11a
40
(K11a
40
)
1
Arc Sequences
4 1 8 2 10 11 3 7 5 6 9
Solving Sequence
5,10
6 11
2,7
4 1 3 9 8
c
5
c
10
c
6
c
4
c
1
c
2
c
9
c
8
c
3
, c
7
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
2
a 1, u 1, b 2ai
I
u
2
= hu
46
3u
45
+ ··· 4u + 1, 4u
45
+ 9u
44
+ ··· + b + 5, 9u
45
+ 20u
44
+ ··· + 2a + 13i
There are 2 irreducible components with 48 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
= ha
2
a 1, u 1, b 2ai
(i) Arc colorings
a
5
=
0
1
a
10
=
a
2a
a
6
=
a 1
2a 1
a
11
=
a 1
a 2
a
2
=
1
0
a
7
=
a
a
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
9
=
a
a
a
8
=
a
a
a
8
=
a
a
(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 = 0.618034
b = 1.23607
2.63189 17.0000
u = 1.00000
a = 1.61803
b = 3.23607
10.5276 17.0000
3
II. I
u
2
=
hu
46
3u
45
+· · ·4u+1, 4u
45
+9u
44
+· · ·+b+5, 9u
45
+20u
44
+· · ·+2a+13i
(i) Arc colorings
a
5
=
0
u
a
10
=
9
2
u
45
10u
44
+ ··· +
31
2
u
13
2
4u
45
9u
44
+ ··· + 12u 5
a
6
=
1
2
u
45
+ u
44
+ ···
3
2
u
1
2
u
45
+ 2u
44
+ ··· 3u + 1
a
11
=
1
2
u
45
+ u
44
+ ···
3
2
u +
1
2
1
2
u
45
+ u
44
+ ···
5
2
u +
1
2
a
2
=
1
0
a
7
=
7
2
u
45
8u
44
+ ··· +
27
2
u
13
2
7
2
u
45
9u
44
+ ··· +
27
2
u
11
2
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
9
=
9
2
u
45
10u
44
+ ··· +
31
2
u
13
2
3
2
u
45
3u
44
+ ··· +
5
2
u
3
2
a
8
=
13
2
u
45
14u
44
+ ··· +
39
2
u
17
2
11
2
u
45
11u
44
+ ··· +
29
2
u
11
2
a
8
=
13
2
u
45
14u
44
+ ··· +
39
2
u
17
2
11
2
u
45
11u
44
+ ··· +
29
2
u
11
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 = 1.267551 0.341014I
a = 1.169162 0.401397I
b = 3.76634 + 0.63602I
12.88443 + 3.59966I 12.40284 2.77016I
u = 1.267551 + 0.341014I
a = 1.169162 + 0.401397I
b = 3.76634 0.63602I
12.88443 3.59966I 12.40284 + 2.77016I
u = 1.24159
a = 1.24865
b = 3.68847
8.97468 7.71594
u = 1.213928 0.324837I
a = 0.494119 + 0.447906I
b = 1.52597 + 0.05798I
4.79298 + 1.73712I 11.07336 4.61384I
u = 1.213928 + 0.324837I
a = 0.494119 0.447906I
b = 1.52597 0.05798I
4.79298 1.73712I 11.07336 + 4.61384I
u = 1.163655 0.487582I
a = 1.219495 + 0.406450I
b = 4.02097 + 1.69098I
12.55172 6.46011I 12.30589 + 3.85763I
u = 1.163655 + 0.487582I
a = 1.219495 0.406450I
b = 4.02097 1.69098I
12.55172 + 6.46011I 12.30589 3.85763I
u = 1.135154 0.447303I
a = 0.688404 + 0.102960I
b = 2.35714 0.64075I
4.51433 4.40744I 10.64051 + 5.57891I
u = 1.135154 + 0.447303I
a = 0.688404 0.102960I
b = 2.35714 + 0.64075I
4.51433 + 4.40744I 10.64051 5.57891I
u = 1.113981 0.361295I
a = 0.097852 0.427648I
b = 0.641025 0.239844I
2.55505 1.20262I 6.54006 + 0.40776I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.113981 + 0.361295I
a = 0.097852 + 0.427648I
b = 0.641025 + 0.239844I
2.55505 + 1.20262I 6.54006 0.40776I
u = 1.09496
a = 0.366495
b = 1.15952
2.16597 1.80756
u = 0.766719 0.461059I
a = 1.49128 + 0.85322I
b = 0.56166 + 1.56838I
7.50480 1.94811I 9.46528 + 3.30327I
u = 0.766719 + 0.461059I
a = 1.49128 0.85322I
b = 0.56166 1.56838I
7.50480 + 1.94811I 9.46528 3.30327I
u = 0.678138 0.225871I
a = 0.670278 + 0.195456I
b = 0.123205 0.963683I
0.893062 1.082038I 6.48995 + 6.28251I
u = 0.678138 + 0.225871I
a = 0.670278 0.195456I
b = 0.123205 + 0.963683I
0.893062 + 1.082038I 6.48995 6.28251I
u = 0.121683 0.704360I
a = 0.25766 + 2.24376I
b = 0.857380 0.869121I
9.57946 + 1.98178I 9.32338 + 0.02436I
u = 0.121683 + 0.704360I
a = 0.25766 2.24376I
b = 0.857380 + 0.869121I
9.57946 1.98178I 9.32338 0.02436I
u = 0.010899 0.557387I
a = 0.488622 1.315606I
b = 0.623869 + 0.439846I
1.56660 + 0.49532I 7.68115 1.36018I
u = 0.010899 + 0.557387I
a = 0.488622 + 1.315606I
b = 0.623869 0.439846I
1.56660 0.49532I 7.68115 + 1.36018I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.178478 0.876402I
a = 0.21137 1.79423I
b = 0.538476 + 1.206468I
8.32029 7.70926I 7.84827 + 4.72284I
u = 0.178478 + 0.876402I
a = 0.21137 + 1.79423I
b = 0.538476 1.206468I
8.32029 + 7.70926I 7.84827 4.72284I
u = 0.203438 0.815815I
a = 0.336446 + 0.956201I
b = 0.394222 0.584610I
0.41848 5.45501I 5.35190 + 6.48052I
u = 0.203438 + 0.815815I
a = 0.336446 0.956201I
b = 0.394222 + 0.584610I
0.41848 + 5.45501I 5.35190 6.48052I
u = 0.241472 0.712682I
a = 0.753217 0.047792I
b = 0.375721 + 0.038766I
1.22062 1.95597I 0.98131 + 1.36818I
u = 0.241472 + 0.712682I
a = 0.753217 + 0.047792I
b = 0.375721 0.038766I
1.22062 + 1.95597I 0.98131 1.36818I
u = 0.519026
a = 1.52049
b = 1.02294
1.45882 5.44130
u = 0.572969 0.701430I
a = 0.97577 1.31372I
b = 0.368735 + 0.188892I
2.87293 1.27541I 4.15275 + 0.42491I
u = 0.572969 + 0.701430I
a = 0.97577 + 1.31372I
b = 0.368735 0.188892I
2.87293 + 1.27541I 4.15275 0.42491I
u = 0.712026 0.604892I
a = 0.035594 + 0.691934I
b = 0.309655 0.162782I
3.12084 + 0.58749I 0.262566 + 0.327262I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.712026 + 0.604892I
a = 0.035594 0.691934I
b = 0.309655 + 0.162782I
3.12084 0.58749I 0.262566 0.327262I
u = 0.784410
a = 2.09377
b = 2.82931
9.96538 1.52926
u = 0.827592 0.600489I
a = 0.694060 0.109448I
b = 1.144526 + 0.361429I
2.79032 + 4.13635I 1.98154 7.56914I
u = 0.827592 + 0.600489I
a = 0.694060 + 0.109448I
b = 1.144526 0.361429I
2.79032 4.13635I 1.98154 + 7.56914I
u = 0.929112 0.623725I
a = 1.182588 0.669173I
b = 2.13102 0.91610I
3.88727 + 6.30351I 6.21237 6.26913I
u = 0.929112 + 0.623725I
a = 1.182588 + 0.669173I
b = 2.13102 + 0.91610I
3.88727 6.30351I 6.21237 + 6.26913I
u = 1.129120 0.456816I
a = 0.555801 0.529655I
b = 1.56062 + 0.12087I
4.45409 + 3.41461I 10.48322 2.63232I
u = 1.129120 + 0.456816I
a = 0.555801 + 0.529655I
b = 1.56062 0.12087I
4.45409 3.41461I 10.48322 + 2.63232I
u = 1.143356 0.519855I
a = 0.128120 + 0.492117I
b = 0.436483 + 0.316438I
1.40155 + 6.64307I 5.04471 5.15805I
u = 1.143356 + 0.519855I
a = 0.128120 0.492117I
b = 0.436483 0.316438I
1.40155 6.64307I 5.04471 + 5.15805I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.160217 0.405425I
a = 1.243709 + 0.509810I
b = 3.56098 0.77770I
13.13773 + 1.79965I 12.67234 3.85113I
u = 1.160217 + 0.405425I
a = 1.243709 0.509810I
b = 3.56098 + 0.77770I
13.13773 1.79965I 12.67234 + 3.85113I
u = 1.180686 0.538180I
a = 0.689296 0.108322I
b = 2.31192 + 0.22869I
3.31037 + 10.45055I 8.69927 9.35917I
u = 1.180686 + 0.538180I
a = 0.689296 + 0.108322I
b = 2.31192 0.22869I
3.31037 10.45055I 8.69927 + 9.35917I
u = 1.209799 0.545440I
a = 1.161122 0.403031I
b = 4.12912 1.14012I
11.4148 + 12.8867I 10.94789 7.74672I
u = 1.209799 + 0.545440I
a = 1.161122 + 0.403031I
b = 4.12912 + 1.14012I
11.4148 12.8867I 10.94789 + 7.74672I
9
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
2
(u
46
+ 3u
45
+ ··· + 4u + 1)
c
2
(u + 1)
2
(u
46
+ 25u
45
+ ··· + 4u + 1)
c
3
, c
7
u
2
(u
46
+ u
45
+ ··· + 8u + 4)
c
4
(u + 1)
2
(u
46
+ 3u
45
+ ··· + 4u + 1)
c
5
, c
6
(u
2
+ u 1)(u
46
+ 2u
45
+ ··· u + 1)
c
8
u
2
(u
46
+ 15u
45
+ ··· + 232u + 16)
c
9
, c
10
(u
2
u 1)(u
46
+ 2u
45
+ ··· u + 1)
c
11
(u
2
u 1)(u
46
+ 14u
45
+ ··· + 885u + 207)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
2
(y
46
25y
45
+ ··· 4y + 1)
c
2
(y 1)
2
(y
46
5y
45
+ ··· 44y + 1)
c
3
, c
7
y
2
(y
46
15y
45
+ ··· 232y + 16)
c
5
, c
6
, c
9
c
10
(y
2
3y + 1)(y
46
54y
45
+ ··· 9y + 1)
c
8
y
2
(y
46
+ 29y
45
+ ··· 9760y + 256)
c
11
(y
2
3y + 1)(y
46
18y
45
+ ··· 704565y + 42849)
11