11a
221
(K11a
221
)
1
Arc Sequences
7 1 10 11 8 2 5 6 3 4 9
Solving Sequence
5,7
8
6,11
4 10 3 9 1 2
c
7
c
5
c
4
c
10
c
3
c
9
c
11
c
2
c
1
, c
6
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
2
+ b 1, b + a, u 1i
I
u
2
= hu
41
3u
40
+ ··· 6u + 1, u
40
2u
39
+ ··· + 2a 5, 5u
40
+ 8u
39
+ ··· + 2b + 3i
There are 2 irreducible components with 43 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
=
1
0
a
7
=
0
1
a
8
=
1
1
a
6
=
0
1
a
11
=
b
b
a
4
=
b
b + 1
a
10
=
1
b + 1
a
3
=
0
b
a
9
=
1
0
a
1
=
0
b
a
2
=
0
b
a
2
=
0
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 3.00000
u = 1.00000
a = 0.618034
b = 0.618034
0.657974 3.00000
3
II.
I
u
2
= hu
41
3u
40
+· · ·6u+1, u
40
2u
39
+· · ·+2a5, 5u
40
+8u
39
+· · ·+2b+3i
(i) Arc colorings
a
5
=
1
0
a
7
=
0
u
a
8
=
u
u
a
6
=
u
2
+ 1
u
2
a
11
=
1
2
u
40
+ u
39
+ ··· +
5
2
u +
5
2
5
2
u
40
4u
39
+ ··· +
23
2
u
3
2
a
4
=
1
2
u
40
+ u
39
+ ··· +
1
2
u +
5
2
1
2
u
40
u
39
+ ··· +
7
2
u
1
2
a
10
=
u
12
5u
10
2u
9
+ 9u
8
+ 8u
7
4u
6
10u
5
6u
4
+ 2u
3
+ 5u
2
+ 2u + 1
1
2
u
40
u
39
+ ··· +
9
2
u
1
2
a
3
=
3u
40
+ 5u
39
+ ··· 7u + 5
3
2
u
40
2u
39
+ ··· +
11
2
u
1
2
a
9
=
u
3
2u
u
3
+ u
a
1
=
u
39
u
38
+ ··· + u + 3
9
2
u
40
8u
39
+ ··· +
43
2
u
7
2
a
2
=
u
39
u
38
+ ··· + u + 3
5
2
u
40
6u
39
+ ··· +
31
2
u
5
2
a
2
=
u
39
u
38
+ ··· + u + 3
5
2
u
40
6u
39
+ ··· +
31
2
u
5
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 = 1.33636
a = 0.921081
b = 1.49684
2.95183 3.34747
u = 1.320325 0.305527I
a = 1.11755 + 2.31035I
b = 1.59868 + 0.12246I
6.66704 5.82869I 5.58070 + 3.39540I
u = 1.320325 + 0.305527I
a = 1.11755 2.31035I
b = 1.59868 0.12246I
6.66704 + 5.82869I 5.58070 3.39540I
u = 1.281478 0.256105I
a = 0.82914 1.45043I
b = 0.672657 0.430516I
1.06929 3.78517I 3.33312 + 5.32313I
u = 1.281478 + 0.256105I
a = 0.82914 + 1.45043I
b = 0.672657 + 0.430516I
1.06929 + 3.78517I 3.33312 5.32313I
u = 1.220599 0.156477I
a = 0.595788 + 0.579512I
b = 0.186386 + 0.436485I
2.45808 0.68070I 0.75996 1.22832I
u = 1.220599 + 0.156477I
a = 0.595788 0.579512I
b = 0.186386 0.436485I
2.45808 + 0.68070I 0.75996 + 1.22832I
u = 1.082591 0.468575I
a = 1.87677 0.36346I
b = 1.59643 + 0.10454I
7.20747 + 3.04463I 5.27357 2.39823I
u = 1.082591 + 0.468575I
a = 1.87677 + 0.36346I
b = 1.59643 0.10454I
7.20747 3.04463I 5.27357 + 2.39823I
u = 1.018830 0.371555I
a = 0.594839 + 0.125685I
b = 0.651786 0.355869I
0.495471 + 1.323545I 2.90171 5.22285I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.018830 + 0.371555I
a = 0.594839 0.125685I
b = 0.651786 + 0.355869I
0.495471 1.323545I 2.90171 + 5.22285I
u = 0.642973 0.323659I
a = 0.753058 + 0.352117I
b = 0.360810 + 0.343666I
1.37623 1.10536I 2.43864 + 5.69625I
u = 0.642973 + 0.323659I
a = 0.753058 0.352117I
b = 0.360810 0.343666I
1.37623 + 1.10536I 2.43864 5.69625I
u = 0.562217 0.598084I
a = 2.01623 1.02652I
b = 1.51159 0.03761I
4.86048 2.17709I 3.44971 + 3.79306I
u = 0.562217 + 0.598084I
a = 2.01623 + 1.02652I
b = 1.51159 + 0.03761I
4.86048 + 2.17709I 3.44971 3.79306I
u = 0.182840 0.800800I
a = 1.54166 + 0.26277I
b = 0.717256 + 0.455564I
2.06145 5.60392I 6.28993 + 7.67426I
u = 0.182840 + 0.800800I
a = 1.54166 0.26277I
b = 0.717256 0.455564I
2.06145 + 5.60392I 6.28993 7.67426I
u = 0.181877 0.689383I
a = 0.100958 + 0.259390I
b = 0.112211 0.532295I
0.32453 2.20665I 2.42130 + 3.15065I
u = 0.181877 + 0.689383I
a = 0.100958 0.259390I
b = 0.112211 + 0.532295I
0.32453 + 2.20665I 2.42130 3.15065I
u = 0.170327 0.865341I
a = 3.04255 1.05733I
b = 1.61126 0.13147I
9.99184 7.80021I 8.36490 + 5.64860I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.170327 + 0.865341I
a = 3.04255 + 1.05733I
b = 1.61126 + 0.13147I
9.99184 + 7.80021I 8.36490 5.64860I
u = 0.001662 0.650682I
a = 1.86849 + 0.48856I
b = 0.761966 + 0.327369I
2.92066 + 0.49867I 9.33255 1.40381I
u = 0.001662 + 0.650682I
a = 1.86849 0.48856I
b = 0.761966 0.327369I
2.92066 0.49867I 9.33255 + 1.40381I
u = 0.085807 0.724003I
a = 3.28026 1.44916I
b = 1.62008 0.09360I
11.08218 + 2.09439I 10.58118 0.49911I
u = 0.085807 + 0.724003I
a = 3.28026 + 1.44916I
b = 1.62008 + 0.09360I
11.08218 2.09439I 10.58118 + 0.49911I
u = 0.132462
a = 2.59836
b = 0.525578
0.783707 12.9615
u = 0.386383
a = 0.544018
b = 1.58785
8.19168 12.7987
u = 1.209876 0.257619I
a = 1.85773 0.18949I
b = 1.65267 + 0.06288I
7.70281 + 1.46253I 4.97551 4.38414I
u = 1.209876 + 0.257619I
a = 1.85773 + 0.18949I
b = 1.65267 0.06288I
7.70281 1.46253I 4.97551 + 4.38414I
u = 1.298389 0.245537I
a = 0.719597 + 0.060854I
b = 0.912702 0.284319I
1.14433 + 2.71303I 3.06796 2.16565I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.298389 + 0.245537I
a = 0.719597 0.060854I
b = 0.912702 + 0.284319I
1.14433 2.71303I 3.06796 + 2.16565I
u = 1.364281 0.291615I
a = 0.353416 + 0.323818I
b = 0.156324 + 0.636184I
4.56137 + 5.79983I 1.81042 3.80578I
u = 1.364281 + 0.291615I
a = 0.353416 0.323818I
b = 0.156324 0.636184I
4.56137 5.79983I 1.81042 + 3.80578I
u = 1.37517 0.33619I
a = 0.98917 1.05405I
b = 0.731269 0.520482I
2.86444 + 9.70772I 1.83000 8.47841I
u = 1.37517 + 0.33619I
a = 0.98917 + 1.05405I
b = 0.731269 + 0.520482I
2.86444 9.70772I 1.83000 + 8.47841I
u = 1.37919 0.37091I
a = 1.49906 + 1.89880I
b = 1.61567 + 0.15360I
5.09814 + 12.24282I 4.42370 7.04565I
u = 1.37919 + 0.37091I
a = 1.49906 1.89880I
b = 1.61567 0.15360I
5.09814 12.24282I 4.42370 + 7.04565I
u = 1.42404 0.03377I
a = 0.132859 0.815243I
b = 0.461377 0.569569I
7.93550 + 1.94462I 3.70499 3.68184I
u = 1.42404 + 0.03377I
a = 0.132859 + 0.815243I
b = 0.461377 + 0.569569I
7.93550 1.94462I 3.70499 + 3.68184I
u = 1.43772 0.11527I
a = 0.325255 + 0.929785I
b = 1.47335 + 0.12656I
1.67461 + 4.38863I 0.33432 3.52334I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.43772 + 0.11527I
a = 0.325255 0.929785I
b = 1.47335 0.12656I
1.67461 4.38863I 0.33432 + 3.52334I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
u
2
(u
41
+ u
40
+ ··· u
2
+ 4)
c
2
u
2
(u
41
+ 15u
40
+ ··· + 8u 16)
c
3
, c
4
, c
9
c
10
(u
2
u 1)(u
41
+ 2u
40
+ ··· u + 1)
c
5
, c
7
, c
8
(u + 1)
2
(u
41
+ 3u
40
+ ··· 6u 1)
c
11
(u
2
+ u 1)(u
41
+ 12u
40
+ ··· 503u 73)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
y
2
(y
41
+ 15y
40
+ ··· + 8y 16)
c
2
y
2
(y
41
+ 19y
40
+ ··· + 16416y 256)
c
3
, c
4
, c
9
c
10
(y
2
3y + 1)(y
41
48y
40
+ ··· + 3y 1)
c
5
, c
7
, c
8
(y 1)
2
(y
41
35y
40
+ ··· + 62y 1)
c
11
(y
2
3y + 1)(y
41
12y
40
+ ··· + 23351y 5329)
10