11a
55
(K11a
55
)
1
Arc Sequences
4 1 8 2 10 11 9 3 5 6 7
Solving Sequence
5,9
10 6 11
2,7
4 1 3 8
c
9
c
5
c
10
c
6
c
4
c
1
c
2
c
8
c
3
, c
7
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
2
+ b 1, b + a, u 1i
I
u
2
= hu
37
3u
36
+ ··· 2u + 1, 5u
36
12u
35
+ ··· + 2a 5, 5u
36
14u
35
+ ··· + 2b 9i
There are 2 irreducible components with 39 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
2
+ b 1, b + a, u 1i
(i) Arc colorings
a
5
=
0
1
a
9
=
b
b
a
10
=
b
0
a
6
=
b + 1
1
a
11
=
b + 1
b
a
2
=
1
0
a
7
=
b
b
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
8
=
b
b
a
8
=
b
b
(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 = 1.61803
b = 1.61803
7.23771 5.00000
u = 1.00000
a = 0.618034
b = 0.618034
0.657974 5.00000
3
II. I
u
2
=
hu
37
3u
36
+· · ·2u+1, 5u
36
12u
35
+· · ·+2a5, 5u
36
14u
35
+· · ·+2b9i
(i) Arc colorings
a
5
=
0
u
a
9
=
5
2
u
36
+ 6u
35
+ ···
5
2
u +
5
2
5
2
u
36
+ 7u
35
+ ···
9
2
u +
9
2
a
10
=
5
2
u
36
+ 6u
35
+ ···
5
2
u +
5
2
2u
36
+ 5u
35
+ ··· 4u + 3
a
6
=
1
2
u
36
+ u
35
+ ···
1
2
u
1
2
u
14
+ 4u
12
2u
11
7u
10
+ 6u
9
+ 4u
8
8u
7
+ 2u
6
+ 4u
5
4u
4
+ u
2
a
11
=
1
2
u
36
+ u
35
+ ···
1
2
u +
1
2
1
2
u
36
+ u
35
+ ···
3
2
u +
1
2
a
2
=
1
0
a
7
=
7
2
u
36
+ 8u
35
+ ···
9
2
u +
5
2
1
2
u
36
+ u
35
+ ···
3
2
u +
1
2
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
8
=
11
2
u
36
+ 12u
35
+ ···
15
2
u +
9
2
9
2
u
36
+ 9u
35
+ ···
11
2
u +
9
2
a
8
=
11
2
u
36
+ 12u
35
+ ···
15
2
u +
9
2
9
2
u
36
+ 9u
35
+ ···
11
2
u +
9
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 = 1.292490 0.226502I
a = 1.266095 + 0.175912I
b = 1.88132 0.28860I
7.80383 + 2.56815I 4.68234 2.67332I
u = 1.292490 + 0.226502I
a = 1.266095 0.175912I
b = 1.88132 + 0.28860I
7.80383 2.56815I 4.68234 + 2.67332I
u = 1.202165 0.240427I
a = 0.698801 + 0.075518I
b = 1.054060 + 0.770930I
1.03449 + 1.41041I 2.89217 4.96755I
u = 1.202165 + 0.240427I
a = 0.698801 0.075518I
b = 1.054060 0.770930I
1.03449 1.41041I 2.89217 + 4.96755I
u = 1.107221 0.345026I
a = 0.093196 0.419052I
b = 0.20139 1.59348I
3.33947 1.17576I 4.43128 + 1.03066I
u = 1.107221 + 0.345026I
a = 0.093196 + 0.419052I
b = 0.20139 + 1.59348I
3.33947 + 1.17576I 4.43128 1.03066I
u = 1.068482 0.534952I
a = 1.234777 0.686344I
b = 1.32008 2.75090I
8.52780 5.28278I 4.22428 + 3.63452I
u = 1.068482 + 0.534952I
a = 1.234777 + 0.686344I
b = 1.32008 + 2.75090I
8.52780 + 5.28278I 4.22428 3.63452I
u = 1.065336 0.455792I
a = 0.600494 + 0.534236I
b = 0.66421 + 2.20189I
0.53133 3.88210I 2.57643 + 5.18911I
u = 1.065336 + 0.455792I
a = 0.600494 0.534236I
b = 0.66421 2.20189I
0.53133 + 3.88210I 2.57643 5.18911I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.895459
a = 0.225285
b = 0.328595
1.30402 9.26699
u = 0.413609 0.613148I
a = 1.30913 1.85786I
b = 0.995830 0.873919I
10.43059 + 0.72718I 6.63678 + 1.09583I
u = 0.413609 + 0.613148I
a = 1.30913 + 1.85786I
b = 0.995830 + 0.873919I
10.43059 0.72718I 6.63678 1.09583I
u = 0.346933 0.362873I
a = 1.27013 + 1.04308I
b = 0.729275 + 0.348350I
1.50853 + 0.14938I 6.45155 + 0.46456I
u = 0.346933 + 0.362873I
a = 1.27013 1.04308I
b = 0.729275 0.348350I
1.50853 0.14938I 6.45155 0.46456I
u = 0.264993 0.644063I
a = 0.763761 0.058524I
b = 0.455045 0.323893I
0.29596 1.82108I 2.47769 + 3.83748I
u = 0.264993 + 0.644063I
a = 0.763761 + 0.058524I
b = 0.455045 + 0.323893I
0.29596 + 1.82108I 2.47769 3.83748I
u = 0.291593 0.810846I
a = 0.717144 0.923118I
b = 0.572017 0.001336I
3.73036 4.62550I 7.76738 + 4.90690I
u = 0.291593 + 0.810846I
a = 0.717144 + 0.923118I
b = 0.572017 + 0.001336I
3.73036 + 4.62550I 7.76738 4.90690I
u = 0.303756 0.900467I
a = 0.65488 + 1.67244I
b = 0.688101 + 0.324075I
13.1327 6.1887I 8.61498 + 3.29397I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.303756 + 0.900467I
a = 0.65488 1.67244I
b = 0.688101 0.324075I
13.1327 + 6.1887I 8.61498 3.29397I
u = 0.524588 0.440300I
a = 0.145703 + 0.839859I
b = 0.523657 + 0.627013I
1.20413 + 1.03970I 6.27276 4.95197I
u = 0.524588 + 0.440300I
a = 0.145703 0.839859I
b = 0.523657 0.627013I
1.20413 1.03970I 6.27276 + 4.95197I
u = 0.765250 0.648990I
a = 0.777976 0.766069I
b = 0.374237 0.818529I
6.20655 + 2.48097I 9.67939 3.72325I
u = 0.765250 + 0.648990I
a = 0.777976 + 0.766069I
b = 0.374237 + 0.818529I
6.20655 2.48097I 9.67939 + 3.72325I
u = 0.808570 0.765765I
a = 1.11915 + 1.07868I
b = 0.275422 + 1.378049I
16.2880 + 2.8364I 9.81426 2.86801I
u = 0.808570 + 0.765765I
a = 1.11915 1.07868I
b = 0.275422 1.378049I
16.2880 2.8364I 9.81426 + 2.86801I
u = 0.985925 0.345436I
a = 1.45751 0.44914I
b = 1.50123 + 0.26577I
7.10974 + 1.19498I 5.28072 5.41154I
u = 0.985925 + 0.345436I
a = 1.45751 + 0.44914I
b = 1.50123 0.26577I
7.10974 1.19498I 5.28072 + 5.41154I
u = 1.051550 0.471969I
a = 0.684498 0.105620I
b = 0.446515 1.072806I
0.46634 + 2.82395I 2.81248 2.07751I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.051550 + 0.471969I
a = 0.684498 + 0.105620I
b = 0.446515 + 1.072806I
0.46634 2.82395I 2.81248 + 2.07751I
u = 1.115345 0.521228I
a = 0.118843 + 0.497110I
b = 0.26744 + 1.73358I
2.10926 + 6.36685I 0.76306 6.73734I
u = 1.115345 + 0.521228I
a = 0.118843 0.497110I
b = 0.26744 1.73358I
2.10926 6.36685I 0.76306 + 6.73734I
u = 1.151514 0.569123I
a = 0.529684 0.596376I
b = 1.11367 2.02963I
1.18638 + 9.75247I 4.12651 8.53256I
u = 1.151514 + 0.569123I
a = 0.529684 + 0.596376I
b = 1.11367 + 2.02963I
1.18638 9.75247I 4.12651 + 8.53256I
u = 1.180879 0.602533I
a = 1.107721 + 0.652171I
b = 1.85086 + 2.25935I
10.4876 + 11.6846I 5.51812 6.88609I
u = 1.180879 + 0.602533I
a = 1.107721 0.652171I
b = 1.85086 2.25935I
10.4876 11.6846I 5.51812 + 6.88609I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
2
(u
37
+ 3u
36
+ ··· 2u 1)
c
2
(u + 1)
2
(u
37
+ 19u
36
+ ··· + 4u + 1)
c
3
, c
8
u
2
(u
37
+ u
36
+ ··· 3u
2
4)
c
4
(u + 1)
2
(u
37
+ 3u
36
+ ··· 2u 1)
c
5
, c
6
(u
2
u 1)(u
37
+ 2u
36
+ ··· + u 1)
c
7
u
2
(u
37
+ 15u
36
+ ··· 24u 16)
c
9
, c
10
, c
11
(u
2
+ u 1)(u
37
+ 2u
36
+ ··· + u 1)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
2
(y
37
19y
36
+ ··· + 4y 1)
c
2
(y 1)
2
(y
37
+ y
36
+ ··· 44y 1)
c
3
, c
8
y
2
(y
37
+ 15y
36
+ ··· 24y 16)
c
5
, c
6
, c
9
(y
2
3y + 1)(y
37
48y
36
+ ··· + 25y 1)
c
7
y
2
(y
37
+ 11y
36
+ ··· + 7712y 256)
c
10
(y
2
3y + 1)(y
37
48y
36
+ ··· + 25y 1)
c
11
(y
2
3y + 1)(y
37
48y
36
+ ··· + 25y 1)
10