11n
116
(K11n
116
)
1
Arc Sequences
7 1 8 10 9 2 10 1 5 8 4
Solving Sequence
4,11 1,8
3 2 10 5 7 9 6
c
11
c
3
c
2
c
10
c
4
c
7
c
9
c
5
c
1
, c
6
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
8
+ 7u
6
+ 4u
5
+ 16u
4
+ 10u
3
+ 11u
2
+ 7u + 3,
56u
7
6u
6
+ 421u
5
+ 108u
4
+ 1168u
3
+ 208u
2
+ 397b + 934u + 377,
418u
7
+ 300u
6
2788u
5
+ 158u
4
4408u
3
475u
2
+ 1191a 251u 985i
I
u
2
= hu
8
+ 2u
7
+ 9u
6
+ 10u
5
+ 20u
4
+ 8u
3
+ 7u
2
+ u + 1,
8u
7
+ 2u
6
+ 43u
5
42u
4
+ 38u
3
164u
2
+ 17b + 20u 27,
28u
7
+ 58u
6
+ 244u
5
+ 278u
4
+ 490u
3
+ 191u
2
+ 17a + 53u + 33i
There are 2 irreducible components with 16 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
8
+ 7u
6
+ 4u
5
+ 16u
4
+ 10u
3
+ 11u
2
+ 7u + 3, 56u
7
6u
6
+ · · · +
397b + 377, 418u
7
+ 300u
6
+ · · · + 1191a 985i
(i) Arc colorings
a
4
=
1
0
a
11
=
0.350966u
7
0.251889u
6
+ ··· + 0.210747u + 0.827036
0.141058u
7
+ 0.0151134u
6
+ ··· 2.35264u 0.949622
a
1
=
0.492024u
7
0.267003u
6
+ ··· + 2.56339u + 1.77666
0.141058u
7
+ 0.0151134u
6
+ ··· 2.35264u 0.949622
a
8
=
0
u
a
3
=
1
u
2
a
2
=
0.306465u
7
0.425693u
6
+ ··· + 2.93283u + 1.58102
0.141058u
7
+ 0.0151134u
6
+ ··· 2.35264u + 0.0503778
a
10
=
0.350966u
7
0.251889u
6
+ ··· + 0.210747u + 0.827036
0.256927u
7
+ 0.241814u
6
+ ··· 1.64232u 0.193955
a
5
=
0.316541u
7
+ 0.141058u
6
+ ··· 2.95802u + 1.13686
0.267003u
7
0.0428212u
6
+ ··· 1.66751u 2.47607
a
7
=
0.0772460u
7
+ 0.151134u
6
+ ··· + 2.14022u + 1.83711
0.141058u
7
+ 0.0151134u
6
+ ··· 1.35264u 0.949622
a
9
=
0.590260u
7
+ 0.241814u
6
+ ··· 5.30898u 2.52729
0.256927u
7
0.241814u
6
+ ··· + 2.64232u + 0.193955
a
6
=
0.518892u
7
0.158690u
6
+ ··· 3.29723u 2.52897
0.241814u
7
0.168766u
6
+ ··· + 1.60453u + 0.770781
a
6
=
0.518892u
7
0.158690u
6
+ ··· 3.29723u 2.52897
0.241814u
7
0.168766u
6
+ ··· + 1.60453u + 0.770781
(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 = 0.456155 0.354859I
a = 1.219070 0.568101I
b = 0.188647 + 1.156400I
6.52667 + 1.60807I 7.81804 3.92468I
u = 0.456155 + 0.354859I
a = 1.219070 + 0.568101I
b = 0.188647 1.156400I
6.52667 1.60807I 7.81804 + 3.92468I
u = 0.26697 1.43177I
a = 0.909444 + 0.062444I
b = 1.377938 0.156538I
2.57180 + 1.45446I 1.40377 1.96166I
u = 0.26697 + 1.43177I
a = 0.909444 0.062444I
b = 1.377938 + 0.156538I
2.57180 1.45446I 1.40377 + 1.96166I
u = 0.186474 0.912486I
a = 0.013874 + 1.371961I
b = 0.307955 + 0.595773I
2.35558 + 2.73711I 0.76054 3.80045I
u = 0.186474 + 0.912486I
a = 0.013874 1.371961I
b = 0.307955 0.595773I
2.35558 2.73711I 0.76054 + 3.80045I
u = 0.53665 2.14327I
a = 0.343167 0.006141I
b = 0.874540 0.277933I
6.31045 2.33823I 5.48234 + 1.59126I
u = 0.53665 + 2.14327I
a = 0.343167 + 0.006141I
b = 0.874540 + 0.277933I
6.31045 + 2.33823I 5.48234 1.59126I
3
II.
I
u
2
= hu
8
+2u
7
+· · ·+u+1, 8u
7
+2u
6
+· · ·+17b27, 28u
7
+58u
6
+· · ·+17a+33i
(i) Arc colorings
a
4
=
1
0
a
11
=
1.64706u
7
3.41176u
6
+ ··· 3.11765u 1.94118
0.470588u
7
0.117647u
6
+ ··· 1.17647u + 1.58824
a
1
=
1.17647u
7
3.29412u
6
+ ··· 1.94118u 3.52941
0.470588u
7
0.117647u
6
+ ··· 1.17647u + 1.58824
a
8
=
0
u
a
3
=
1
u
2
a
2
=
7.58824u
7
14.6471u
6
+ ··· 17.4706u 8.76471
2u
7
+ 6u
6
+ 21u
5
+ 34u
4
+ 46u
3
+ 37u
2
+ 2u + 8
a
10
=
1.64706u
7
3.41176u
6
+ ··· 3.11765u 1.94118
0.235294u
7
+ 1.05882u
6
+ ··· + 0.588235u + 1.70588
a
5
=
1.35294u
7
+ 0.588235u
6
+ ··· + 6.88235u 2.94118
2u
7
3u
6
16u
5
12u
4
31u
3
u
2
6u 1
a
7
=
1.64706u
7
1.41176u
6
+ ··· 7.11765u + 3.05882
2.11765u
7
+ 3.52941u
6
+ ··· + 6.29412u + 1.35294
a
9
=
6.52941u
7
+ 11.8824u
6
+ ··· + 15.8235u + 6.58824
2.47059u
7
6.11765u
6
+ ··· 2.17647u 6.41176
a
6
=
3.64706u
7
+ 12.4118u
6
+ ··· + 1.11765u + 20.9412
4u
7
+ 3u
6
+ 25u
5
3u
4
+ 35u
3
44u
2
+ 13u 8
a
6
=
3.64706u
7
+ 12.4118u
6
+ ··· + 1.11765u + 20.9412
4u
7
+ 3u
6
+ 25u
5
3u
4
+ 35u
3
44u
2
+ 13u 8
(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 = 0.66203 1.74906I
a = 0.565751 + 0.130138I
b = 0.274053 + 0.658590I
5.40704 + 2.79901I 1.66145 4.10976I
u = 0.66203 + 1.74906I
a = 0.565751 0.130138I
b = 0.274053 0.658590I
5.40704 2.79901I 1.66145 + 4.10976I
u = 0.270939 0.522049I
a = 3.45142 1.23452I
b = 0.44567 + 2.88321I
4.85825 + 1.12061I 1.61393 0.60117I
u = 0.270939 + 0.522049I
a = 3.45142 + 1.23452I
b = 0.44567 2.88321I
4.85825 1.12061I 1.61393 + 0.60117I
u = 0.17191 2.00694I
a = 1.16757 0.88058I
b = 2.85593 2.12606I
19.3281 + 7.7545I 1.70851 2.41364I
u = 0.17191 + 2.00694I
a = 1.16757 + 0.88058I
b = 2.85593 + 2.12606I
19.3281 7.7545I 1.70851 + 2.41364I
u = 0.104875 0.438980I
a = 0.849602 0.031731I
b = 0.136204 0.426385I
0.137633 1.005542I 2.51610 + 6.63610I
u = 0.104875 + 0.438980I
a = 0.849602 + 0.031731I
b = 0.136204 + 0.426385I
0.137633 + 1.005542I 2.51610 6.63610I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
8
+ u
7
+ 4u
6
+ 3u
5
+ 6u
4
+ 3u
3
+ 5u
2
+ u + 1)
(u
8
+ 3u
7
+ 12u
6
+ 15u
5
+ 32u
4
+ 9u
3
+ 27u
2
21u + 11)
c
2
(u
8
+ 7u
7
+ 22u
6
+ 43u
5
+ 58u
4
+ 53u
3
+ 31u
2
+ 9u + 1)
(u
8
+ 15u
7
+ ··· + 153u + 121)
c
3
(u
8
+ 7u
6
+ 4u
5
+ 16u
4
+ 10u
3
+ 11u
2
+ 7u + 3)
(u
8
+ 2u
7
+ 9u
6
+ 10u
5
+ 20u
4
+ 8u
3
+ 7u
2
+ u + 1)
c
4
, c
5
(u
8
+ 4u
6
+ 6u
4
u
3
+ 5u
2
u + 1)
(u
8
+ 8u
6
+ 4u
5
+ 36u
4
+ 29u
3
+ 41u
2
17u + 19)
c
6
(u
8
u
7
+ 4u
6
3u
5
+ 6u
4
3u
3
+ 5u
2
u + 1)
(u
8
+ 3u
7
+ 12u
6
+ 15u
5
+ 32u
4
+ 9u
3
+ 27u
2
21u + 11)
c
7
(u
8
+ 2u
7
+ 2u
6
+ u
5
2u
4
2u
3
+ 1)
(u
8
+ 2u
7
+ 8u
6
+ 3u
5
+ 38u
4
38u
3
+ 58u
2
12u + 7)
c
8
(u
8
2u
5
2u
4
+ u
3
+ 2u
2
+ 2u + 1)
(u
8
+ 2u
7
+ 10u
6
6u
5
+ 168u
4
139u
3
+ 116u
2
+ 60u + 47)
c
9
(u
8
+ 4u
6
+ 6u
4
+ u
3
+ 5u
2
+ u + 1)
(u
8
+ 8u
6
+ 4u
5
+ 36u
4
+ 29u
3
+ 41u
2
17u + 19)
c
10
(u
8
2u
7
+ 2u
6
u
5
2u
4
+ 2u
3
+ 1)
(u
8
+ 2u
7
+ 8u
6
+ 3u
5
+ 38u
4
38u
3
+ 58u
2
12u + 7)
c
11
(u
8
2u
7
+ u
6
2u
4
+ 2u
3
+ u
2
u + 1)
(u
8
+ 4u
7
+ 13u
6
+ 28u
5
+ 90u
4
60u
3
+ 85u
2
23u + 11)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
8
+ 7y
7
+ 22y
6
+ 43y
5
+ 58y
4
+ 53y
3
+ 31y
2
+ 9y + 1)
(y
8
+ 15y
7
+ ··· + 153y + 121)
c
2
(y
8
5y
7
2y
6
+ 23y
5
+ 46y
4
+ 57y
3
+ 123y
2
19y + 1)
(y
8
+ 11y
7
+ ··· + 414853y + 14641)
c
3
(y
8
+ 14y
7
+ 81y
6
+ 230y
5
+ 336y
4
+ 238y
3
+ 77y
2
+ 17y + 9)
(y
8
+ 14y
7
+ 81y
6
+ 242y
5
+ 364y
4
+ 214y
3
+ 73y
2
+ 13y + 1)
c
4
, c
5
, c
9
(y
8
+ 8y
7
+ 28y
6
+ 58y
5
+ 78y
4
+ 67y
3
+ 35y
2
+ 9y + 1)
(y
8
+ 16y
7
+ ··· + 1269y + 361)
c
7
, c
10
(y
8
4y
6
+ ··· 4y
2
+ 1)(y
8
+ 12y
7
+ ··· + 668y + 49)
c
8
(y
8
4y
6
+ ··· 4y
2
+ 1)(y
8
+ 16y
7
+ ··· + 7304y + 2209)
c
11
(y
8
2y
7
3y
6
+ 6y
5
+ 4y
4
6y
3
+ y
2
+ y + 1)
(y
8
+ 10y
7
+ ··· + 1341y + 121)
7