11n
67
(K11n
67
)
1
Arc Sequences
4 1 9 2 10 9 11 3 6 1 7
Solving Sequence
3,8
9
4,11
7 1 2 5 6 10
c
8
c
3
c
7
c
11
c
2
c
4
c
6
c
10
c
1
, c
5
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
6
3u
4
+ 2u
2
+ 1, u
4
2u
2
+ a + 1, u
5
+ 3u
3
u
2
+ b 2u + 2i
I
u
2
= hu
14
+ 8u
13
+ ··· + 80u + 64,
1245614045971u
13
9003912184492u
12
+ ··· + 402125952084352a 110658132853216,
1648650555445u
13
12281755209843u
12
+ ··· + 201062976042176b 8756113999552i
I
v
1
= h2b
4
+ 5b
3
+ 6b
2
+ 3b + 1, 4b
3
+ 10b
2
+ 10b + v + 3, ai
There are 3 irreducible components with 24 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
6
3u
4
+ 2u
2
+ 1, u
4
2u
2
+ a + 1, u
5
+ 3u
3
u
2
+ b 2u + 2i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
9
=
u
u
a
4
=
u
2
+ 1
u
2
a
11
=
u
4
+ 2u
2
1
u
5
3u
3
+ u
2
+ 2u 2
a
7
=
u
5
+ 3u
3
2u
u
5
u
4
+ 3u
3
+ 2u
2
2u 1
a
1
=
0
u
4
u
2
1
a
2
=
1
u
2
a
5
=
u
4
+ u
2
+ 1
u
4
+ u
2
+ 1
a
6
=
u
5
u
4
+ 3u
3
+ u
2
2u + 1
u
5
2u
4
+ 3u
3
+ 3u
2
2u
a
10
=
u
4
+ 2u
2
1
u
5
u
4
3u
3
+ 2u
2
+ 2u 1
a
10
=
u
4
+ 2u
2
1
u
5
u
4
3u
3
+ 2u
2
+ 2u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.307141 0.215080I
a = 0.122561 0.744862I
b = 0.407221 + 0.439718I
3.02413 + 2.82812I 0.49024 2.97945I
u = 1.307141 + 0.215080I
a = 0.122561 + 0.744862I
b = 0.407221 0.439718I
3.02413 2.82812I 0.49024 + 2.97945I
u = 0.569840I
a = 1.75488
b = 2.32472 1.75488I
1.11345 7.01951
u = 0.569840I
a = 1.75488
b = 2.32472 + 1.75488I
1.11345 7.01951
u = 1.307141 0.215080I
a = 0.122561 + 0.744862I
b = 1.082503 0.684841I
3.02413 2.82812I 0.49024 + 2.97945I
u = 1.307141 + 0.215080I
a = 0.122561 0.744862I
b = 1.082503 + 0.684841I
3.02413 + 2.82812I 0.49024 2.97945I
3
II. I
u
2
=
hu
14
+8u
13
+· · ·+80u+64, 1.25×10
12
u
13
9.00×10
12
u
12
+· · ·+4.02×10
14
a
1.11 × 10
14
, 1.65× 10
12
u
13
1.23 ×10
13
u
12
+ · · · +2.01 × 10
14
b 8.76 × 10
12
i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
9
=
u
u
a
4
=
u
2
+ 1
u
2
a
11
=
0.00309757u
13
+ 0.0223908u
12
+ ··· + 1.25004u + 0.275183
0.00819967u
13
+ 0.0610841u
12
+ ··· + 1.75934u + 0.0435491
a
7
=
0.00528928u
13
0.0422327u
12
+ ··· 0.631112u 0.962607
0.00258478u
13
0.0234953u
12
+ ··· + 1.11393u 0.669300
a
1
=
0.00358617u
13
+ 0.0252008u
12
+ ··· + 0.982198u + 0.266347
0.00283357u
13
+ 0.0184213u
12
+ ··· + 0.863729u 0.364400
a
2
=
0.00285456u
13
+ 0.0222774u
12
+ ··· + 0.382633u + 0.879103
0.00357770u
13
+ 0.0241819u
12
+ ··· + 1.01903u 0.164599
a
5
=
0.00242537u
13
0.0186477u
12
+ ··· + 0.0688957u + 0.407477
0.00601154u
13
0.0438485u
12
+ ··· 0.913303u + 0.141130
a
6
=
0.00125179u
13
0.0106673u
12
+ ··· 0.572311u 0.777096
0.00145271u
13
+ 0.00807009u
12
+ ··· + 1.17273u 0.483788
a
10
=
0.00129492u
13
+ 0.00726798u
12
+ ··· + 1.06422u + 0.119244
0.00577594u
13
+ 0.0445625u
12
+ ··· + 1.18034u + 0.00752302
a
10
=
0.00129492u
13
+ 0.00726798u
12
+ ··· + 1.06422u + 0.119244
0.00577594u
13
+ 0.0445625u
12
+ ··· + 1.18034u + 0.00752302
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 3.29126 0.20537I
a = 0.526182 + 0.474565I
b = 1.73878 0.18839I
17.5696 0.3825I 0.199766 + 0.045547I
u = 3.29126 + 0.20537I
a = 0.526182 0.474565I
b = 1.73878 + 0.18839I
17.5696 + 0.3825I 0.199766 0.045547I
u = 2.61296 0.90964I
a = 0.626029 0.276929I
b = 1.67859 + 0.02512I
9.20003 2.06852I 0.364251 + 1.127832I
u = 2.61296 + 0.90964I
a = 0.626029 + 0.276929I
b = 1.67859 0.02512I
9.20003 + 2.06852I 0.364251 1.127832I
u = 0.967277 0.435469I
a = 0.045288 + 1.073159I
b = 0.403328 0.434208I
0.02319 2.21939I 1.77809 + 3.53992I
u = 0.967277 + 0.435469I
a = 0.045288 1.073159I
b = 0.403328 + 0.434208I
0.02319 + 2.21939I 1.77809 3.53992I
u = 0.397047 0.435424I
a = 0.759045 0.931962I
b = 0.91950 1.34836I
2.12302 0.75753I 7.75042 3.06748I
u = 0.397047 + 0.435424I
a = 0.759045 + 0.931962I
b = 0.91950 + 1.34836I
2.12302 + 0.75753I 7.75042 + 3.06748I
u = 0.254175 0.519094I
a = 1.039701 + 0.048648I
b = 0.016234 0.474107I
0.212568 1.285476I 1.55268 + 6.08941I
u = 0.254175 + 0.519094I
a = 1.039701 0.048648I
b = 0.016234 + 0.474107I
0.212568 + 1.285476I 1.55268 6.08941I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.919489 0.424732I
a = 0.929806 + 0.312103I
b = 0.581539 0.520300I
5.01039 4.24504I 1.99936 + 6.80413I
u = 0.919489 + 0.424732I
a = 0.929806 0.312103I
b = 0.581539 + 0.520300I
5.01039 + 4.24504I 1.99936 6.80413I
u = 2.09487 1.16319I
a = 0.719591 0.363540I
b = 2.23079 + 0.02826I
19.4276 10.6503I 1.06301 + 4.03963I
u = 2.09487 + 1.16319I
a = 0.719591 + 0.363540I
b = 2.23079 0.02826I
19.4276 + 10.6503I 1.06301 4.03963I
6
III. I
v
1
= h2b
4
+ 5b
3
+ 6b
2
+ 3b + 1, 4b
3
+ 10b
2
+ 10b + v + 3, ai
(i) Arc colorings
a
3
=
1
0
a
8
=
4b
3
10b
2
10b 3
0
a
9
=
4b
3
10b
2
10b 3
0
a
4
=
1
0
a
11
=
0
b
a
7
=
4b
3
10b
2
10b 3
2b
3
+ 3b
2
+ 2b
a
1
=
4b
3
8b
2
7b
1
a
2
=
4b
3
8b
2
7b + 1
1
a
5
=
4b
3
+ 8b
2
+ 7b
1
a
6
=
6b
3
11b
2
8b + 1
2b
3
+ 3b
2
+ 2b
a
10
=
4b
3
+ 12b
2
+ 15b + 7
2b
3
5b
2
5b 2
a
10
=
4b
3
+ 12b
2
+ 15b + 7
2b
3
5b
2
5b 2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 0.152300 0.614030I
a = 0
b = 0.971274 0.813859I
1.85594 1.41510I 3.26394 + 5.88934I
v = 0.152300 + 0.614030I
a = 0
b = 0.971274 + 0.813859I
1.85594 + 1.41510I 3.26394 5.88934I
v = 0.65230 + 2.13814I
a = 0
b = 0.278726 0.483420I
5.14581 3.16396I 2.13894 0.11292I
v = 0.65230 2.13814I
a = 0
b = 0.278726 + 0.483420I
5.14581 + 3.16396I 2.13894 + 0.11292I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
3
+ u
2
1)
2
(u
14
+ 7u
13
+ ··· 3u + 4)
c
2
(u + 1)
4
(1 + 2u + u
2
+ u
3
)
2
(u
14
+ 3u
13
+ ··· + 127u + 16)
c
3
, c
8
u
4
(u
6
3u
4
+ 2u
2
+ 1)(u
14
+ 8u
13
+ ··· + 80u + 64)
c
4
(u + 1)
4
(u
3
u
2
+ 1)
2
(u
14
+ 7u
13
+ ··· 3u + 4)
c
5
, c
6
(u
2
+ 1)
3
(u
4
u
3
+ ··· 2u + 1)(u
14
+ 2u
13
+ ··· 24u + 17)
c
7
(u
2
+ 1)
3
(u
4
u
3
+ u
2
+ 1)(u
14
+ 2u
13
+ ··· + 12u + 17)
c
9
(u
2
+ 1)
3
(u
4
+ u
3
+ ··· + 2u + 1)(u
14
+ 2u
13
+ ··· 24u + 17)
c
10
(u 1)
6
(u
4
u
3
+ ··· 2u + 1)(u
14
+ 4u
13
+ ··· 2066u + 289)
c
11
(u
2
+ 1)
3
(u
4
+ u
3
+ u
2
+ 1)(u
14
+ 2u
13
+ ··· + 12u + 17)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
4
(1 + 2y y
2
+ y
3
)
2
(y
14
+ 3y
13
+ ··· + 127y + 16)
c
2
(y 1)
4
(1 + 2y + 3y
2
+ y
3
)
2
(y
14
+ 47y
13
+ ··· + 32223y + 256)
c
3
, c
8
y
4
(1 + 2y 3y
2
+ y
3
)
2
(y
14
42y
13
+ ··· + 4864y + 4096)
c
5
, c
6
, c
9
(y + 1)
6
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
14
+ 28y
13
+ ··· + 2994y + 289)
c
7
, c
11
(y + 1)
6
(y
4
+ y
3
+ ··· + 2y + 1)(y
14
4y
13
+ ··· + 2066y + 289)
c
10
(y 1)
6
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
(y
14
+ 52y
13
+ ··· + 189758y + 83521)
10