11n
138
(K11n
138
)
1
Arc Sequences
8 1 9 8 10 11 2 4 1 6 5
Solving Sequence
5,10
6 11
1,8
2 3 4 7 9
c
5
c
10
c
11
c
1
c
2
c
4
c
7
c
9
c
3
, c
6
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
4
+ u
3
+ 4u
2
+ 5, u
3
+ 3b u + 1, u
3
u
2
+ 5a 4ui
I
u
2
= hb
6
3b
4
+ 2b
2
+ 1, b
5
+ 2b
3
b + u, b
5
+ b
4
+ 2b
3
2b
2
b + ai
I
u
3
= hu
9
+ 9u
7
3u
6
+ 23u
5
15u
4
+ 7u
3
5u
2
1,
u
8
+ u
7
8u
6
+ 11u
5
18u
4
+ 27u
3
4u
2
+ 8a 15u 1,
3u
8
u
7
+ 26u
6
19u
5
+ 62u
4
71u
3
+ 10u
2
+ 8b 13u + 3i
There are 3 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
4
+ u
3
+ 4u
2
+ 5, u
3
+ 3b u + 1, u
3
u
2
+ 5a 4ui
(i) Arc colorings
a
5
=
1
0
a
10
=
1
5
u
3
+
1
5
u
2
+
4
5
u
1
3
u
3
+
1
3
u
1
3
a
6
=
1
15
u
3
2
5
u
2
4
15
u +
2
3
1
3
u
3
+
1
3
u +
2
3
a
11
=
4
15
u
3
+
2
5
u
2
1
15
u +
2
3
1
3
u
3
1
3
u
2
3
a
1
=
1
15
u
3
+
2
5
u
2
+
4
15
u +
4
3
1
3
u
3
1
3
u
2
3
a
8
=
0
u
a
2
=
1
15
u
3
+
2
5
u
2
+
4
15
u +
4
3
1
a
3
=
u
2
1
u
3
+ 2u
2
+ 5
a
4
=
1
u
2
a
7
=
8
15
u
3
1
5
u
2
17
15
u +
1
3
1
3
u
3
1
3
u +
1
3
a
9
=
u
u
3
+ u
a
9
=
u
u
3
+ u
(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 = 0.80902 1.72149I
a = 0.223607 0.475808I
b = 1.61803
12.1725 6.00000
u = 0.80902 + 1.72149I
a = 0.223607 + 0.475808I
b = 1.61803
12.1725 6.00000
u = 0.309017 1.134229I
a = 0.223607 0.820736I
b = 0.618034
4.27683 6.00000
u = 0.309017 + 1.134229I
a = 0.223607 + 0.820736I
b = 0.618034
4.27683 6.00000
3
II. I
u
2
= hb
6
3b
4
+ 2b
2
+ 1, b
5
+ 2b
3
b + u, b
5
+ b
4
+ 2b
3
2b
2
b + ai
(i) Arc colorings
a
5
=
1
0
a
10
=
b
5
b
4
2b
3
+ 2b
2
+ b
b
a
6
=
b
5
+ b
4
+ 2b
3
b
2
b
2
a
11
=
b
3
+ b 1
b
3
+ b
a
1
=
1
b
3
+ b
a
8
=
0
b
5
2b
3
+ b
a
2
=
1
b
3
+ b 1
a
3
=
0
1
a
4
=
1
1
a
7
=
b
5
+ 2b
3
b
b
4
+ 2b
2
a
9
=
b
5
2b
3
+ b
0
a
9
=
b
5
2b
3
+ b
0
(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 = 1.00000I
a = 0.87744 1.74486I
b = 1.307141 0.215080I
6.31400 2.82812I 7.50976 + 2.97945I
u = 1.00000I
a = 0.87744 + 1.74486I
b = 1.307141 + 0.215080I
6.31400 + 2.82812I 7.50976 2.97945I
u = 1.00000I
a = 0.754878 1.000000I
b = 0.569840I
2.17641 0.980489
u = 1.00000I
a = 0.754878 + 1.000000I
b = 0.569840I
2.17641 0.980489
u = 1.00000I
a = 0.877439 0.255138I
b = 1.307141 0.215080I
6.31400 + 2.82812I 7.50976 2.97945I
u = 1.00000I
a = 0.877439 + 0.255138I
b = 1.307141 + 0.215080I
6.31400 2.82812I 7.50976 + 2.97945I
5
III. I
u
3
= hu
9
+ 9u
7
3u
6
+ 23u
5
15u
4
+ 7u
3
5u
2
1, u
8
+ u
7
+ · · · +
8a 1, 3u
8
u
7
+ · · · + 8b + 3i
(i) Arc colorings
a
5
=
1
0
a
10
=
1
8
u
8
1
8
u
7
+ ··· +
15
8
u +
1
8
3
8
u
8
+
1
8
u
7
+ ··· +
13
8
u
3
8
a
6
=
1
8
u
8
+
3
8
u
7
+ ···
7
8
u
1
8
5
8
u
8
+
1
8
u
7
+ ···
5
8
u +
3
8
a
11
=
1
8
u
8
+
1
8
u
7
+ ···
1
8
u
9
8
1
8
u
8
+
1
8
u
7
+ ···
1
8
u
1
8
a
1
=
1
1
8
u
8
+
1
8
u
7
+ ···
1
8
u
1
8
a
8
=
0
u
a
2
=
1
1
8
u
8
+
1
8
u
7
+ ···
1
8
u
1
8
a
3
=
u
2
1
u
4
2u
2
a
4
=
1
u
2
a
7
=
u
1
8
u
8
1
8
u
7
+ ··· +
7
8
u +
1
8
a
9
=
u
u
3
+ u
a
9
=
u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.46861 2.14498I
a = 0.352513 + 1.059132I
b = 1.60286 + 0.34454I
14.7600 + 7.7767I 5.49011 2.86525I
u = 0.46861 + 2.14498I
a = 0.352513 1.059132I
b = 1.60286 0.34454I
14.7600 7.7767I 5.49011 + 2.86525I
u = 0.198901 0.378443I
a = 0.569562 0.748755I
b = 0.170870 0.458690I
0.131099 + 0.964036I 2.44921 7.22651I
u = 0.198901 + 0.378443I
a = 0.569562 + 0.748755I
b = 0.170870 + 0.458690I
0.131099 0.964036I 2.44921 + 7.22651I
u = 0.14689 2.12129I
a = 0.129905 + 1.196459I
b = 0.598501 + 0.925231I
17.5620 3.0332I 3.21143 + 2.16261I
u = 0.14689 + 2.12129I
a = 0.129905 1.196459I
b = 0.598501 0.925231I
17.5620 + 3.0332I 3.21143 2.16261I
u = 0.159982 0.567821I
a = 0.81772 1.40375I
b = 1.351862 0.195890I
4.68408 3.45373I 2.44779 + 5.78928I
u = 0.159982 + 0.567821I
a = 0.81772 + 1.40375I
b = 1.351862 + 0.195890I
4.68408 + 3.45373I 2.44779 5.78928I
u = 0.721273
a = 0.948897
b = 1.37071
3.38429 0.599763
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
3
(u
4
+ u
3
+ 4u
2
+ 5)
(u
9
+ 9u
7
3u
6
+ 23u
5
15u
4
+ 7u
3
5u
2
1)
c
2
(u + 1)
6
(u
4
+ 7u
3
+ ··· + 40u + 25)(u
9
+ 18u
8
+ ··· 10u 1)
c
5
, c
6
, c
10
(u
2
u 1)
2
(u
6
3u
4
+ 2u
2
+ 1)
(u
9
+ 3u
8
5u
6
+ u
5
+ 2u
4
9u
3
5u
2
+ u 2)
c
9
(u
2
+ u 1)
2
(u
3
u
2
+ 1)
2
(u
9
u
8
+ 22u
7
19u
6
+ 127u
5
84u
4
+ 67u
3
+ 41u
2
+ 23u 8)
c
11
(u
2
3u + 1)
2
(u
6
+ u
4
+ 2u
2
+ 1)
(u
9
+ 9u
8
+ 38u
7
+ 85u
6
+ 87u
5
18u
4
147u
3
167u
2
85u 26)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
7
, c
8
(y + 1)
6
(y
4
+ 7y
3
+ ··· + 40y + 25)(y
9
+ 18y
8
+ ··· 10y 1)
c
2
(y 1)
6
(y
4
+ 3y
3
+ ··· 300y + 625)(y
9
70y
8
+ ··· 10y 1)
c
5
, c
6
, c
10
(y
2
3y + 1)
2
(y
3
3y
2
+ 2y + 1)
2
(y
9
9y
8
+ 32y
7
55y
6
+ 53y
5
60y
4
+ 83y
3
35y
2
19y 4)
c
9
(y
2
3y + 1)
2
(1 + 2y y
2
+ y
3
)
2
(y
9
+ 43y
8
+ ··· + 1185y 64)
c
11
(y
2
7y + 1)
2
(1 + 2y + y
2
+ y
3
)
2
(y
9
5y
8
+ ··· 1459y 676)
9