11a
161
(K11a
161
)
1
Arc Sequences
6 1 10 7 2 5 4 11 3 8 9
Solving Sequence
1,6
2 3 5 7
4,9
11 8 10
c
1
c
2
c
5
c
6
c
4
c
11
c
8
c
10
c
3
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
4
u
3
+ u
2
+ 1, b 1, u
2
+ a + ui
I
u
2
= hu
32
2u
31
+ ··· 4u + 1, u
28
+ u
27
+ ··· + a 2, u
30
u
29
+ ··· + b 2ui
There are 2 irreducible components with 36 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
+ u
2
+ 1, b 1, u
2
+ a + ui
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
7
=
u
3
u
3
u
2
1
a
4
=
u
2
+ 1
u
2
a
9
=
u
2
u
1
a
11
=
u
2
u + 1
1
a
8
=
1
0
a
10
=
u
2
u
1
a
10
=
u
2
u
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 = 0.351808 0.720342I
a = 0.043315 + 1.227186I
b = 1.00000
1.43393 + 1.41510I 6.86477 6.85627I
u = 0.351808 + 0.720342I
a = 0.043315 1.227186I
b = 1.00000
1.43393 1.41510I 6.86477 + 6.85627I
u = 0.851808 0.911292I
a = 0.956685 0.641200I
b = 1.00000
8.43568 3.16396I 12.63523 + 2.29471I
u = 0.851808 + 0.911292I
a = 0.956685 + 0.641200I
b = 1.00000
8.43568 + 3.16396I 12.63523 2.29471I
3
II.
I
u
2
= hu
32
2u
31
+· · ·4u+1, u
28
+u
27
+· · ·+a2, u
30
u
29
+· · ·+b2ui
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
7
=
u
3
u
5
+ u
3
+ u
a
4
=
u
5
u
u
7
+ u
5
+ 2u
3
+ u
a
9
=
u
28
u
27
+ ··· 6u + 2
u
30
+ u
29
+ ··· 4u
2
+ 2u
a
11
=
u
30
u
29
+ ··· 7u + 3
u
30
+ u
29
+ ··· 3u
2
+ u
a
8
=
u
7
2u
3
u
9
+ u
7
+ 3u
5
+ 2u
3
+ u
a
10
=
2u
30
u
29
+ ··· 7u + 3
u
31
u
30
+ ··· u
2
+ u
a
10
=
2u
30
u
29
+ ··· 7u + 3
u
31
u
30
+ ··· u
2
+ u
(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 = 0.915023 0.941366I
a = 2.39534 + 0.89971I
b = 1.55384 + 0.01176I
14.7168 + 3.3681I 10.32984 2.30184I
u = 0.915023 + 0.941366I
a = 2.39534 0.89971I
b = 1.55384 0.01176I
14.7168 3.3681I 10.32984 + 2.30184I
u = 0.866691 0.917191I
a = 0.900315 0.319705I
b = 0.516773 0.024573I
7.59385 + 3.21086I 2.30282 2.66372I
u = 0.866691 + 0.917191I
a = 0.900315 + 0.319705I
b = 0.516773 + 0.024573I
7.59385 3.21086I 2.30282 + 2.66372I
u = 0.798476 0.579251I
a = 1.76126 0.16137I
b = 1.56838 + 0.16195I
10.60128 2.72339I 11.99326 + 0.76740I
u = 0.798476 + 0.579251I
a = 1.76126 + 0.16137I
b = 1.56838 0.16195I
10.60128 + 2.72339I 11.99326 0.76740I
u = 0.645468 0.683845I
a = 0.0195996 + 0.1036640I
b = 0.629638 0.624160I
3.24411 + 0.05063I 9.93222 + 0.12660I
u = 0.645468 + 0.683845I
a = 0.0195996 0.1036640I
b = 0.629638 + 0.624160I
3.24411 0.05063I 9.93222 0.12660I
u = 0.614078 0.966359I
a = 1.02310 2.02918I
b = 1.52314 0.21166I
9.31106 + 7.91275I 9.40540 6.63145I
u = 0.614078 + 0.966359I
a = 1.02310 + 2.02918I
b = 1.52314 + 0.21166I
9.31106 7.91275I 9.40540 + 6.63145I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.601316 0.841196I
a = 0.867822 + 0.956388I
b = 0.473466 + 0.702927I
2.73490 + 4.63620I 7.44058 7.48323I
u = 0.601316 + 0.841196I
a = 0.867822 0.956388I
b = 0.473466 0.702927I
2.73490 4.63620I 7.44058 + 7.48323I
u = 0.161526 0.565105I
a = 0.28634 + 2.32243I
b = 1.053374 + 0.143956I
1.29270 + 0.72541I 4.11413 + 2.96939I
u = 0.161526 + 0.565105I
a = 0.28634 2.32243I
b = 1.053374 0.143956I
1.29270 0.72541I 4.11413 2.96939I
u = 0.145430 0.769393I
a = 0.613814 1.208164I
b = 0.193912 0.402278I
1.08342 1.49550I 1.76412 + 6.31671I
u = 0.145430 + 0.769393I
a = 0.613814 + 1.208164I
b = 0.193912 + 0.402278I
1.08342 + 1.49550I 1.76412 6.31671I
u = 0.213719 0.980461I
a = 0.92583 + 1.26296I
b = 1.45934 + 0.07190I
4.40938 2.90320I 5.62644 + 3.88291I
u = 0.213719 + 0.980461I
a = 0.92583 1.26296I
b = 1.45934 0.07190I
4.40938 + 2.90320I 5.62644 3.88291I
u = 0.309046
a = 0.604076
b = 0.507683
0.870053 11.8553
u = 0.423229 0.733264I
a = 0.896141 + 0.367650I
b = 0.201219 + 0.141382I
0.00047 1.65514I 0.39437 + 4.54470I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.423229 + 0.733264I
a = 0.896141 0.367650I
b = 0.201219 0.141382I
0.00047 + 1.65514I 0.39437 4.54470I
u = 0.644830 0.775185I
a = 2.23380 1.34488I
b = 1.360182 0.056984I
4.97796 2.41324I 9.06378 + 3.46829I
u = 0.644830 + 0.775185I
a = 2.23380 + 1.34488I
b = 1.360182 + 0.056984I
4.97796 + 2.41324I 9.06378 3.46829I
u = 0.705788
a = 1.72336
b = 1.53223
7.74304 12.4359
u = 0.900364 0.984023I
a = 1.99724 + 1.52148I
b = 1.60853 + 0.31544I
19.6887 11.0126I 11.01976 + 5.54593I
u = 0.900364 + 0.984023I
a = 1.99724 1.52148I
b = 1.60853 0.31544I
19.6887 + 11.0126I 11.01976 5.54593I
u = 0.903860 0.952544I
a = 1.222938 0.282890I
b = 0.608396 0.924301I
12.41047 6.40086I 9.40627 + 4.90251I
u = 0.903860 + 0.952544I
a = 1.222938 + 0.282890I
b = 0.608396 + 0.924301I
12.41047 + 6.40086I 9.40627 4.90251I
u = 0.918056 0.925441I
a = 0.220309 0.450379I
b = 0.638176 + 0.913496I
12.49898 0.30826I 9.65224 0.25325I
u = 0.918056 + 0.925441I
a = 0.220309 + 0.450379I
b = 0.638176 0.913496I
12.49898 + 0.30826I 9.65224 + 0.25325I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.945672 0.902976I
a = 1.81639 + 0.29726I
b = 1.62133 0.30141I
19.5223 + 4.2287I 11.43737 1.00370I
u = 0.945672 + 0.902976I
a = 1.81639 0.29726I
b = 1.62133 + 0.30141I
19.5223 4.2287I 11.43737 + 1.00370I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
4
u
3
+ u
2
+ 1)(u
32
+ 2u
31
+ ··· + 4u + 1)
c
2
, c
6
, c
7
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
32
+ 6u
31
+ ··· 56u
2
+ 1)
c
3
, c
9
u
4
(u
32
+ u
31
+ ··· 24u 16)
c
4
(u
4
u
3
+ 3u
2
2u + 1)(u
32
+ 6u
31
+ ··· 56u
2
+ 1)
c
5
(u
4
+ u
3
+ u
2
+ 1)(u
32
+ 2u
31
+ ··· + 4u + 1)
c
8
(u + 1)
4
(u
32
+ 5u
31
+ ··· 4u 1)
c
10
, c
11
(u 1)
4
(u
32
+ 5u
31
+ ··· 4u 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
32
+ 6y
31
+ ··· 56y
2
+ 1)
c
2
, c
4
, c
6
c
7
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
32
+ 42y
31
+ ··· 112y + 1)
c
3
, c
9
y
4
(y
32
27y
31
+ ··· + 448y + 256)
c
8
, c
10
, c
11
(y 1)
4
(y
32
35y
31
+ ··· 22y + 1)
9