11a
58
(K11a
58
)
1
Arc Sequences
4 1 8 2 11 10 9 3 7 5 6
Solving Sequence
1,4
2
3,6
11 5 10 7 9 8
c
1
c
2
c
11
c
5
c
10
c
6
c
9
c
8
c
3
, c
4
, c
7
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu 1, a, b 1i
I
u
2
= hu
12
4u
10
+ u
9
+ 6u
8
3u
7
u
6
+ 3u
5
5u
4
+ u
3
+ 3u
2
2u + 1, a, b ui
I
u
3
= hu
29
2u
28
+ ··· u + 1, u
27
+ u
26
+ ··· + b 1, u
28
u
27
+ ··· + a 2ui
There are 3 irreducible components with 42 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 1, a, b 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
3
=
0
1
a
6
=
0
1
a
11
=
1
1
a
5
=
1
0
a
10
=
0
1
a
7
=
0
1
a
9
=
0
1
a
8
=
0
1
a
8
=
0
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
a = 0
b = 1.00000
0 0
3
II. I
u
2
= hu
12
4u
10
+ · · · 2u + 1, a, b ui
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
0
u
a
11
=
1
u
2
a
5
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
+ 2u
2
a
7
=
u
5
2u
3
+ u
u
7
3u
5
+ 2u
3
+ u
a
9
=
u
8
+ 3u
6
3u
4
+ 1
u
10
+ 4u
8
5u
6
+ 3u
2
a
8
=
u
11
4u
9
+ 6u
7
2u
5
3u
3
+ 2u
u
11
u
10
+ 3u
9
+ 3u
8
3u
7
3u
6
u
5
u
4
+ 2u
3
+ 2u
2
a
8
=
u
11
4u
9
+ 6u
7
2u
5
3u
3
+ 2u
u
11
u
10
+ 3u
9
+ 3u
8
3u
7
3u
6
u
5
u
4
+ 2u
3
+ 2u
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 = 1.271939 0.422443I
a = 0
b = 1.271939 0.422443I
6.79074 + 3.16396I 1.82674 2.56480I
u = 1.271939 + 0.422443I
a = 0
b = 1.271939 + 0.422443I
6.79074 3.16396I 1.82674 + 2.56480I
u = 1.186693 0.158407I
a = 0
b = 1.186693 0.158407I
0.21101 + 1.41510I 1.82674 4.90874I
u = 1.186693 + 0.158407I
a = 0
b = 1.186693 + 0.158407I
0.21101 1.41510I 1.82674 + 4.90874I
u = 0.031664 0.878090I
a = 0
b = 0.031664 0.878090I
6.79074 3.16396I 1.82674 + 2.56480I
u = 0.031664 + 0.878090I
a = 0
b = 0.031664 + 0.878090I
6.79074 + 3.16396I 1.82674 2.56480I
u = 0.241868 0.480324I
a = 0
b = 0.241868 0.480324I
0.21101 1.41510I 1.82674 + 4.90874I
u = 0.241868 + 0.480324I
a = 0
b = 0.241868 + 0.480324I
0.21101 + 1.41510I 1.82674 4.90874I
u = 0.944825 0.321917I
a = 0
b = 0.944825 0.321917I
0.21101 + 1.41510I 1.82674 4.90874I
u = 0.944825 + 0.321917I
a = 0
b = 0.944825 + 0.321917I
0.21101 1.41510I 1.82674 + 4.90874I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.240275 0.455646I
a = 0
b = 1.240275 0.455646I
6.79074 + 3.16396I 1.82674 2.56480I
u = 1.240275 + 0.455646I
a = 0
b = 1.240275 + 0.455646I
6.79074 3.16396I 1.82674 + 2.56480I
6
III.
I
u
3
= hu
29
2u
28
+· · · u + 1, u
27
+u
26
+· · · + b 1, u
28
u
27
+· · · + a 2ui
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
28
+ u
27
+ ··· 6u
2
+ 2u
u
27
u
26
+ ··· u
2
+ 1
a
11
=
u
25
+ 8u
23
+ ··· 4u
2
+ 3u
u
28
+ u
27
+ ··· u + 1
a
5
=
u
u
3
+ u
a
10
=
u
28
+ u
27
+ ··· 4u
2
+ 2u
u
28
+ 2u
27
+ ··· 5u
2
+ 2
a
7
=
u
28
+ u
27
+ ··· 4u
2
+ u
2u
28
+ 3u
27
+ ··· 8u
2
+ 3
a
9
=
u
28
+ u
27
+ ··· 3u
2
+ u
3u
28
+ 4u
27
+ ··· 12u
2
+ 4
a
8
=
u
28
+ u
27
+ ··· 2u
3
3u
2
4u
28
+ 5u
27
+ ··· 15u
2
+ 5
a
8
=
u
28
+ u
27
+ ··· 2u
3
3u
2
4u
28
+ 5u
27
+ ··· 15u
2
+ 5
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.257115 0.448584I
a = 0.704471 + 0.741241I
b = 0.008721 + 0.887960I
10.71028 1.52343I 5.35413 + 0.68771I
u = 1.257115 + 0.448584I
a = 0.704471 0.741241I
b = 0.008721 0.887960I
10.71028 + 1.52343I 5.35413 0.68771I
u = 1.237428 0.470287I
a = 0.82863 1.97410I
b = 1.286004 0.418935I
6.68464 6.20004I 1.73580 + 3.81481I
u = 1.237428 + 0.470287I
a = 0.82863 + 1.97410I
b = 1.286004 + 0.418935I
6.68464 + 6.20004I 1.73580 3.81481I
u = 1.062616 0.280483I
a = 0.822036 + 0.588542I
b = 0.063245 + 0.516212I
3.02142 1.01433I 6.77496 + 0.83339I
u = 1.062616 + 0.280483I
a = 0.822036 0.588542I
b = 0.063245 0.516212I
3.02142 + 1.01433I 6.77496 0.83339I
u = 0.972190 0.408832I
a = 2.12668 2.40827I
b = 1.242321 0.189774I
0.91595 3.56420I 0.67873 + 4.99863I
u = 0.972190 + 0.408832I
a = 2.12668 + 2.40827I
b = 1.242321 + 0.189774I
0.91595 + 3.56420I 0.67873 4.99863I
u = 0.888826
a = 0.722046
b = 0.193938
1.29813 8.53893
u = 0.426002 0.324266I
a = 3.81749 + 0.31265I
b = 1.189732 + 0.056062I
2.39907 + 0.12369I 3.50407 + 1.07759I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.426002 + 0.324266I
a = 3.81749 0.31265I
b = 1.189732 0.056062I
2.39907 0.12369I 3.50407 1.07759I
u = 0.014978 0.851014I
a = 1.46295 + 0.66868I
b = 1.249689 + 0.417811I
3.02235 + 1.47420I 1.47993 0.60903I
u = 0.014978 + 0.851014I
a = 1.46295 0.66868I
b = 1.249689 0.417811I
3.02235 1.47420I 1.47993 + 0.60903I
u = 0.074758 0.894541I
a = 1.50134 + 0.55093I
b = 1.301316 + 0.407588I
2.63518 7.77071I 2.10858 + 5.30383I
u = 0.074758 + 0.894541I
a = 1.50134 0.55093I
b = 1.301316 0.407588I
2.63518 + 7.77071I 2.10858 5.30383I
u = 0.359435 0.685270I
a = 2.23815 + 0.27468I
b = 1.311938 + 0.179476I
4.98921 3.78682I 7.27007 + 4.16727I
u = 0.359435 + 0.685270I
a = 2.23815 0.27468I
b = 1.311938 0.179476I
4.98921 + 3.78682I 7.27007 4.16727I
u = 0.678336 0.303222I
a = 0.425800 + 0.584678I
b = 0.529946 + 0.329108I
0.93542 + 1.41053I 5.39446 5.74020I
u = 0.678336 + 0.303222I
a = 0.425800 0.584678I
b = 0.529946 0.329108I
0.93542 1.41053I 5.39446 + 5.74020I
u = 0.739200 0.575207I
a = 2.46695 1.00249I
b = 1.342149 0.040293I
6.61715 + 2.27209I 8.89752 3.80982I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.739200 + 0.575207I
a = 2.46695 + 1.00249I
b = 1.342149 + 0.040293I
6.61715 2.27209I 8.89752 + 3.80982I
u = 1.029202 0.437563I
a = 0.711833 + 0.635826I
b = 0.230236 + 0.672244I
1.85869 + 5.19499I 2.04173 8.30480I
u = 1.029202 + 0.437563I
a = 0.711833 0.635826I
b = 0.230236 0.672244I
1.85869 5.19499I 2.04173 + 8.30480I
u = 1.040505 0.532887I
a = 1.61127 1.86164I
b = 1.333982 0.244603I
3.04589 + 8.42692I 3.52830 8.66921I
u = 1.040505 + 0.532887I
a = 1.61127 + 1.86164I
b = 1.333982 + 0.244603I
3.04589 8.42692I 3.52830 + 8.66921I
u = 1.245918 0.505749I
a = 0.86036 1.85625I
b = 1.320522 0.424615I
6.1783 + 12.8069I 0.92308 8.12569I
u = 1.245918 + 0.505749I
a = 0.86036 + 1.85625I
b = 1.320522 + 0.424615I
6.1783 12.8069I 0.92308 + 8.12569I
u = 1.247387 0.482397I
a = 0.692230 + 0.727568I
b = 0.050913 + 0.910185I
10.46167 + 8.03356I 4.76249 5.59744I
u = 1.247387 + 0.482397I
a = 0.692230 0.727568I
b = 0.050913 0.910185I
10.46167 8.03356I 4.76249 + 5.59744I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u + 1)(u
12
4u
10
+ ··· + 2 u + 1)
(u
29
+ 2u
28
+ ··· u 1)
c
2
(u + 1)(u
12
+ 8u
11
+ ··· 2u + 1)(u
29
+ 16u
28
+ ··· + 7u + 1)
c
3
, c
8
(u)(1 + u
2
+ u
3
+ u
4
)
3
(u
29
2u
28
+ ··· + 2u 2)
c
5
, c
10
, c
11
(u 1)(u
12
4u
10
+ ··· 2u + 1)
(u
29
+ 2u
28
+ ··· + 3 u + 1)
c
6
, c
7
, c
9
(u)(1 + 2u + 3u
2
+ u
3
+ u
4
)
3
(u
29
+ 6u
28
+ ··· + 8 u 4)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)(y
12
8y
11
+ ··· + 2 y + 1)(y
29
16y
28
+ ··· + 7 y 1)
c
2
(y 1)(y
12
8y
11
+ ··· + 2 y + 1)(y
29
4y
28
+ ··· 17y 1)
c
3
, c
8
(y)(1 + 2y + 3y
2
+ y
3
+ y
4
)
3
(y
29
+ 6y
28
+ ··· + 8 y 4)
c
5
, c
10
, c
11
(y 1)(y
12
8y
11
+ ··· + 2 y + 1)(y
29
24y
28
+ ··· + 23 y 1)
c
6
, c
7
, c
9
(y)(1 + 2y + 7y
2
+ 5y
3
+ y
4
)
3
(y
29
+ 30y
28
+ ··· + 504y 16)
12