11n
126
(K11n
126
)
1
Arc Sequences
5 1 10 7 2 9 11 5 1 8 4
Solving Sequence
1,5
2 3
6,9
7 10 4 8 11
c
1
c
2
c
5
c
6
c
9
c
4
c
8
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
8
\
i=1
I
u
i
I
u
1
= hu
2
+ 2u + 2, b + u + 1, 2a u 2i
I
u
2
= hu
4
+ 3u
3
+ 2u
2
1, a + 1, u
2
+ b + ui
I
u
3
= hu
5
4u
4
+ 3u
3
+ 2u
2
+ u 1, a 1, u
2
+ b + ui
I
u
4
= hu
6
4u
5
+ u
4
+ 9u
3
16u + 10, 8u
5
+ 23u
4
+ 14u
3
33u
2
+ 31b 41u + 47,
27u
5
128u
4
+ 7u
3
+ 433u
2
+ 310a + 150u 612i
I
u
5
= hu
6
+ 3u
5
3u
3
+ 2u
2
+ 1, a 1, u
5
4u
4
2u
3
+ 3u
2
+ 2b u 1i
I
u
6
= hb
2
+ 1, a + 1, u 1i
I
u
7
= hb
2
+ 1, u 1, b + a 1i
I
u
8
= hu
6
+ 3u
5
3u
3
+ 2u
2
+ 1, u
5
+ 2u
4
2u
3
u
2
+ 2b + 3u 1, u
5
2u
4
+ 4u
3
+ 5u
2
+ 2a 7u + 1i
There are 8 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
2
+ 2u + 2, b + u + 1, 2a u 2i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
2u 2
a
3
=
2u + 3
2u 2
a
6
=
u
u 4
a
9
=
1
2
u + 1
u 1
a
7
=
1
2
u
u 3
a
10
=
3
2
u + 2
u 1
a
4
=
1
2
u + 1
u 1
a
8
=
1
2
u + 1
1
a
11
=
1
2
u + 1
1
a
11
=
1
2
u + 1
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 = 1.00000 1.00000I
a = 0.500000 0.500000I
b = 1.00000I
1.64493 8.00000
u = 1.00000 + 1.00000I
a = 0.500000 + 0.500000I
b = 1.00000I
1.64493 8.00000
3
II. I
u
2
= hu
4
+ 3u
3
+ 2u
2
1, a + 1, u
2
+ b + ui
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
9
=
1
u
2
u
a
7
=
u
2
2u
u
2
+ u 1
a
10
=
u
2
+ u 1
u
2
u
a
4
=
u
3
2u
2
+ u + 1
u
3
+ 2u
2
a
8
=
1
u
a
11
=
u
3
+ 3u
2
+ u 1
u
3
3u
2
u + 1
a
11
=
u
3
+ 3u
2
+ u 1
u
3
3u
2
u + 1
(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 = 2.17872
a = 1.00000
b = 2.56811
13.8089 12.1771
u = 0.667076 0.670769I
a = 1.00000
b = 0.672017 0.224139I
3.04056 6.31855I 7.20042 + 6.94067I
u = 0.667076 + 0.670769I
a = 1.00000
b = 0.672017 + 0.224139I
3.04056 + 6.31855I 7.20042 6.94067I
u = 0.512876
a = 1.00000
b = 0.775919
2.14179 18.4220
5
III. I
u
3
= hu
5
4u
4
+ 3u
3
+ 2u
2
+ u 1, a 1, u
2
+ b + ui
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
9
=
1
u
2
u
a
7
=
u
2
2u
u
4
3u
3
+ u
2
+ u
a
10
=
u
2
+ u + 1
u
2
u
a
4
=
u
3
2u
2
u + 1
u
3
+ 2u
2
a
8
=
1
u
a
11
=
u
3
3u
2
+ u + 1
u
4
+ 2u
3
+ u
2
u
a
11
=
u
3
3u
2
+ u + 1
u
4
+ 2u
3
+ u
2
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 = 0.456518 0.478939I
a = 1.00000
b = 0.435545 + 0.916228I
4.11394 0.50358I 4.17139 + 2.42983I
u = 0.456518 + 0.478939I
a = 1.00000
b = 0.435545 0.916228I
4.11394 + 0.50358I 4.17139 2.42983I
u = 0.455088
a = 1.00000
b = 0.247983
0.742760 13.4344
u = 2.22897 0.22563I
a = 1.00000
b = 2.68845 0.78020I
11.1448 + 10.7639I 11.61144 5.00628I
u = 2.22897 + 0.22563I
a = 1.00000
b = 2.68845 + 0.78020I
11.1448 10.7639I 11.61144 + 5.00628I
7
IV. I
u
4
= hu
6
4u
5
+ u
4
+ 9u
3
16u + 10, 8u
5
+ 23u
4
+ · · · + 31b +
47, 27u
5
128u
4
+ · · · + 310a 612i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
9
=
0.0870968u
5
+ 0.412903u
4
+ ··· 0.483871u + 1.97419
0.258065u
5
0.741935u
4
+ ··· + 1.32258u 1.51613
a
7
=
0.164516u
5
+ 0.335484u
4
+ ··· 1.58065u + 0.729032
0.290323u
5
+ 0.709677u
4
+ ··· 0.612903u + 1.58065
a
10
=
0.345161u
5
+ 1.15484u
4
+ ··· 1.80645u + 3.49032
0.258065u
5
0.741935u
4
+ ··· + 1.32258u 1.51613
a
4
=
0.0935484u
5
0.406452u
4
+ ··· + 0.741935u 0.787097
0.322581u
5
0.677419u
4
+ ··· + 1.90323u 1.64516
a
8
=
0.0870968u
5
+ 0.412903u
4
+ ··· 0.483871u + 1.97419
0.0645161u
5
0.0645161u
4
+ ··· 0.580645u 0.870968
a
11
=
1
10
u
5
+
2
5
u
4
+ ···
9
10
u
2
+
8
5
0.0322581u
5
+ 0.0322581u
4
+ ··· 0.709677u + 0.935484
a
11
=
1
10
u
5
+
2
5
u
4
+ ···
9
10
u
2
+
8
5
0.0322581u
5
+ 0.0322581u
4
+ ··· 0.709677u + 0.935484
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.221244 0.712792I
a = 0.093681 0.373158I
b = 0.709117 1.052541I
2.26485 4.47692I 9.60193 + 3.00061I
u = 1.221244 + 0.712792I
a = 0.093681 + 0.373158I
b = 0.709117 + 1.052541I
2.26485 + 4.47692I 9.60193 3.00061I
u = 0.912842 0.311096I
a = 0.588546 + 0.687760I
b = 0.072877 0.272459I
0.389538 + 0.233200I 13.01274 1.15455I
u = 0.912842 + 0.311096I
a = 0.588546 0.687760I
b = 0.072877 + 0.272459I
0.389538 0.233200I 13.01274 + 1.15455I
u = 2.30840 0.22043I
a = 0.894865 0.153901I
b = 2.78199 + 0.25607I
13.38985 2.37783I 11.38532 + 2.96944I
u = 2.30840 + 0.22043I
a = 0.894865 + 0.153901I
b = 2.78199 0.25607I
13.38985 + 2.37783I 11.38532 2.96944I
9
V.
I
u
5
= hu
6
+ 3u
5
3u
3
+ 2u
2
+ 1, a 1, u
5
4u
4
2u
3
+ 3u
2
+ 2b u 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
9
=
1
1
2
u
5
+ 2u
4
+ ··· +
1
2
u +
1
2
a
7
=
1
2
u
5
u
4
+ ···
5
2
u +
1
2
1
a
10
=
1
2
u
5
2u
4
+ ···
1
2
u +
1
2
1
2
u
5
+ 2u
4
+ ··· +
1
2
u +
1
2
a
4
=
u
4
+ 2u
3
u
2
u + 1
1
2
u
5
+ u
4
+ ··· +
3
2
u +
1
2
a
8
=
1
1
2
u
5
+ 2u
4
+ ··· +
1
2
u +
1
2
a
11
=
u
5
3u
4
+ 3u
2
2u
u
3
+ u
2
2u + 1
a
11
=
u
5
3u
4
+ 3u
2
2u
u
3
+ u
2
2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 2.09963 0.15801I
a = 1.00000
b = 2.07508 + 0.88397I
13.38985 2.37783I 11.38532 + 2.96944I
u = 2.09963 + 0.15801I
a = 1.00000
b = 2.07508 0.88397I
13.38985 + 2.37783I 11.38532 2.96944I
u = 0.151576 0.522493I
a = 1.00000
b = 0.971221 0.554396I
2.26485 + 4.47692I 9.60193 3.00061I
u = 0.151576 + 0.522493I
a = 1.00000
b = 0.971221 + 0.554396I
2.26485 4.47692I 9.60193 + 3.00061I
u = 0.751209 0.444722I
a = 1.00000
b = 0.546305 0.979253I
0.389538 0.233200I 13.01274 + 1.15455I
u = 0.751209 + 0.444722I
a = 1.00000
b = 0.546305 + 0.979253I
0.389538 + 0.233200I 13.01274 1.15455I
11
VI. I
u
6
= hb
2
+ 1, a + 1, u 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
6
=
1
0
a
9
=
1
b
a
7
=
b 1
1
a
10
=
b 1
b
a
4
=
2b
b
a
8
=
1
b + 1
a
11
=
1
1
a
11
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.00000I
1.64493 8.00000
u = 1.00000
a = 1.00000
b = 1.00000I
1.64493 8.00000
13
VII. I
u
7
= hb
2
+ 1, u 1, b + a 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
6
=
1
0
a
9
=
b + 1
b
a
7
=
b
1
a
10
=
1
b
a
4
=
1
b + 1
a
8
=
b + 1
1
a
11
=
b + 2
2b
a
11
=
b + 2
2b
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
14
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000 1.00000I
b = 1.00000I
1.64493 8.00000
u = 1.00000
a = 1.00000 + 1.00000I
b = 1.00000I
1.64493 8.00000
15
VIII. I
u
8
= hu
6
+ 3u
5
3u
3
+ 2u
2
+ 1, u
5
+ 2u
4
2u
3
u
2
+ 2b + 3u
1, u
5
2u
4
+ 4u
3
+ 5u
2
+ 2a 7u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
9
=
1
2
u
5
+ u
4
+ ··· +
7
2
u
1
2
1
2
u
5
u
4
+ ···
3
2
u +
1
2
a
7
=
1
2
u
5
+ u
4
+ ··· +
5
2
u
3
2
1
a
10
=
u
5
+ 2u
4
3u
3
3u
2
+ 5u 1
1
2
u
5
u
4
+ ···
3
2
u +
1
2
a
4
=
u
5
+ 3u
4
4u
2
+ u + 2
1
2
u
5
u
4
+ ···
1
2
u
1
2
a
8
=
1
2
u
5
+ u
4
+ ··· +
7
2
u
1
2
u
a
11
=
3
2
u
5
+ 4u
4
+ ··· +
7
2
u +
3
2
u
5
2u
4
+ u
3
+ u
2
u
a
11
=
3
2
u
5
+ 4u
4
+ ··· +
7
2
u +
3
2
u
5
2u
4
+ u
3
+ u
2
u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
16
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
1(vol +
1CS) Cusp shape
u = 2.09963 0.15801I
a = 1.085384 + 0.186667I
b = 2.78199 + 0.25607I
13.38985 2.37783I 11.38532 + 2.96944I
u = 2.09963 + 0.15801I
a = 1.085384 0.186667I
b = 2.78199 0.25607I
13.38985 + 2.37783I 11.38532 2.96944I
u = 0.151576 0.522493I
a = 0.63288 2.52094I
b = 0.709117 + 1.052541I
2.26485 + 4.47692I 9.60193 3.00061I
u = 0.151576 + 0.522493I
a = 0.63288 + 2.52094I
b = 0.709117 1.052541I
2.26485 4.47692I 9.60193 + 3.00061I
u = 0.751209 0.444722I
a = 0.718265 + 0.839346I
b = 0.072877 + 0.272459I
0.389538 0.233200I 13.01274 + 1.15455I
u = 0.751209 + 0.444722I
a = 0.718265 0.839346I
b = 0.072877 0.272459I
0.389538 + 0.233200I 13.01274 1.15455I
17
IX. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
2
+ 2u + 2)(u
4
+ 3u
3
+ 2u
2
1)(u
5
+ 4u
4
+ ··· + u + 1)
(u
6
3u
5
+ 3u
3
+ 2u
2
+ 1)
2
(u
6
+ 4u
5
+ u
4
9u
3
+ 16u + 10)
c
2
(u + 1)
4
(u
2
+ 4)(u
4
+ 5u
3
+ 2u
2
+ 4u + 1)
(u
5
+ 10u
4
+ 27u
3
+ 6u
2
+ 5u + 1)
(u
6
+ 9u
5
+ 22u
4
+ 7u
3
+ 4u
2
4u + 1)
2
(u
6
+ 14u
5
+ 73u
4
+ 189u
3
+ 308u
2
+ 256u + 100)
c
3
, c
5
, c
8
(u + 1)
4
(u
2
2u + 2)(u
4
3u
3
+ 2u
2
1)(u
5
+ 4u
4
+ ··· + u + 1)
(u
6
3u
5
+ 3u
3
+ 2u
2
+ 1)
2
(u
6
+ 4u
5
+ u
4
9u
3
+ 16u + 10)
c
4
, c
7
, c
11
(u
2
+ 1)
2
(u
2
2u + 2)(u
4
u
3
+ u
2
+ u 1)
(u
5
+ 2u
4
+ 4u
3
+ 2u
2
+ 2u 1)(u
6
u
5
+ 2u
4
u
3
+ 2u
2
2u + 1)
2
(u
6
+ 4u
5
+ 7u
4
+ 7u
3
+ 6u
2
+ 4u + 2)
c
6
, c
9
(u
2
+ 1)
3
(u
4
2u
3
2u
2
+ u + 1)(u
5
+ 6u
4
+ 12u
3
+ 9u
2
+ 5u 2)
(u
6
4u
5
+ u
4
+ 2u
3
+ 13u
2
+ 2u + 1)
2
(u
6
+ 5u
5
+ 9u
4
+ 7u
3
+ 8u
2
+ 12u + 8)
c
10
(u
2
+ 1)
2
(u
2
+ 2u + 2)(u
4
+ u
3
+ u
2
u 1)
(u
5
+ 2u
4
+ 4u
3
+ 2u
2
+ 2u 1)(u
6
u
5
+ 2u
4
u
3
+ 2u
2
2u + 1)
2
(u
6
+ 4u
5
+ 7u
4
+ 7u
3
+ 6u
2
+ 4u + 2)
18
X. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
5
c
8
(y 1)
4
(y
2
+ 4)(y
4
5y
3
+ 2y
2
4y + 1)
(y
5
10y
4
+ 27y
3
6y
2
+ 5y 1)
(y
6
14y
5
+ 73y
4
189y
3
+ 308y
2
256y + 100)
(y
6
9y
5
+ 22y
4
7y
3
+ 4y
2
+ 4y + 1)
2
c
2
(y 1)
4
(y + 4)
2
(y
4
21y
3
34y
2
12y + 1)
(y
5
46y
4
+ 619y
3
+ 214y
2
+ 13y 1)
(y
6
50y
5
+ 653y
4
+ 2279y
3
+ 12696y
2
3936y + 10000)
(y
6
37y
5
+ 366y
4
+ 201y
3
+ 116y
2
8y + 1)
2
c
4
, c
7
, c
10
c
11
(y + 1)
4
(y
2
+ 4)(y
4
+ y
3
+ ··· 3y + 1)(y
5
+ 4y
4
+ ··· + 8y 1)
(y
6
2y
5
+ ··· + 8y + 4)(1 + 4y
2
+ 5y
3
+ 6y
4
+ 3y
5
+ y
6
)
2
c
6
, c
9
(y + 1)
6
(y
4
8y
3
+ ··· 5y + 1)(y
5
12y
4
+ ··· + 61y 4)
(y
6
14y
5
+ 43y
4
+ 40y
3
+ 163y
2
+ 22y + 1)
2
(y
6
7y
5
+ 27y
4
9y
3
+ 40y
2
16y + 64)
19