II. I
u
2
= hu
9
6u
8
+ · · · + 2u 1, u
7
5u
6
+ 10u
5
8u
4
u
3
+ 5u
2
+ a
u 1, u
8
+ 5u
7
+ · · · + b + ui
(i) Arc colorings
a
5
=
0
u
a
10
=
u
7
+ 5u
6
10u
5
+ 8u
4
+ u
3
5u
2
+ u + 1
u
8
5u
7
+ 10u
6
8u
5
u
4
+ 5u
3
u
2
u
a
6
=
1
0
a
1
=
u
8
6u
7
+ 15u
6
18u
5
+ 7u
4
+ 6u
3
6u
2
+ 1
u
8
5u
7
+ 10u
6
8u
5
u
4
+ 5u
3
u
2
u
a
4
=
1
0
a
2
=
u
7
+ 5u
6
10u
5
+ 8u
4
+ u
3
5u
2
+ u + 1
u
8
5u
7
+ 10u
6
8u
5
u
4
+ 5u
3
u
2
u
a
7
=
u + 2
u
a
3
=
1
u
2
a
9
=
u
6
+ 4u
5
7u
4
+ 6u
3
2u
2
2u + 3
u
8
4u
7
+ 6u
6
2u
5
4u
4
+ 4u
3
2u
a
8
=
u
4
+ 4u
3
5u
2
+ 3
u
4
+ 2u
3
2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
6
16u
5
+ 24u
4
8u
3
12u
2
+ 8u + 6
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.639060 0.117940I
a = 0.113036 + 1.235328I
b = 0.073457 + 0.802780I
3.02413 2.82812I 2.49024 + 2.97945I
u = 0.639060 + 0.117940I
a = 0.113036 1.235328I
b = 0.073457 0.802780I
3.02413 + 2.82812I 2.49024 2.97945I
u = 0.265002 0.388164I
a = 1.28983 + 0.68118I
b = 0.606217 + 0.320153I
1.11345 9.01951
u = 0.265002 + 0.388164I
a = 1.28983 0.68118I
b = 0.606217 0.320153I
1.11345 9.01951
u = 1.20097 1.03313I
a = 0.744906 + 0.276315I
b = 1.180077 + 0.437737I
3.02413 + 2.82812I 2.49024 2.97945I
u = 1.20097 + 1.03313I
a = 0.744906 0.276315I
b = 1.180077 0.437737I
3.02413 2.82812I 2.49024 + 2.97945I
u = 1.43809 0.91519I
a = 0.735381 0.214151I
b = 1.253534 0.365043I
3.02413 2.82812I 2.49024 + 2.97945I
u = 1.43809 + 0.91519I
a = 0.735381 + 0.214151I
b = 1.253534 + 0.365043I
3.02413 + 2.82812I 2.49024 2.97945I
u = 1.47000
a = 0.824787
b = 1.21243
1.11345 9.01951
5
III. I
u
3
= hu
23
12u
22
+ · · · + 7u 1, 9375u
22
108217u
21
+ · · · + 5969b +
14071, 1.24 × 10
4
u
22
+ 1.45 × 10
5
u
21
+ · · · + 1.19 × 10
4
a 5.51 × 10
4
i
(i) Arc colorings
a
5
=
0
u
a
10
=
1.04096u
22
12.1554u
21
+ ··· 4.58201u + 4.61476
1.57061u
22
+ 18.1298u
21
+ ··· + 12.3408u 2.35735
a
6
=
0.836153u
22
9.37846u
21
+ ··· + 1.32803u + 2.54096
1.56215u
22
+ 18.4945u
21
+ ··· + 14.3248u 2.23446
a
1
=
0.590384u
22
7.44614u
21
+ ··· 12.1799u + 3.85240
0.172307u
22
2.25691u
21
+ ··· 1.34394u 0.418077
a
4
=
1
0
a
2
=
0.418077u
22
5.18923u
21
+ ··· 10.8360u + 4.27048
0.172307u
22
2.25691u
21
+ ··· 1.34394u 0.418077
a
7
=
1.65539u
22
19.4862u
21
+ ··· 11.3121u + 5.83615
0.906768u
22
+ 11.0083u
21
+ ··· + 12.0127u 2.39831
a
3
=
1
u
2
a
9
=
1.71595u
22
19.9552u
21
+ ··· 15.9211u + 7.69601
1.17524u
22
+ 13.8038u
21
+ ··· + 10.5574u 2.45619
a
8
=
2.93600u
22
33.9760u
21
+ ··· 23.7410u + 9.51600
1.19610u
22
+ 14.6375u
21
+ ··· + 13.6758u 3.07598
(ii) Obstruction class = 1
(iii) Cusp Shapes =
37028
5969
u
22
413378
5969
u
21
+ ···
191046
5969
u +
92216
5969
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.724852 0.256047I
a = 1.46390 1.00680I
b = 0.265783 1.134045I
1.33811 7.00485I 7.04339 + 5.13787I
u = 0.724852 + 0.256047I
a = 1.46390 + 1.00680I
b = 0.265783 + 1.134045I
1.33811 + 7.00485I 7.04339 5.13787I
u = 0.517401 0.075244I
a = 2.18707 + 0.96872I
b = 0.409400 + 0.905706I
0.429871 1.292378I 5.93678 + 0.45977I
u = 0.517401 + 0.075244I
a = 2.18707 0.96872I
b = 0.409400 0.905706I
0.429871 + 1.292378I 5.93678 0.45977I
u = 0.169134 0.760322I
a = 0.27717 1.73915I
b = 0.76382 1.77584I
6.97398 1.20490I 11.80214 + 0.58796I
u = 0.169134 + 0.760322I
a = 0.27717 + 1.73915I
b = 0.76382 + 1.77584I
6.97398 + 1.20490I 11.80214 0.58796I
u = 0.199940
a = 3.49308
b = 0.600331
1.01631 10.3718
u = 0.431420 0.662786I
a = 0.79226 + 1.38758I
b = 1.52923 + 2.58060I
2.49785 + 1.83570I 6.37573 3.60335I
u = 0.431420 + 0.662786I
a = 0.79226 1.38758I
b = 1.52923 2.58060I
2.49785 1.83570I 6.37573 + 3.60335I
u = 0.529906 1.130869I
a = 0.346261 1.056189I
b = 0.68241 2.74013I
5.85182 + 6.12354I 9.22962 6.59776I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.529906 + 1.130869I
a = 0.346261 + 1.056189I
b = 0.68241 + 2.74013I
5.85182 6.12354I 9.22962 + 6.59776I
u = 0.595376 0.789505I
a = 0.236651 + 0.660824I
b = 0.284932 + 0.304128I
0.26922 + 3.59706I 4.75645 7.79597I
u = 0.595376 + 0.789505I
a = 0.236651 0.660824I
b = 0.284932 0.304128I
0.26922 3.59706I 4.75645 + 7.79597I
u = 0.864881 0.315143I
a = 0.051470 0.334706I
b = 0.714844 0.282502I
1.67067 + 0.60932I 3.84266 0.84402I
u = 0.864881 + 0.315143I
a = 0.051470 + 0.334706I
b = 0.714844 + 0.282502I
1.67067 0.60932I 3.84266 + 0.84402I
u = 1.17508 1.11119I
a = 0.421192 + 0.709925I
b = 0.18568 + 3.19242I
2.78844 + 5.69706I 2.62032 4.06061I
u = 1.17508 + 1.11119I
a = 0.421192 0.709925I
b = 0.18568 3.19242I
2.78844 5.69706I 2.62032 + 4.06061I
u = 1.19423 1.28438I
a = 0.356000 0.711972I
b = 0.16442 3.05178I
1.85559 + 12.07471I 3.82521 8.06520I
u = 1.19423 + 1.28438I
a = 0.356000 + 0.711972I
b = 0.16442 + 3.05178I
1.85559 12.07471I 3.82521 + 8.06520I
u = 1.21632 1.17599I
a = 0.319423 + 0.411130I
b = 0.111007 + 0.809298I
6.35503 + 7.52364I 0.34364 6.02284I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.21632 + 1.17599I
a = 0.319423 0.411130I
b = 0.111007 0.809298I
6.35503 7.52364I 0.34364 + 6.02284I
u = 1.30420 1.02033I
a = 0.295814 0.396904I
b = 0.229133 0.874444I
6.84422 + 1.43226I 1.58922 0.72835I
u = 1.30420 + 1.02033I
a = 0.295814 + 0.396904I
b = 0.229133 + 0.874444I
6.84422 1.43226I 1.58922 + 0.72835I
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)(u
9
3u
7
u
6
+ 3u
5
+ 2u
4
+ u
3
u
2
2u 1)
(u
23
+ 2u
22
+ ··· + 3u + 1)
c
2
, c
8
(u)(u
3
+ u
2
1)
3
(u
23
2u
22
+ ··· + 2u 2)
c
3
(u 1)(u
9
+ 6u
8
+ 15u
7
+ 17u
6
+ 3u
5
12u
4
9u
3
+ u
2
+ 2u + 1)
(u
23
+ 12u
22
+ ··· + 7u + 1)
c
4
(u + 1)(u
9
3u
7
u
6
+ 3u
5
+ 2u
4
+ u
3
u
2
2u 1)
(u
23
+ 2u
22
+ ··· + 3u + 1)
c
5
, c
6
(u + 1)(u
9
3u
7
+ u
6
+ 3u
5
2u
4
+ u
3
+ u
2
2u + 1)
(u
23
+ 2u
22
+ ··· u 1)
c
7
, c
9
(u)(1 + 2u + u
2
+ u
3
)
3
(u
23
+ 6u
22
+ ··· + 8u + 4)
c
10
(u 1)(u
9
3u
7
+ u
6
+ 3u
5
2u
4
+ u
3
+ u
2
2u + 1)
(u
23
+ 2u
22
+ ··· u 1)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)(y
9
6y
8
+ 15y
7
17y
6
+ 3y
5
+ 12y
4
9y
3
y
2
+ 2y 1)
(y
23
12y
22
+ ··· + 7y 1)
c
2
1y(1y
3
y
2
+ 2.00000y 1.00000)
3
(1y
23
6.00000y
22
+ ··· + 8.0000y 4.00000)
c
3
(y 1)
(y
9
6y
8
+ 27y
7
73y
6
+ 139y
5
184y
4
+ 83y
3
13y
2
+ 2y 1)
(y
23
+ 32y
21
+ ··· + 31y 1)
c
5
(y 1)(y
9
6y
8
+ 15y
7
17y
6
+ 3y
5
+ 12y
4
9y
3
y
2
+ 2y 1)
(y
23
20y
22
+ ··· 9y 1)
c
6
+ 1.00000(1y 1.00000)
(1y
9
6.0000y
8
+ ··· + 2.00000y 1.00000)
(1y
23
20.0000y
22
+ ··· 9.0000y 1.00000)
c
7
, c
9
(y)(1 + 2y + 3y
2
+ y
3
)
3
(y
23
+ 18y
22
+ ··· 8y 16)
c
8
(y)(1 + 2y y
2
+ y
3
)
3
(y
23
6y
22
+ ··· + 8y 4)
c
10
+ 1.00000(1y 1.000000)
(1y
9
6.00000y
8
+ ··· + 2.00000y 1.000000)
(1y
23
20.0000y
22
+ ··· 9.00000y 1.000000)
11