11n
102
(K11n
102
)
1
Arc Sequences
7 1 8 7 10 2 4 1 11 5 9
Solving Sequence
5,7 4,11
10 6 9 1 8 3 2
c
4
c
10
c
5
c
9
c
11
c
8
c
3
c
2
c
1
, c
6
, c
7
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
6
+ b
4
+ 2b
2
+ 1, b
5
b + u, b
5
+ b
4
b
3
b + a + 1i
I
u
2
= hu
7
+ u
6
3u
5
15u
4
+ 7u
3
9u
2
+ 3u 1, 2u
6
u
5
+ 8u
4
+ 28u
3
32u
2
+ 4a + 9u 6,
5u
6
+ 6u
5
15u
4
80u
3
+ 23u
2
+ 8b 22u + 3i
There are 2 irreducible components with 13 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
6
+ b
4
+ 2b
2
+ 1, b
5
b + u, b
5
+ b
4
b
3
b + a + 1i
(i) Arc colorings
a
5
=
1
0
a
7
=
0
b
5
+ b
a
4
=
1
1
a
11
=
b
5
b
4
+ b
3
+ b 1
b
a
10
=
b
5
b
4
+ b
3
+ 2b 1
b
a
6
=
b
5
b
b
2
a
9
=
b
5
+ b
2
+ b
b
3
+ b
a
1
=
1
b
5
+ b
3
+ b
a
8
=
b
5
+ b
0
a
3
=
0
1
a
2
=
1
b
5
+ b
3
+ b 1
a
2
=
1
b
5
+ b
3
+ b 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 = 1.00000I
a = 1.96950 + 0.77736I
b = 0.744862 0.877439I
6.31400 2.82812I 3.50976 + 2.97945I
u = 1.00000I
a = 1.96950 0.77736I
b = 0.744862 + 0.877439I
6.31400 + 2.82812I 3.50976 2.97945I
u = 1.00000I
a = 1.32472 0.56984I
b = 0.754878I
2.17641 3.01951
u = 1.00000I
a = 1.32472 + 0.56984I
b = 0.754878I
2.17641 3.01951
u = 1.00000I
a = 0.644782 0.347200I
b = 0.744862 0.877439I
6.31400 + 2.82812I 3.50976 2.97945I
u = 1.00000I
a = 0.644782 + 0.347200I
b = 0.744862 + 0.877439I
6.31400 2.82812I 3.50976 + 2.97945I
3
II. I
u
2
= hu
7
+ u
6
3u
5
15u
4
+ 7u
3
9u
2
+ 3u 1, 2u
6
u
5
+ · · · +
4a 6, 5u
6
+ 6u
5
+ · · · + 8b + 3i
(i) Arc colorings
a
5
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
11
=
1
2
u
6
+
1
4
u
5
+ ···
9
4
u +
3
2
5
8
u
6
3
4
u
5
+ ··· +
11
4
u
3
8
a
10
=
1
8
u
6
1
2
u
5
+ ··· +
1
2
u +
9
8
5
8
u
6
3
4
u
5
+ ··· +
11
4
u
3
8
a
6
=
u
1
8
u
6
1
4
u
5
+ ···
5
4
u +
1
8
a
9
=
1
8
u
6
+
1
4
u
5
+ ··· +
9
4
u
1
8
3
8
u
6
3
4
u
5
+ ···
1
4
u +
5
8
a
1
=
1
1
8
u
6
5
8
u
4
+ ···
1
2
u +
1
8
a
8
=
u
u
3
+ u
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
1
1
8
u
6
5
8
u
4
+ ···
1
2
u +
1
8
a
2
=
1
1
8
u
6
5
8
u
4
+ ···
1
2
u +
1
8
(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 = 1.98465 1.73888I
a = 0.709221 0.766981I
b = 0.76621 1.39847I
8.20573 + 7.24432I 2.98176 3.02276I
u = 1.98465 + 1.73888I
a = 0.709221 + 0.766981I
b = 0.76621 + 1.39847I
8.20573 7.24432I 2.98176 + 3.02276I
u = 0.027229 0.619612I
a = 1.72115 0.61285I
b = 0.780622 + 0.883029I
4.84599 2.93728I 3.03833 + 3.35250I
u = 0.027229 + 0.619612I
a = 1.72115 + 0.61285I
b = 0.780622 0.883029I
4.84599 + 2.93728I 3.03833 3.35250I
u = 0.229714 0.315113I
a = 1.044055 0.342650I
b = 0.209546 0.619099I
0.336860 0.937443I 6.03846 + 7.34722I
u = 0.229714 + 0.315113I
a = 1.044055 + 0.342650I
b = 0.209546 + 0.619099I
0.336860 + 0.937443I 6.03846 7.34722I
u = 2.45542
a = 0.227363
b = 1.32543
4.12134 1.88291
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
4
c
6
, c
7
(u
2
+ 1)
3
(u
7
+ u
6
3u
5
15u
4
+ 7u
3
9u
2
+ 3u 1)
c
2
(u 1)
6
(u
7
+ 7u
6
+ 53u
5
+ 243u
4
237u
3
+ 69u
2
9u + 1)
c
5
, c
10
(u
6
+ u
4
+ 2u
2
+ 1)(u
7
+ 4u
6
+ 9u
5
+ 12u
4
+ 10u
3
+ 6u
2
+ 3u + 2)
c
8
, c
9
, c
11
(u
3
+ u
2
+ 2u + 1)
2
(u
7
+ 2u
6
+ 5u
5
6u
4
6u
3
24u
2
15u 4)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
6
, c
7
(y + 1)
6
(y
7
7y
6
+ 53y
5
243y
4
237y
3
69y
2
9y 1)
c
2
(y 1)
6
(y
7
+ 57y
6
1067y
5
85155y
4
+ 21667y
3
981y
2
57y 1)
c
5
, c
10
(y
3
+ y
2
+ 2y + 1)
2
(y
7
+ 2y
6
+ 5y
5
6y
4
6y
3
24y
2
15y 4)
c
8
, c
9
, c
11
(y
3
+ 3y
2
+ 2y 1)
2
(y
7
+ 6y
6
+ 37y
5
30y
4
386y
3
444y
2
+ 33y 16)
7