11n
49
(K11n
49
)
1
Arc Sequences
6 1 7 9 2 10 1 11 6 4 8
Solving Sequence
1,7 8,10
6 2 5 9 4 3 11
c
7
c
6
c
1
c
5
c
9
c
4
c
3
c
11
c
2
, c
8
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
\
I
v
1
I
u
1
= hb
4
+ 2b
3
+ 7b
2
+ 6b + 3, 2b
3
+ 3b
2
+ 15b + 5u + 7, 2b
3
3b
2
15b + 10a 7i
I
u
2
= hu
6
2u
5
+ 8u
4
4u
3
+ 12u
2
+ 8u + 4, u
4
2u
3
+ 4u
2
+ 4b 2u,
u
5
+ 4u
4
10u
3
+ 12u
2
+ 12a 12u + 4i
I
v
1
= hb + v 1, v
2
v + 1, ai
There are 3 irreducible components with 12 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
= hb
4
+2b
3
+7b
2
+6b+3, 2b
3
+3b
2
+15b+5u+7, 2b
3
3b
2
15b+10a7i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
2
5
b
3
3
5
b
2
3b
7
5
a
8
=
2
5
b
3
3
5
b
2
3b
7
5
2
5
b
3
3
5
b
2
3b
7
5
a
10
=
1
5
b
3
+
3
10
b
2
+
3
2
b +
7
10
b
a
6
=
1
5
b
3
+
3
10
b
2
+
3
2
b +
7
10
2
5
b
3
3
5
b
2
2b
7
5
a
2
=
1
10
b
3
1
10
b
2
+
1
2
b +
3
5
2
5
b
3
3
5
b
2
2b
2
5
a
5
=
1
2
b
3
+
1
2
b
2
+
5
2
b + 1
2
5
b
3
3
5
b
2
2b
2
5
a
9
=
0
2
5
b
3
+
3
5
b
2
+ 3b +
7
5
a
4
=
1
2
b
3
+
1
2
b
2
+
5
2
b + 1
3
5
b
3
+
2
5
b
2
+ 3b +
8
5
a
3
=
1
2
b
3
+
1
2
b
2
+
5
2
b + 1
2
5
b
3
3
5
b
2
2b
2
5
a
11
=
1
2
a
11
=
1
2
(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 = 1.41421I
a = 0.707107I
b = 0.500000 0.548188I
6.57974 2.02988I 6.00000 + 3.46410I
u = 1.41421I
a = 0.707107I
b = 0.500000 + 0.548188I
6.57974 + 2.02988I 6.00000 3.46410I
u = 1.41421I
a = 0.707107I
b = 0.50000 2.28024I
6.57974 + 2.02988I 6.00000 3.46410I
u = 1.41421I
a = 0.707107I
b = 0.50000 + 2.28024I
6.57974 2.02988I 6.00000 + 3.46410I
3
II. I
u
2
= hu
6
2u
5
+ 8u
4
4u
3
+ 12u
2
+ 8u + 4, u
4
2u
3
+ 4u
2
+ 4b
2u, u
5
+ 4u
4
10u
3
+ 12u
2
+ 12a 12u + 4i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
8
=
u
u
a
10
=
1
12
u
5
1
3
u
4
+ ··· + u
1
3
1
4
u
4
+
1
2
u
3
u
2
+
1
2
u
a
6
=
1
12
u
5
+
1
6
u
4
+ ··· + u +
2
3
1
6
u
5
+
1
12
u
4
+ ··· +
3
2
u +
1
3
a
2
=
1
4
u
5
+
1
2
u
4
+ ··· + u +
3
2
1
3
u
5
+
5
12
u
4
+ ··· +
3
2
u +
2
3
a
5
=
11
12
u
5
17
12
u
4
+ ···
7
2
u
7
6
5
6
u
5
19
12
u
4
+ ···
5
2
u
4
3
a
9
=
u
3
+ 2u
u
3
+ u
a
4
=
1
12
u
5
+
1
12
u
4
+ ···
1
2
u +
5
6
1
12
u
5
+
1
12
u
4
+ ··· +
1
2
u +
1
3
a
3
=
1
12
u
5
+
1
12
u
4
+ ···
1
2
u +
5
6
1
3
u
5
+
5
12
u
4
+ ··· +
3
2
u +
2
3
a
11
=
u
2
+ 1
u
2
a
11
=
u
2
+ 1
u
2
(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 = 0.327848 0.380167I
a = 0.513717 0.677222I
b = 0.058235 0.468561I
0.080134 + 1.031469I 1.24075 6.28341I
u = 0.327848 + 0.380167I
a = 0.513717 + 0.677222I
b = 0.058235 + 0.468561I
0.080134 1.031469I 1.24075 + 6.28341I
u = 0.31945 1.74021I
a = 0.782599 + 0.260942I
b = 0.180552 + 0.983566I
4.41014 + 1.50896I 1.48189 1.11182I
u = 0.31945 + 1.74021I
a = 0.782599 0.260942I
b = 0.180552 0.983566I
4.41014 1.50896I 1.48189 + 1.11182I
u = 1.00840 2.01334I
a = 0.296316 1.198936I
b = 0.26121 2.10173I
9.26481 6.90911I 1.75886 + 2.47219I
u = 1.00840 + 2.01334I
a = 0.296316 + 1.198936I
b = 0.26121 + 2.10173I
9.26481 + 6.90911I 1.75886 2.47219I
5
III. I
v
1
= hb + v 1, v
2
v + 1, ai
(i) Arc colorings
a
1
=
1
0
a
7
=
v
0
a
8
=
v
0
a
10
=
0
v + 1
a
6
=
v
v 1
a
2
=
2
v
a
5
=
v + 2
v
a
9
=
v
0
a
4
=
v + 1
v
a
3
=
v + 2
v
a
11
=
1
0
a
11
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 0.500000 0.866025I
a = 0
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 3.46410I
v = 0.500000 + 0.866025I
a = 0
b = 0.500000 0.866025I
1.64493 2.02988I 3.46410I
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
3
(u
6
+ 7u
5
+ 30u
4
+ 59u
3
+ 78u
2
+ 23u + 9)
c
2
(u
2
+ u + 1)
3
(u
6
+ 11u
5
+ 230u
4
+ 895u
3
+ 3910u
2
+ 875u + 81)
c
3
(u
2
u + 1)(u
4
2u
3
+ u
2
6u + 9)
(u
6
+ u
5
+ 4u
4
203u
3
+ 402u
2
+ 199u + 127)
c
4
(u
2
u + 1)(u
4
+ 2u
3
+ u
2
+ 6u + 9)
(u
6
+ 13u
5
+ 64u
4
+ 127u
3
+ 74u
2
17u + 41)
c
5
(u
2
u + 1)
3
(u
6
+ 7u
5
+ 30u
4
+ 59u
3
+ 78u
2
+ 23u + 9)
c
6
(u 1)
2
(u + 1)
4
(u
6
+ 4u
5
+ 9u
4
+ 8u
3
+ 19u
2
+ 4u + 3)
c
7
, c
8
, c
11
u
2
(u
2
+ 2)
2
(u
6
+ 2u
5
+ 8u
4
+ 4u
3
+ 12u
2
8u + 4)
c
9
(u 1)
4
(u + 1)
2
(u
6
+ 4u
5
+ 9u
4
+ 8u
3
+ 19u
2
+ 4u + 3)
c
10
(u
2
u + 1)(u
2
+ u + 1)
2
(u
6
+ u
5
+ 4u
4
+ u
3
+ 8u
2
+ 5u + 3)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
3
(y
6
+ 11y
5
+ 230y
4
+ 895y
3
+ 3910y
2
+ 875y + 81)
c
2
(y
2
+ y + 1)
3
(y
6
+ 339y
5
+ ··· 132205y + 6561)
c
3
(y
2
+ y + 1)(y
4
2y
3
5y
2
18y + 81)
(y
6
+ 7y
5
+ 1226y
4
38137y
3
+ 243414y
2
+ 62507y + 16129)
c
4
(y
2
+ y + 1)(y
4
2y
3
5y
2
18y + 81)
(y
6
41y
5
+ 942y
4
6133y
3
+ 15042y
2
+ 5779y + 1681)
c
6
, c
9
(y 1)
6
(y
6
+ 2y
5
+ 55y
4
+ 252y
3
+ 351y
2
+ 98y + 9)
c
7
, c
8
, c
11
y
2
(y + 2)
4
(y
6
+ 12y
5
+ 72y
4
+ 216y
3
+ 272y
2
+ 32y + 16)
c
10
(y
2
+ y + 1)
3
(y
6
+ 7y
5
+ 30y
4
+ 59y
3
+ 78y
2
+ 23y + 9)
9