11n
113
(K11n
113
)
1
Arc Sequences
6 1 7 9 2 10 1 5 4 7 4
Solving Sequence
7,10 4,11
1 3 2 6 9 5 8
c
10
c
11
c
3
c
2
c
6
c
9
c
4
c
8
c
1
, c
5
, c
7
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
8
+ 5u
6
u
5
+ 10u
4
3u
3
+ 8u
2
2u + 1, b + u, 2u
7
+ 9u
5
2u
4
+ 16u
3
5u
2
+ a + 10u 2i
I
u
2
= hu
12
u
11
+ 8u
10
7u
9
+ 26u
8
13u
7
+ 53u
6
+ 7u
5
+ 80u
4
+ 38u
3
+ 72u
2
+ 24u + 19,
15310u
11
9930u
10
+ ··· + 116501b + 188420,
26132u
11
+ 71846u
10
+ ··· + 316217a + 554233i
I
u
3
= hu
13
+ 9u
11
+ u
10
+ 29u
9
+ 6u
8
+ 37u
7
+ 10u
6
+ 14u
5
+ 3u
4
+ 5u
3
2u
2
+ u 1, b u,
10u
12
51u
11
+ ··· + 77a + 69i
There are 3 irreducible components with 33 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
+ 5u
6
u
5
+ 10u
4
3u
3
+ 8u
2
2u + 1, b + u, 2u
7
+ 9u
5
2u
4
+ 16u
3
5u
2
+ a + 10u 2i
(i) Arc colorings
a
7
=
0
u
a
10
=
2u
7
9u
5
+ 2u
4
16u
3
+ 5u
2
10u + 2
u
a
4
=
1
0
a
11
=
2u
7
9u
5
+ 2u
4
16u
3
+ 5u
2
10u + 2
u
7
4u
5
+ u
4
6u
3
+ 2u
2
3u
a
1
=
u
7
5u
5
+ u
4
10u
3
+ 3u
2
7u + 2
u
7
4u
5
+ u
4
6u
3
+ 2u
2
3u
a
3
=
1
u
2
a
2
=
u
4
+ 3u
2
+ 4
u
7
4u
5
+ u
4
6u
3
+ 3u
2
3u + 1
a
6
=
u
7
+ u
6
+ 6u
5
+ 3u
4
+ 12u
3
+ 2u
2
+ 9u 1
u
7
+ 4u
5
u
4
+ 6u
3
2u
2
+ 3u
a
9
=
2u
7
9u
5
+ 2u
4
16u
3
+ 5u
2
9u + 2
u
a
5
=
u
6
+ 4u
4
u
3
+ 7u
2
2u + 3
u
2
a
8
=
u
7
5u
5
+ u
4
9u
3
+ 3u
2
6u + 2
u
3
u
a
8
=
u
7
5u
5
+ u
4
9u
3
+ 3u
2
6u + 2
u
3
u
(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 = 0.53500 1.45526I
a = 0.361874 0.050334I
b = 0.53500 + 1.45526I
8.16060 + 3.11503I 0.52239 1.94780I
u = 0.53500 + 1.45526I
a = 0.361874 + 0.050334I
b = 0.53500 1.45526I
8.16060 3.11503I 0.52239 + 1.94780I
u = 0.050347 1.305133I
a = 0.238431 1.046938I
b = 0.050347 + 1.305133I
1.51415 3.01964I 3.37685 + 3.22289I
u = 0.050347 + 1.305133I
a = 0.238431 + 1.046938I
b = 0.050347 1.305133I
1.51415 + 3.01964I 3.37685 3.22289I
u = 0.151478 0.362294I
a = 0.76573 + 2.72911I
b = 0.151478 + 0.362294I
1.88148 + 2.34966I 12.61375 3.05058I
u = 0.151478 + 0.362294I
a = 0.76573 2.72911I
b = 0.151478 0.362294I
1.88148 2.34966I 12.61375 + 3.05058I
u = 0.333178 1.212557I
a = 0.889168 0.134614I
b = 0.333178 + 1.212557I
5.36621 1.77463I 0.03179 + 1.94850I
u = 0.333178 + 1.212557I
a = 0.889168 + 0.134614I
b = 0.333178 1.212557I
5.36621 + 1.77463I 0.03179 1.94850I
3
II. I
u
2
= hu
12
u
11
+ · · · + 24u + 19, 15310u
11
9930u
10
+ · · · + 116501b +
188420, 2.61 × 10
4
u
11
+ 7.18 × 10
4
u
10
+ · · · + 3.16 × 10
5
a + 5.54 × 10
5
i
(i) Arc colorings
a
7
=
0
u
a
10
=
0.0826395u
11
0.227205u
10
+ ··· + 3.37538u 1.75270
0.131415u
11
+ 0.0852353u
10
+ ··· 5.94685u 1.61733
a
4
=
1
0
a
11
=
0.0826395u
11
0.227205u
10
+ ··· + 3.37538u 1.75270
0.234762u
11
+ 0.0895615u
10
+ ··· 7.84627u 4.36407
a
1
=
0.317401u
11
0.316766u
10
+ ··· + 11.2217u + 2.61137
0.234762u
11
+ 0.0895615u
10
+ ··· 7.84627u 4.36407
a
3
=
1
u
2
a
2
=
0.537344u
11
0.633335u
10
+ ··· + 18.5569u + 3.49184
0.413104u
11
+ 0.358821u
10
+ ··· 14.9788u 5.59521
a
6
=
0.0300079u
11
0.174573u
10
+ ··· 0.414089u 3.01586
0.0241200u
11
+ 0.170445u
10
+ ··· 0.572871u + 4.03170
a
9
=
0.214055u
11
0.312440u
10
+ ··· + 9.32223u 0.135373
0.131415u
11
+ 0.0852353u
10
+ ··· 5.94685u 1.61733
a
5
=
0.118075u
11
0.244437u
10
+ ··· 10.4369u 12.7712
0.0941194u
11
+ 0.432511u
10
+ ··· + 2.19875u + 7.10565
a
8
=
0.674354u
11
+ 0.991579u
10
+ ··· 19.6620u + 0.118551
0.516880u
11
0.642235u
10
+ ··· + 16.6038u + 4.18226
a
8
=
0.674354u
11
+ 0.991579u
10
+ ··· 19.6620u + 0.118551
0.516880u
11
0.642235u
10
+ ··· + 16.6038u + 4.18226
(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 = 0.538956 0.733343I
a = 1.092128 + 0.120828I
b = 0.161517 + 1.387086I
3.82135 + 2.02988I 7.01951 3.46410I
u = 0.538956 + 0.733343I
a = 1.092128 0.120828I
b = 0.161517 1.387086I
3.82135 2.02988I 7.01951 + 3.46410I
u = 0.230918 0.703145I
a = 0.900950 1.006970I
b = 0.02457 + 1.83546I
7.95893 + 0.79824I 0.490245 + 0.484655I
u = 0.230918 + 0.703145I
a = 0.900950 + 1.006970I
b = 0.02457 1.83546I
7.95893 0.79824I 0.490245 0.484655I
u = 0.22870 1.46755I
a = 0.627962 0.242842I
b = 1.31249 + 1.08009I
7.95893 + 4.85801I 0.49024 6.44355I
u = 0.22870 + 1.46755I
a = 0.627962 + 0.242842I
b = 1.31249 1.08009I
7.95893 4.85801I 0.49024 + 6.44355I
u = 0.02457 1.83546I
a = 0.468100 0.278677I
b = 0.230918 + 0.703145I
7.95893 0.79824I 0.490245 0.484655I
u = 0.02457 + 1.83546I
a = 0.468100 + 0.278677I
b = 0.230918 0.703145I
7.95893 + 0.79824I 0.490245 + 0.484655I
u = 0.161517 1.387086I
a = 0.657409 0.283917I
b = 0.538956 + 0.733343I
3.82135 2.02988I 7.01951 + 3.46410I
u = 0.161517 + 1.387086I
a = 0.657409 + 0.283917I
b = 0.538956 0.733343I
3.82135 + 2.02988I 7.01951 3.46410I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.31249 1.08009I
a = 0.550884 + 0.206492I
b = 0.22870 + 1.46755I
7.95893 4.85801I 0.49024 + 6.44355I
u = 1.31249 + 1.08009I
a = 0.550884 0.206492I
b = 0.22870 1.46755I
7.95893 + 4.85801I 0.49024 6.44355I
6
III. I
u
3
= hu
13
+ 9u
11
+ · · · + u 1, b u, 10u
12
51u
11
+ · · · + 77a + 69i
(i) Arc colorings
a
7
=
0
u
a
10
=
0.129870u
12
+ 0.662338u
11
+ ··· + 1.77922u 0.896104
u
a
4
=
1
0
a
11
=
0.129870u
12
+ 0.662338u
11
+ ··· + 1.77922u 0.896104
0.0779221u
12
+ 0.402597u
11
+ ··· + 1.53247u 0.662338
a
1
=
0.207792u
12
+ 0.259740u
11
+ ··· + 0.246753u 0.233766
0.0779221u
12
+ 0.402597u
11
+ ··· + 1.53247u 0.662338
a
3
=
1
u
2
a
2
=
0.259740u
12
+ 0.324675u
11
+ ··· 0.441558u + 1.20779
0.376623u
12
0.220779u
11
+ ··· + 0.740260u + 0.298701
a
6
=
3
7
u
12
2
7
u
11
+ ··· +
10
7
u
1
7
0.0779221u
12
+ 0.402597u
11
+ ··· + 1.53247u 0.662338
a
9
=
0.129870u
12
+ 0.662338u
11
+ ··· + 0.779221u 0.896104
u
a
5
=
0.662338u
12
0.0779221u
11
+ ··· + 1.02597u + 0.870130
u
2
a
8
=
0.207792u
12
+ 0.259740u
11
+ ··· 0.753247u 0.233766
u
3
+ u
a
8
=
0.207792u
12
+ 0.259740u
11
+ ··· 0.753247u 0.233766
u
3
+ u
(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 = 0.589003 0.398981I
a = 0.857122 0.581889I
b = 0.589003 0.398981I
1.89853 + 1.81856I 3.56685 4.77795I
u = 0.589003 + 0.398981I
a = 0.857122 + 0.581889I
b = 0.589003 + 0.398981I
1.89853 1.81856I 3.56685 + 4.77795I
u = 0.15240 1.67572I
a = 0.823933 + 0.649212I
b = 0.15240 1.67572I
12.27388 + 4.64857I 3.70723 2.31166I
u = 0.15240 + 1.67572I
a = 0.823933 0.649212I
b = 0.15240 + 1.67572I
12.27388 4.64857I 3.70723 + 2.31166I
u = 0.07312 1.68645I
a = 0.661995 + 0.792914I
b = 0.07312 1.68645I
16.5929 + 2.0437I 1.74217 0.86236I
u = 0.07312 + 1.68645I
a = 0.661995 0.792914I
b = 0.07312 + 1.68645I
16.5929 2.0437I 1.74217 + 0.86236I
u = 0.067844 0.603023I
a = 0.27200 1.41650I
b = 0.067844 0.603023I
1.10712 2.64289I 2.15723 + 4.40909I
u = 0.067844 + 0.603023I
a = 0.27200 + 1.41650I
b = 0.067844 + 0.603023I
1.10712 + 2.64289I 2.15723 4.40909I
u = 0.313260 0.606523I
a = 0.47039 + 1.79422I
b = 0.313260 0.606523I
1.23789 + 1.81241I 3.13803 + 1.59537I
u = 0.313260 + 0.606523I
a = 0.47039 1.79422I
b = 0.313260 + 0.606523I
1.23789 1.81241I 3.13803 1.59537I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.35135 1.77586I
a = 0.805718 + 0.460521I
b = 0.35135 1.77586I
17.1576 11.0164I 1.87407 + 4.91060I
u = 0.35135 + 1.77586I
a = 0.805718 0.460521I
b = 0.35135 + 1.77586I
17.1576 + 11.0164I 1.87407 4.91060I
u = 0.435514
a = 1.02992
b = 0.435514
0.684474 14.6288
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
3
u
2
+ 2u 1)
4
(u
8
u
7
+ 3u
6
2u
5
+ 4u
4
2u
3
+ 4u
2
u + 1)
(u
13
+ 6u
12
+ ··· + 26u + 4)
c
2
(u
3
+ 3u
2
+ 2u 1)
4
(u
8
+ 5u
7
+ 13u
6
+ 24u
5
+ 32u
4
+ 30u
3
+ 20u
2
+ 7u + 1)
(u
13
+ 8u
12
+ ··· + 124u 16)
c
3
, c
8
, c
9
(u
8
+ 5u
6
+ ··· 2u + 1)(u
12
+ u
11
+ ··· 24u + 19)
(u
13
+ 9u
11
+ ··· + u + 1)
c
4
(u
8
+ 5u
6
+ ··· + 2u + 1)(u
12
+ u
11
+ ··· 24u + 19)
(u
13
+ 9u
11
+ ··· + u + 1)
c
5
(u
3
u
2
+ 2u 1)
4
(u
8
+ u
7
+ 3u
6
+ 2u
5
+ 4u
4
+ 2u
3
+ 4u
2
+ u + 1)
(u
13
+ 6u
12
+ ··· + 26u + 4)
c
6
(u
2
u + 1)
6
(u
8
2u
7
+ 3u
5
4u
4
+ u
3
+ 3u
2
2u + 1)
(u
13
+ 9u
12
+ ··· + 40u + 8)
c
7
(u
8
2u
7
+ ··· 2u + 1)(u
12
+ 5u
11
+ ··· + 96u + 37)
(u
13
2u
12
+ ··· 7u 1)
c
10
(u
2
u + 1)
6
(u
8
+ 2u
7
3u
5
4u
4
u
3
+ 3u
2
+ 2u + 1)
(u
13
+ 9u
12
+ ··· + 40u + 8)
c
11
(u
8
2u
7
+ ··· 3u + 1)(u
12
+ 3u
11
+ ··· 18u + 19)
(u
13
+ 2u
12
+ ··· 6u + 1)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
3
+ 3y
2
+ 2y 1)
4
(y
8
+ 5y
7
+ 13y
6
+ 24y
5
+ 32y
4
+ 30y
3
+ 20y
2
+ 7y + 1)
(y
13
+ 8y
12
+ ··· + 124y 16)
c
2
(y
3
5y
2
+ 10y 1)
4
(y
8
+ y
7
7y
6
4y
5
+ 36y
4
+ 70y
3
+ 44y
2
9y + 1)
(y
13
4y
12
+ ··· + 31600y 256)
c
3
, c
4
, c
8
c
9
(y
8
+ 10y
7
+ 45y
6
+ 115y
5
+ 176y
4
+ 157y
3
+ 72y
2
+ 12y + 1)
(y
12
+ 15y
11
+ ··· + 2160y + 361)(y
13
+ 18y
12
+ ··· 3y 1)
c
6
, c
10
(y
2
+ y + 1)
6
(y
8
4y
7
+ 4y
6
+ y
5
+ 4y
4
13y
3
+ 5y
2
+ 2y + 1)
(y
13
3y
12
+ ··· + 352y 64)
c
7
(y
8
+ 2y
7
+ 5y
6
13y
5
+ 4y
4
+ y
3
+ 4y
2
4y + 1)
(y
12
+ 19y
11
+ ··· 1224y + 1369)(y
13
+ 26y
12
+ ··· + 53y 1)
c
11
(y
8
4y
7
+ 8y
6
9y
5
+ 4y
4
+ 5y
3
+ 10y
2
+ 3y + 1)
(y
12
17y
11
+ ··· 2224y + 361)(y
13
24y
12
+ ··· + 50y 1)
11