11a
6
(K11a
6
)
1
Arc Sequences
5 1 8 2 3 11 9 4 6 7 10
Solving Sequence
1,5
2 3
6,9
10 4 8 7 11
c
1
c
2
c
5
c
9
c
4
c
8
c
7
c
11
c
3
, c
6
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu
17
u
16
+ ··· + u 1, u
8
2u
6
2u
4
+ b,
u
16
+ u
15
4u
14
+ 3u
13
7u
12
+ 5u
11
6u
10
+ 4u
9
u
8
+ 2u
7
+ u
6
+ u
5
+ u
3
+ a + 1i
I
u
2
= hu
54
2u
53
+ ··· u + 1, 3u
53
5u
52
+ ··· + a + 2u, u
53
+ 4u
52
+ ··· + b + 3i
I
u
3
= hu
2
+ u + 1, b u, a 1i
I
u
4
= hu
2
+ u + 1, a u, b + u + 1i
There are 4 irreducible components with 75 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
17
u
16
+ · · · + u 1, u
8
2u
6
2u
4
+ b, u
16
+ u
15
+ · · · + a + 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
5
2u
3
u
u
5
+ u
3
+ u
a
9
=
u
16
u
15
+ ··· u
3
1
u
8
+ 2u
6
+ 2u
4
a
10
=
u
16
u
15
+ ··· + u
2
1
u
2
a
4
=
u
u
3
+ u
a
8
=
u
12
3u
10
5u
8
4u
6
2u
4
u
2
1
u
13
+ 3u
11
+ 5u
9
+ 4u
7
+ 2u
5
+ u
3
+ u
a
7
=
u
13
u
12
+ ··· + u 1
u
3
+ u
a
11
=
u
16
+ u
15
+ ··· + u + 1
u
4
a
11
=
u
16
+ u
15
+ ··· + u + 1
u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
16
+ 6u
15
18u
14
+ 22u
13
40u
12
+ 46u
11
56u
10
+ 54u
9
54u
8
+ 44u
7
44u
6
+ 28u
5
26u
4
+ 22u
3
12u
2
+ 10u 8
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.701375 0.463501I
a = 1.082843 + 0.269282I
b = 1.362306 + 0.221163I
3.82884 + 0.36538I 2.43059 2.03934I
u = 0.701375 + 0.463501I
a = 1.082843 0.269282I
b = 1.362306 0.221163I
3.82884 0.36538I 2.43059 + 2.03934I
u = 0.572214 1.088104I
a = 0.79850 + 1.62987I
b = 1.19525 + 0.83888I
0.06925 + 9.48553I 4.16847 7.65622I
u = 0.572214 + 1.088104I
a = 0.79850 1.62987I
b = 1.19525 0.83888I
0.06925 9.48553I 4.16847 + 7.65622I
u = 0.364031 1.042941I
a = 1.27288 0.62337I
b = 0.241558 0.594295I
3.22508 + 4.12748I 8.88917 5.53460I
u = 0.364031 + 1.042941I
a = 1.27288 + 0.62337I
b = 0.241558 + 0.594295I
3.22508 4.12748I 8.88917 + 5.53460I
u = 0.237306 0.655876I
a = 1.199701 0.284065I
b = 0.146257 0.284741I
0.38141 + 1.45461I 4.03529 4.09951I
u = 0.237306 + 0.655876I
a = 1.199701 + 0.284065I
b = 0.146257 + 0.284741I
0.38141 1.45461I 4.03529 + 4.09951I
u = 0.228107 1.129705I
a = 0.725979 + 0.091052I
b = 0.67587 + 1.27508I
6.66052 + 0.33441I 12.27972 + 0.14725I
u = 0.228107 + 1.129705I
a = 0.725979 0.091052I
b = 0.67587 1.27508I
6.66052 0.33441I 12.27972 0.14725I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.451021 1.148487I
a = 0.818290 + 0.431998I
b = 0.563925 0.097421I
8.03366 8.05681I 11.19202 + 7.52466I
u = 0.451021 + 1.148487I
a = 0.818290 0.431998I
b = 0.563925 + 0.097421I
8.03366 + 8.05681I 11.19202 7.52466I
u = 0.601205 1.123527I
a = 2.11914 1.02041I
b = 1.47816 1.71279I
1.4738 15.8440I 5.36436 + 10.86165I
u = 0.601205 + 1.123527I
a = 2.11914 + 1.02041I
b = 1.47816 + 1.71279I
1.4738 + 15.8440I 5.36436 10.86165I
u = 0.621462
a = 1.44826
b = 0.435791
1.88169 4.17973
u = 0.783861 0.397949I
a = 1.44148 + 1.32259I
b = 1.52877 1.22485I
2.83523 + 5.37992I 0.58832 3.10862I
u = 0.783861 + 0.397949I
a = 1.44148 1.32259I
b = 1.52877 + 1.22485I
2.83523 5.37992I 0.58832 + 3.10862I
4
II.
I
u
2
= hu
54
2u
53
+· · ·u+1, 3u
53
5u
52
+· · ·+a+2u, u
53
+4u
52
+· · ·+b+3i
(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
5
2u
3
u
u
5
+ u
3
+ u
a
9
=
3u
53
+ 5u
52
+ ··· + u
2
2u
u
53
4u
52
+ ··· + 3u 3
a
10
=
2u
53
+ 3u
52
+ ··· 2u 2
u
53
3u
52
+ ··· + 3u 1
a
4
=
u
u
3
+ u
a
8
=
5u
53
+ 9u
52
+ ··· 4u + 2
u
53
6u
52
+ ··· + 3u 5
a
7
=
u
53
+ u
52
+ ··· 6u
2
2
u
53
2u
52
+ ··· + 3u 1
a
11
=
2u
53
3u
52
+ ··· + 2u + 1
u
53
+ 2u
52
+ ··· 2u + 1
a
11
=
2u
53
3u
52
+ ··· + 2u + 1
u
53
+ 2u
52
+ ··· 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
53
8u
52
+ ··· + 15u 5
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.748850 0.603065I
a = 1.257701 0.521171I
b = 1.36566 + 0.63238I
1.90449 + 7.58447I 1.82775 8.11380I
u = 0.748850 + 0.603065I
a = 1.257701 + 0.521171I
b = 1.36566 0.63238I
1.90449 7.58447I 1.82775 + 8.11380I
u = 0.725685 0.567873I
a = 1.198207 + 0.442030I
b = 1.344202 0.366111I
3.74831 + 2.63920I 1.74271 3.37289I
u = 0.725685 + 0.567873I
a = 1.198207 0.442030I
b = 1.344202 + 0.366111I
3.74831 2.63920I 1.74271 + 3.37289I
u = 0.709728 0.409816I
a = 1.025253 0.181668I
b = 1.39491 0.49640I
2.05766 4.56252I 0.66909 + 3.14948I
u = 0.709728 + 0.409816I
a = 1.025253 + 0.181668I
b = 1.39491 + 0.49640I
2.05766 + 4.56252I 0.66909 3.14948I
u = 0.645440 0.992497I
a = 0.03712 + 1.57156I
b = 1.224477 0.290802I
0.74861 2.30237I 3.82172 + 3.52235I
u = 0.645440 + 0.992497I
a = 0.03712 1.57156I
b = 1.224477 + 0.290802I
0.74861 + 2.30237I 3.82172 3.52235I
u = 0.626309 0.880271I
a = 0.369236 + 1.037847I
b = 0.401870 0.581601I
1.01872 + 3.73043I 9.17515 3.57270I
u = 0.626309 + 0.880271I
a = 0.369236 1.037847I
b = 0.401870 + 0.581601I
1.01872 3.73043I 9.17515 + 3.57270I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.622213 0.701611I
a = 0.912937 0.621751I
b = 0.496987 + 0.515968I
0.538859 + 1.164106I 6.09920 3.89817I
u = 0.622213 + 0.701611I
a = 0.912937 + 0.621751I
b = 0.496987 0.515968I
0.538859 1.164106I 6.09920 + 3.89817I
u = 0.616080 1.011226I
a = 0.24972 1.52010I
b = 1.168791 0.018643I
2.43288 + 2.48370I 0.17393 1.70527I
u = 0.616080 + 1.011226I
a = 0.24972 + 1.52010I
b = 1.168791 + 0.018643I
2.43288 2.48370I 0.17393 + 1.70527I
u = 0.578440 1.064248I
a = 0.65704 1.56537I
b = 1.159007 0.604773I
2.05766 + 4.56252I 0.66909 3.14948I
u = 0.578440 + 1.064248I
a = 0.65704 + 1.56537I
b = 1.159007 + 0.604773I
2.05766 4.56252I 0.66909 + 3.14948I
u = 0.511133 1.056253I
a = 0.87815 + 1.22158I
b = 0.612467 + 0.733542I
2.23363 + 2.56106I 7.40701 2.25118I
u = 0.511133 + 1.056253I
a = 0.87815 1.22158I
b = 0.612467 0.733542I
2.23363 2.56106I 7.40701 + 2.25118I
u = 0.347654 0.291444I
a = 0.845131 0.726072I
b = 0.369832 0.532221I
0.36265 + 1.51655I 2.55288 3.58996I
u = 0.347654 + 0.291444I
a = 0.845131 + 0.726072I
b = 0.369832 + 0.532221I
0.36265 1.51655I 2.55288 + 3.58996I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.312561 0.860769I
a = 1.073772 + 0.083607I
b = 0.0869365 + 0.0128869I
0.36265 + 1.51655I 2.55288 3.58996I
u = 0.312561 + 0.860769I
a = 1.073772 0.083607I
b = 0.0869365 0.0128869I
0.36265 1.51655I 2.55288 + 3.58996I
u = 0.227574 1.026587I
a = 1.44381 0.24788I
b = 0.643193 0.129967I
2.23363 2.56106I 7.40701 + 2.25118I
u = 0.227574 + 1.026587I
a = 1.44381 + 0.24788I
b = 0.643193 + 0.129967I
2.23363 + 2.56106I 7.40701 2.25118I
u = 0.151780 1.152756I
a = 1.023369 + 0.287473I
b = 0.86925 + 1.32297I
4.41486 + 8.03203I 8.93584 6.03539I
u = 0.151780 + 1.152756I
a = 1.023369 0.287473I
b = 0.86925 1.32297I
4.41486 8.03203I 8.93584 + 6.03539I
u = 0.155842 1.113422I
a = 1.042462 0.137530I
b = 0.81342 1.22920I
2.12405 + 3.04478I 5.96953 2.28005I
u = 0.155842 + 1.113422I
a = 1.042462 + 0.137530I
b = 0.81342 + 1.22920I
2.12405 3.04478I 5.96953 + 2.28005I
u = 0.239448 0.923216I
a = 1.40550 + 0.49796I
b = 0.49339 1.36458I
0.538859 + 1.164106I 6.09920 3.89817I
u = 0.239448 + 0.923216I
a = 1.40550 0.49796I
b = 0.49339 + 1.36458I
0.538859 1.164106I 6.09920 + 3.89817I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.349223 0.956682I
a = 1.08643 1.19944I
b = 0.97199 + 1.59556I
1.01872 3.73043I 9.17515 + 3.57270I
u = 0.349223 + 0.956682I
a = 1.08643 + 1.19944I
b = 0.97199 1.59556I
1.01872 + 3.73043I 9.17515 3.57270I
u = 0.387985 1.148028I
a = 0.299689 + 0.237241I
b = 0.487015 0.657483I
8.45612 12.3906
u = 0.387985 + 1.148028I
a = 0.299689 0.237241I
b = 0.487015 + 0.657483I
8.45612 12.3906
u = 0.425486 1.115576I
a = 0.487994 0.628916I
b = 0.810221 + 0.417244I
4.91302 3.80494I 7.92053 + 4.08050I
u = 0.425486 + 1.115576I
a = 0.487994 + 0.628916I
b = 0.810221 0.417244I
4.91302 + 3.80494I 7.92053 4.08050I
u = 0.564050 1.052374I
a = 1.68598 1.63791I
b = 2.21082 0.79468I
0.74861 2.30237I 3.82172 + 3.52235I
u = 0.564050 + 1.052374I
a = 1.68598 + 1.63791I
b = 2.21082 + 0.79468I
0.74861 + 2.30237I 3.82172 3.52235I
u = 0.567778 1.119963I
a = 1.81424 0.99171I
b = 1.34130 1.22618I
4.41486 8.03203I 8.93584 + 6.03539I
u = 0.567778 + 1.119963I
a = 1.81424 + 0.99171I
b = 1.34130 + 1.22618I
4.41486 + 8.03203I 8.93584 6.03539I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.578445 1.073080I
a = 1.84435 + 1.44447I
b = 2.02639 + 1.10433I
1.90449 7.58447I 1.82775 + 8.11380I
u = 0.578445 + 1.073080I
a = 1.84435 1.44447I
b = 2.02639 1.10433I
1.90449 + 7.58447I 1.82775 8.11380I
u = 0.595230 1.111931I
a = 2.04735 + 1.11182I
b = 1.60523 + 1.55963I
0.71725 10.56857I 2.49476 + 7.09212I
u = 0.595230 + 1.111931I
a = 2.04735 1.11182I
b = 1.60523 1.55963I
0.71725 + 10.56857I 2.49476 7.09212I
u = 0.664646 0.482177I
a = 1.94963 1.20862I
b = 1.90963 + 0.33212I
2.43288 2.48370I 0.17393 + 1.70527I
u = 0.664646 + 0.482177I
a = 1.94963 + 1.20862I
b = 1.90963 0.33212I
2.43288 + 2.48370I 0.17393 1.70527I
u = 0.708611 0.447215I
a = 1.76213 + 1.25492I
b = 1.79693 0.66978I
3.74831 + 2.63920I 1.74271 3.37289I
u = 0.708611 + 0.447215I
a = 1.76213 1.25492I
b = 1.79693 + 0.66978I
3.74831 2.63920I 1.74271 + 3.37289I
u = 0.721512 0.069386I
a = 1.165027 + 0.222644I
b = 0.219703 0.235284I
4.91302 + 3.80494I 7.92053 4.08050I
u = 0.721512 + 0.069386I
a = 1.165027 0.222644I
b = 0.219703 + 0.235284I
4.91302 3.80494I 7.92053 + 4.08050I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.751654 0.335841I
a = 1.39110 1.07181I
b = 1.13024 + 0.98401I
2.12405 + 3.04478I 5.96953 2.28005I
u = 0.751654 + 0.335841I
a = 1.39110 + 1.07181I
b = 1.13024 0.98401I
2.12405 3.04478I 5.96953 + 2.28005I
u = 0.809979 0.388169I
a = 1.33938 1.35984I
b = 1.44816 + 1.42006I
0.71725 + 10.56857I 2.49476 7.09212I
u = 0.809979 + 0.388169I
a = 1.33938 + 1.35984I
b = 1.44816 1.42006I
0.71725 10.56857I 2.49476 + 7.09212I
11
III. I
u
3
= hu
2
+ u + 1, b u, a 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
6
=
1
0
a
9
=
1
u
a
10
=
u + 1
u
a
4
=
u
u + 1
a
8
=
1
u
a
7
=
1
u
a
11
=
2u
u + 1
a
11
=
2u
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 1.00000
b = 0.500000 0.866025I
0 3.00000
u = 0.500000 + 0.866025I
a = 1.00000
b = 0.500000 + 0.866025I
0 3.00000
13
IV. I
u
4
= hu
2
+ u + 1, a u, b + u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
6
=
1
0
a
9
=
u
u 1
a
10
=
2u + 1
u 1
a
4
=
u
u + 1
a
8
=
u
u 1
a
7
=
u
u 1
a
11
=
u
u
a
11
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u + 4
14
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 0.500000 + 0.866025I
4.05977I 6.92820I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0.500000 0.866025I
4.05977I 6.92820I
15
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
2
+ u + 1)
2
(u
17
+ u
16
+ ··· + u + 1)(u
54
+ 2u
53
+ ··· + u + 1)
c
2
, c
11
(u
2
+ u + 1)
2
(u
17
+ 9u
16
+ ··· 3u 1)(u
54
+ 24u
53
+ ··· + 5u + 1)
c
3
, c
8
u
4
(u
17
+ 5u
16
+ ··· 8u 4)
(2 + 3u + 4u
2
15u
3
+ 4u
4
+ 29u
5
28u
6
34u
7
+ 50u
8
+ 34u
9
66u
10
36u
11
+ 68u
12
+ 46u
13
72u
14
52u
15
+ 76u
16
+ 41u
17
70u
18
21u
19
+ 52u
20
+ 3u
21
28u
22
+ 4u
23
+ 10u
24
3u
25
2u
26
+ u
27
)
2
c
4
, c
10
(u
2
u + 1)
2
(u
17
+ u
16
+ ··· + u + 1)(u
54
+ 2u
53
+ ··· + u + 1)
c
5
, c
9
(u
2
+ u + 1)
2
(u
17
+ u
16
+ ··· u 2)(u
54
+ 2u
53
+ ··· + 145u + 17)
c
7
u
4
(u
17
+ 5u
16
+ ··· + 56u
2
+ 16)
(4 + 25u + 90u
2
+ 255u
3
+ 666u
4
+ 1625u
5
+ 3592u
6
+ 6980u
7
+ 1.20 × 10
4
u
8
+ 1.82 × 10
4
u
9
+ 2.50 × 10
4
u
10
+ 3.15 × 10
4
u
11
+ 3.67 × 10
4
u
12
+ 4.00 × 10
4
u
13
+ 4.06 × 10
4
u
14
+ 3.84 × 10
4
u
15
+ 3.36 × 10
4
u
16
+ 2.70 × 10
4
u
17
+ 1.98 × 10
4
u
18
+ 1.32 × 10
4
u
19
+ 7826u
20
+ 4107u
21
+ 1872u
22
+ 724u
23
+ 230u
24
+ 57u
25
+ 10u
26
+ u
27
)
2
16
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
6
c
10
(y
2
+ y + 1)
2
(y
17
+ 9y
16
+ ··· 3y 1)(y
54
+ 24y
53
+ ··· + 5y + 1)
c
2
, c
11
(y
2
+ y + 1)
2
(y
17
+ y
16
+ ··· + 5y 1)(y
54
+ 12y
53
+ ··· + 53y + 1)
c
3
, c
8
y
4
(y
17
5y
16
+ ··· 56y
2
16)
(4 + 25y 90y
2
+ 255y
3
666y
4
+ 1625y
5
3592y
6
+ 6980y
7
1.20 × 10
4
y
8
+ 1.82 × 10
4
y
9
2.50 × 10
4
y
10
+ 3.15 × 10
4
y
11
3.67 × 10
4
y
12
+ 4.00 × 10
4
y
13
4.06 × 10
4
y
14
+ 3.84 × 10
4
y
15
3.36 × 10
4
y
16
+ 2.70 × 10
4
y
17
1.98 × 10
4
y
18
+ 1.32 × 10
4
y
19
7826y
20
+ 4107y
21
1872y
22
+ 724y
23
230y
24
+ 57y
25
10y
26
+ y
27
)
2
c
5
, c
9
(y
2
+ y + 1)
2
(y
17
7y
16
+ ··· + y 4)
(y
54
+ 38y
52
+ ··· 4637y + 289)
c
7
y
4
(y
17
+ 7y
16
+ ··· 1792y 256)
(16 95y 678y
2
2341y
3
7982y
4
2.66 × 10
4
y
5
8.57 × 10
4
y
6
2.41 × 10
5
y
7
5.37 × 10
5
y
8
9.24 × 10
5
y
9
1.23 × 10
6
y
10
1.26 × 10
6
y
11
9.68 × 10
5
y
12
4.95 × 10
5
y
13
9.53 × 10
4
y
14
+ 7.83 × 10
4
y
15
+ 6.42 × 10
4
y
16
4739y
17
3.92 × 10
4
y
18
3.25 × 10
4
y
19
1.38 × 10
4
y
20
2029y
21
+ 1320y
22
+ 1076y
23
+ 410y
24
+ 97y
25
+ 14y
26
+ y
27
)
2
17