11a
108
(K11a
108
)
1
Arc Sequences
5 1 9 10 2 11 3 4 8 6 7
Solving Sequence
3,9
4 8 10 5
7,11
1 2 6
c
3
c
8
c
9
c
4
c
7
c
11
c
2
c
6
c
1
, c
5
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
4
+ 2u
2
+ 2, u
3
+ b + u + 1, u
3
2u
2
+ 2a 4i
I
u
2
= ha
15
6a
14
+ ··· + 12a 1, 1.40541 × 10
15
u 4.67787 × 10
14
a
14
+ ··· + 2.95507 × 10
16
a 5.05185 × 10
15
,
1.40541 × 10
15
b + 1.09318 × 10
15
a
14
+ ··· 5.77602 × 10
16
a + 1.24718 × 10
16
i
I
u
3
= hu
41
+ 2u
40
+ ··· 4u 2, u
36
+ 9u
34
+ ··· + 4a 2, u
38
+ 10u
36
+ ··· + 4b 4ui
I
v
1
= hb + 1, v 1, ai
There are 4 irreducible components with 61 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
4
+ 2u
2
+ 2, u
3
+ b + u + 1, u
3
2u
2
+ 2a 4i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
8
=
u
u
3
+ u
a
10
=
u
3
u
3
u
a
5
=
1
0
a
7
=
u
3
u
3
+ u
a
11
=
1
2
u
3
+ u
2
+ 2
u
3
u 1
a
1
=
1
2
u
3
+ u
2
+ 2
1
a
2
=
1
2
u
3
+ u
2
+ 3
1
a
6
=
1
2
u
3
+ u
2
+ 2
1
a
6
=
1
2
u
3
+ u
2
+ 2
1
(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 = 0.455090 1.098684I
a = 1.77689 + 1.32180I
b = 2.09868 + 0.45509I
2.46740 3.66386I 4.00000 + 4.00000I
u = 0.455090 + 1.098684I
a = 1.77689 1.32180I
b = 2.09868 0.45509I
2.46740 + 3.66386I 4.00000 4.00000I
u = 0.455090 1.098684I
a = 0.223113 0.678203I
b = 0.098684 + 0.455090I
2.46740 + 3.66386I 4.00000 4.00000I
u = 0.455090 + 1.098684I
a = 0.223113 + 0.678203I
b = 0.098684 0.455090I
2.46740 3.66386I 4.00000 + 4.00000I
3
II.
I
u
2
= ha
15
6a
14
+ · · · + 12a 1, 1.41 × 10
15
u 4.68 × 10
14
a
14
+ · · · + 2.96 ×
10
16
a5.05×10
15
, 1.41×10
15
b+1.09×10
15
a
14
+· · ·5.78×10
16
a+1.25×10
16
i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
0.332848a
14
2.06455a
13
+ ··· 21.0265a + 3.59459
a
4
=
1
1.37553a
14
7.91877a
13
+ ··· 41.2702a + 4.94693
a
8
=
0.332848a
14
2.06455a
13
+ ··· 21.0265a + 3.59459
0.367720a
14
2.22927a
13
+ ··· 18.9499a + 3.66237
a
10
=
0.0348720a
14
0.164720a
13
+ ··· + 2.07663a + 0.0677866
1.16881a
14
6.84435a
13
+ ··· 44.7664a + 6.72752
a
5
=
0.0348720a
14
0.164720a
13
+ ··· + 2.07663a + 0.0677866
0.367720a
14
+ 2.22927a
13
+ ··· + 18.9499a 3.66237
a
7
=
0.0348720a
14
+ 0.164720a
13
+ ··· 2.07663a 0.0677866
0.367720a
14
2.22927a
13
+ ··· 18.9499a + 3.66237
a
11
=
a
0.777842a
14
+ 4.66489a
13
+ ··· + 41.0986a 8.87416
a
1
=
0.0489636a
14
+ 0.201322a
13
+ ··· 0.943041a 0.281136
0.213349a
14
+ 1.26600a
13
+ ··· + 11.6055a 3.42660
a
2
=
a
0.777842a
14
+ 4.66489a
13
+ ··· + 41.0986a 8.87416
a
6
=
a
1.11223a
14
+ 6.60775a
13
+ ··· + 52.6580a 10.2497
a
6
=
a
1.11223a
14
+ 6.60775a
13
+ ··· + 52.6580a 10.2497
(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 = 0.455697 + 1.200152I
a = 1.104768 0.122674I
b = 1.128110 0.258049I
5.87256 4.40083I 6.74431 + 3.49859I
u = 0.455697 1.200152I
a = 1.104768 + 0.122674I
b = 1.128110 + 0.258049I
5.87256 + 4.40083I 6.74431 3.49859I
u = 0.339110 + 0.822375I
a = 0.76442 2.16929I
b = 0.398738 + 1.045420I
0.32910 + 1.53058I 2.51511 4.43065I
u = 0.339110 0.822375I
a = 0.76442 + 2.16929I
b = 0.398738 1.045420I
0.32910 1.53058I 2.51511 + 4.43065I
u = 0.455697 + 1.200152I
a = 0.728228 0.922547I
b = 0.877625 0.238115I
5.87256 4.40083I 6.74431 + 3.49859I
u = 0.455697 1.200152I
a = 0.728228 + 0.922547I
b = 0.877625 + 0.238115I
5.87256 + 4.40083I 6.74431 3.49859I
u = 0.766826
a = 0.189975
b = 3.55917
2.40108 3.48114
u = 0.766826
a = 0.275298 0.566134I
b = 0.779585 + 0.100353I
2.40108 3.48114
u = 0.766826
a = 0.275298 + 0.566134I
b = 0.779585 0.100353I
2.40108 3.48114
u = 0.339110 + 0.822375I
a = 0.287040 0.149550I
b = 1.59308 1.11787I
0.32910 + 1.53058I 2.51511 4.43065I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.339110 0.822375I
a = 0.287040 + 0.149550I
b = 1.59308 + 1.11787I
0.32910 1.53058I 2.51511 + 4.43065I
u = 0.339110 0.822375I
a = 0.734937 0.103511I
b = 0.0081865 0.0724486I
0.32910 1.53058I 2.51511 + 4.43065I
u = 0.339110 + 0.822375I
a = 0.734937 + 0.103511I
b = 0.0081865 + 0.0724486I
0.32910 + 1.53058I 2.51511 4.43065I
u = 0.455697 1.200152I
a = 4.20515 2.79451I
b = 4.00574 0.49616I
5.87256 + 4.40083I 6.74431 3.49859I
u = 0.455697 + 1.200152I
a = 4.20515 + 2.79451I
b = 4.00574 + 0.49616I
5.87256 4.40083I 6.74431 + 3.49859I
6
III.
I
u
3
= hu
41
+2u
40
+· · ·−4u2, u
36
+9u
34
+· · ·+4a2, u
38
+10u
36
+· · ·+4b4ui
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
8
=
u
u
3
+ u
a
10
=
u
3
u
5
+ u
3
+ u
a
5
=
u
6
u
4
+ 1
u
8
2u
6
2u
4
a
7
=
u
3
u
3
+ u
a
11
=
1
4
u
36
9
4
u
34
+ ··· +
1
2
u +
1
2
1
4
u
38
5
2
u
36
+ ··· + u
3
+ u
a
1
=
1
2
u
40
+ u
39
+ ···
1
2
u +
1
2
1
4
u
38
+
5
2
u
36
+ ··· +
1
2
u 1
a
2
=
1
4
u
33
+ 2u
31
+ ··· +
1
2
u
3
+ 1
1
4
u
33
9
4
u
31
+ ···
1
2
u
2
+
1
2
u
a
6
=
1
4
u
36
+
9
4
u
34
+ ···
1
2
u
1
2
1
4
u
36
5
2
u
34
+ ···
1
2
u
2
+ u
a
6
=
1
4
u
36
+
9
4
u
34
+ ···
1
2
u
1
2
1
4
u
36
5
2
u
34
+ ···
1
2
u
2
+ 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 = 0.861859 0.085768I
a = 0.248599 0.056751I
b = 2.94023 0.35593I
9.69240 + 2.90753I 6.01430 0.84016I
u = 0.861859 + 0.085768I
a = 0.248599 + 0.056751I
b = 2.94023 + 0.35593I
9.69240 2.90753I 6.01430 + 0.84016I
u = 0.800093 0.101020I
a = 0.116434 0.508788I
b = 0.861217 + 0.055943I
1.51266 + 5.04411I 1.12155 5.34007I
u = 0.800093 + 0.101020I
a = 0.116434 + 0.508788I
b = 0.861217 0.055943I
1.51266 5.04411I 1.12155 + 5.34007I
u = 0.662080 0.589897I
a = 0.33936 + 1.59786I
b = 0.677786 0.487958I
2.46953 + 3.52956I 2.12209 2.66433I
u = 0.662080 + 0.589897I
a = 0.33936 1.59786I
b = 0.677786 + 0.487958I
2.46953 3.52956I 2.12209 + 2.66433I
u = 0.595879 0.924278I
a = 0.548741 0.029668I
b = 1.41235 + 0.44480I
3.43135 8.38206I 3.46275 + 8.23571I
u = 0.595879 + 0.924278I
a = 0.548741 + 0.029668I
b = 1.41235 0.44480I
3.43135 + 8.38206I 3.46275 8.23571I
u = 0.526691 0.260209I
a = 0.004261 0.175323I
b = 0.698592 0.366853I
1.66678 0.31810I 5.86833 + 0.81571I
u = 0.526691 + 0.260209I
a = 0.004261 + 0.175323I
b = 0.698592 + 0.366853I
1.66678 + 0.31810I 5.86833 0.81571I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.502214 1.228005I
a = 2.75872 2.76110I
b = 3.31424 + 0.19337I
13.1184 7.8365I 8.99343 + 4.06275I
u = 0.502214 + 1.228005I
a = 2.75872 + 2.76110I
b = 3.31424 0.19337I
13.1184 + 7.8365I 8.99343 4.06275I
u = 0.497123 1.201302I
a = 0.762353 + 1.067676I
b = 0.912263 + 0.143760I
4.75069 9.79224I 4.22259 + 8.17334I
u = 0.497123 + 1.201302I
a = 0.762353 1.067676I
b = 0.912263 0.143760I
4.75069 + 9.79224I 4.22259 8.17334I
u = 0.465698 1.112627I
a = 1.38595 + 0.80238I
b = 1.121693 + 0.363954I
0.77731 3.72567I 1.87519 + 3.76789I
u = 0.465698 + 1.112627I
a = 1.38595 0.80238I
b = 1.121693 0.363954I
0.77731 + 3.72567I 1.87519 3.76789I
u = 0.410502 1.250475I
a = 3.57103 1.31456I
b = 2.96417 0.86390I
13.77940 1.51388I 9.75388 + 2.25000I
u = 0.410502 + 1.250475I
a = 3.57103 + 1.31456I
b = 2.96417 + 0.86390I
13.77940 + 1.51388I 9.75388 2.25000I
u = 0.405488 1.211449I
a = 1.204427 + 0.118457I
b = 1.085779 + 0.193096I
5.40261 + 0.90249I 5.70583 2.02309I
u = 0.405488 + 1.211449I
a = 1.204427 0.118457I
b = 1.085779 0.193096I
5.40261 0.90249I 5.70583 + 2.02309I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.012198 0.896132I
a = 0.936428 + 0.866831I
b = 0.318753 0.611958I
1.35320 1.46651I 7.22861 + 4.90542I
u = 0.012198 + 0.896132I
a = 0.936428 0.866831I
b = 0.318753 + 0.611958I
1.35320 + 1.46651I 7.22861 4.90542I
u = 0.053677 1.146818I
a = 0.064449 + 0.602376I
b = 0.141641 + 0.844172I
8.20281 + 2.94250I 9.69079 2.88880I
u = 0.053677 + 1.146818I
a = 0.064449 0.602376I
b = 0.141641 0.844172I
8.20281 2.94250I 9.69079 + 2.88880I
u = 0.373001 1.250030I
a = 3.54123 0.76594I
b = 2.66124 1.09978I
12.10738 4.99417I 8.18458 + 2.23571I
u = 0.373001 + 1.250030I
a = 3.54123 + 0.76594I
b = 2.66124 + 1.09978I
12.10738 + 4.99417I 8.18458 2.23571I
u = 0.423966 1.085319I
a = 0.674123 + 0.382193I
b = 1.054242 + 0.520798I
4.21251 + 3.60145I 9.92786 4.45844I
u = 0.423966 + 1.085319I
a = 0.674123 0.382193I
b = 1.054242 0.520798I
4.21251 3.60145I 9.92786 + 4.45844I
u = 0.500958 0.624025I
a = 0.252123 0.307538I
b = 0.457537 0.492796I
2.57302 1.04199I 5.23599 + 1.20683I
u = 0.500958 + 0.624025I
a = 0.252123 + 0.307538I
b = 0.457537 + 0.492796I
2.57302 + 1.04199I 5.23599 1.20683I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.503278 0.876227I
a = 0.769222 0.665393I
b = 0.398861 + 0.338027I
1.85795 + 5.19311I 2.06190 8.35313I
u = 0.503278 + 0.876227I
a = 0.769222 + 0.665393I
b = 0.398861 0.338027I
1.85795 5.19311I 2.06190 + 8.35313I
u = 0.526084 1.215281I
a = 2.29289 3.01255I
b = 3.21584 + 0.52769I
11.0125 + 14.2368I 6.63638 8.24449I
u = 0.526084 + 1.215281I
a = 2.29289 + 3.01255I
b = 3.21584 0.52769I
11.0125 14.2368I 6.63638 + 8.24449I
u = 0.534614
a = 1.13888
b = 0.542878
1.42690 6.75837
u = 0.547888 1.013946I
a = 0.648893 + 0.065471I
b = 1.297057 + 0.468918I
4.68586 + 3.37196I 6.53621 2.75945I
u = 0.547888 + 1.013946I
a = 0.648893 0.065471I
b = 1.297057 0.468918I
4.68586 3.37196I 6.53621 + 2.75945I
u = 0.683105 0.455196I
a = 0.45229 + 1.40012I
b = 0.652018 0.398386I
3.06237 + 1.36624I 3.26071 2.82351I
u = 0.683105 + 0.455196I
a = 0.45229 1.40012I
b = 0.652018 + 0.398386I
3.06237 1.36624I 3.26071 + 2.82351I
u = 0.860560 0.141859I
a = 0.244942 0.094552I
b = 2.81312 0.55313I
7.79923 9.18843I 3.80065 + 5.17633I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.860560 + 0.141859I
a = 0.244942 + 0.094552I
b = 2.81312 + 0.55313I
7.79923 + 9.18843I 3.80065 5.17633I
11
IV. I
v
1
= hb + 1, v 1, ai
(i) Arc colorings
a
3
=
1
0
a
9
=
1
0
a
4
=
1
0
a
8
=
1
0
a
10
=
1
0
a
5
=
1
0
a
7
=
1
0
a
11
=
0
1
a
1
=
1
1
a
2
=
0
1
a
6
=
1
1
a
6
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
13
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)(u + 1)
4
(u
15
5u
13
+ ··· u + 1)(u
41
+ 2u
40
+ ··· + 5u 1)
c
2
(u + 1)
5
(u
15
+ 10u
14
+ ··· u + 1)(u
41
+ 14u
40
+ ··· + u + 1)
c
3
, c
8
(u)(u
4
+ 2u
2
+ 2)(1 + u + u
2
+ 2u
3
+ u
4
+ u
5
)
3
(u
41
2u
40
+ ··· 4u + 2)
c
4
, c
7
u(u
4
2u
2
+ 2)(u
5
+ u
4
2u
3
u
2
+ u 1)
3
(u
41
2u
40
+ ··· 24u 16)
c
5
(u 1)
4
(u + 1)(u
15
5u
13
+ ··· u + 1)(u
41
+ 2u
40
+ ··· + 5u 1)
c
6
(u 1)
4
(u + 1)(u
15
5u
13
+ ··· u 1)(u
41
+ 2u
40
+ ··· 7u + 1)
c
9
u(u
2
2u + 2)
2
(u
5
+ 3u
4
+ 4u
3
+ u
2
u 1)
3
(u
41
+ 22u
40
+ ··· + 8u 4)
c
10
, c
11
(u 1)(u + 1)
4
(u
15
5u
13
+ ··· u 1)(u
41
+ 2u
40
+ ··· 7u + 1)
14
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)
5
(y
15
10y
14
+ ··· y 1)(y
41
14y
40
+ ··· + y 1)
c
2
(y 1)
5
(y
15
10y
14
+ ··· y 1)(y
41
+ 34y
40
+ ··· + 737y 1)
c
3
, c
8
y(y
2
+ 2y + 2)
2
(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
3
(y
41
+ 22y
40
+ ··· + 8y 4)
c
4
, c
7
y(y
2
2y + 2)
2
(y
5
5y
4
+ 8y
3
3y
2
y 1)
3
(y
41
34y
40
+ ··· 16448y 256)
c
6
(y 1)
5
(y
15
10y
14
+ ··· y 1)(y
41
46y
40
+ ··· 47y 1)
c
9
y(y
2
+ 4)
2
(y
5
y
4
+ 8y
3
3y
2
+ 3y 1)
3
(y
41
6y
40
+ ··· + 160y 16)
c
10
, c
11
(y 1)
5
(y
15
10y
14
+ ··· y 1)(y
41
46y
40
+ ··· 47y 1)
15