11a
51
(K11a
51
)
1
Arc Sequences
5 1 8 2 4 10 3 7 11 6 9
Solving Sequence
3,7
8 4
9,11
10 1 2 6 5
c
7
c
3
c
8
c
9
c
11
c
2
c
6
c
5
c
1
, c
4
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
2
\
j=1
I
v
j
I
u
1
= hu
15
+ 5u
14
+ 10u
13
+ 5u
12
18u
11
44u
10
40u
9
+ 3u
8
+ 49u
7
+ 55u
6
+ 26u
5
+ u
4
+ 2u
3
+ 12u
2
+ 12u + 4,
3u
14
+ 16u
13
+ ··· + 4a + 18, 7u
14
+ 29u
13
+ ··· + 4b + 20i
I
u
2
= h4a
46
10a
45
+ ··· + 6a + 1, 8.37389 × 10
138
u + 1.09324 × 10
139
a
45
+ ··· 7.08984 × 10
139
a 2.09181 × 10
138
,
1.51567 × 10
141
b 7.73405 × 10
140
a
45
+ ··· 3.51283 × 10
141
a + 6.25773 × 10
140
i
I
v
1
= hv + 1, b
2
+ b + 1, ai
I
v
2
= hb + v, b
2
+ b + 1, ai
There are 4 irreducible components with 65 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
15
+5u
14
+· · ·+12u+4, 3u
14
+16u
13
+· · ·+4a+18, 7u
14
+29u
13
+· · ·+4b+20i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
8
=
u
u
a
4
=
u
2
+ 1
u
2
a
9
=
u
u
3
+ u
a
11
=
3
4
u
14
4u
13
+ ··· 9u
9
2
7
4
u
14
29
4
u
13
+ ···
25
2
u 5
a
10
=
1
4
u
13
+
3
4
u
12
+ ··· + 2u +
1
2
3
4
u
14
+
13
4
u
13
+ ··· +
13
2
u + 2
a
1
=
7
4
u
14
15
2
u
13
+ ··· 16u
15
2
3
4
u
14
11
4
u
13
+ ···
11
2
u 2
a
2
=
1
4
u
13
+
3
4
u
12
+ ··· + u +
3
2
1
4
u
14
+
5
4
u
13
+ ··· +
5
2
u + 1
a
6
=
1
2
u
14
+
7
4
u
13
+ ··· +
5
2
u +
1
2
5
4
u
14
23
4
u
13
+ ···
27
2
u 7
a
5
=
5
4
u
14
+
9
2
u
13
+ ··· + 6u +
5
2
1
4
u
14
1
4
u
13
+ ···
9
2
u 3
a
5
=
5
4
u
14
+
9
2
u
13
+ ··· + 6u +
5
2
1
4
u
14
1
4
u
13
+ ···
9
2
u 3
(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 = 1.194601 0.597734I
a = 0.398040 1.279833I
b = 2.00980 0.17282I
6.11311 + 3.45523I 8.74146 0.79948I
u = 1.194601 + 0.597734I
a = 0.398040 + 1.279833I
b = 2.00980 + 0.17282I
6.11311 3.45523I 8.74146 + 0.79948I
u = 1.173581 0.723559I
a = 0.19142 + 1.41760I
b = 2.59824 + 1.67352I
3.8210 + 15.1159I 4.84980 10.19781I
u = 1.173581 + 0.723559I
a = 0.19142 1.41760I
b = 2.59824 1.67352I
3.8210 15.1159I 4.84980 + 10.19781I
u = 0.843039 0.715120I
a = 0.161620 + 0.023497I
b = 0.561167 + 0.559468I
6.27477 + 2.71677I 5.40032 3.41816I
u = 0.843039 + 0.715120I
a = 0.161620 0.023497I
b = 0.561167 0.559468I
6.27477 2.71677I 5.40032 + 3.41816I
u = 0.700518
a = 0.683952
b = 0.108199
0.940705 11.2759
u = 0.528547 1.045589I
a = 1.49622 0.20229I
b = 1.25501 2.17532I
1.75577 8.71874I 3.93323 + 7.24615I
u = 0.528547 + 1.045589I
a = 1.49622 + 0.20229I
b = 1.25501 + 2.17532I
1.75577 + 8.71874I 3.93323 7.24615I
u = 0.297110 1.013619I
a = 1.36351 0.49939I
b = 1.95749 + 0.16426I
3.26489 + 2.24335I 7.04256 3.44027I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.297110 + 1.013619I
a = 1.36351 + 0.49939I
b = 1.95749 0.16426I
3.26489 2.24335I 7.04256 + 3.44027I
u = 0.548950 0.445559I
a = 0.293206 0.521857I
b = 0.812618 + 0.338797I
1.34006 1.53790I 1.51731 + 5.00908I
u = 0.548950 + 0.445559I
a = 0.293206 + 0.521857I
b = 0.812618 0.338797I
1.34006 + 1.53790I 1.51731 5.00908I
u = 1.338188 0.093539I
a = 0.34390 + 1.43052I
b = 0.583320 + 0.031795I
9.46149 5.98215I 9.71265 + 5.53392I
u = 1.338188 + 0.093539I
a = 0.34390 1.43052I
b = 0.583320 0.031795I
9.46149 + 5.98215I 9.71265 5.53392I
4
II. I
u
2
=
h4a
46
10a
45
+· · ·+6a+1, 8.37×10
138
u+1.09×10
139
a
45
+· · ·7.09×10
139
a
2.09×10
138
, 1.52×10
141
b7.73× 10
140
a
45
+· · ·3.51× 10
141
a+6.26× 10
140
i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
1.30554a
45
+ 2.93699a
44
+ ··· + 8.46661a + 0.249801
a
8
=
1.30554a
45
+ 2.93699a
44
+ ··· + 8.46661a + 0.249801
1.30554a
45
+ 2.93699a
44
+ ··· + 8.46661a + 0.249801
a
4
=
2.58976a
45
+ 5.71078a
44
+ ··· + 9.25135a + 2.33559
2.58976a
45
+ 5.71078a
44
+ ··· + 9.25135a + 1.33559
a
9
=
1.30554a
45
+ 2.93699a
44
+ ··· + 8.46661a + 0.249801
1.58965a
45
3.27790a
44
+ ··· 6.47871a 2.16306
a
11
=
a
0.510271a
45
1.12136a
44
+ ··· + 2.31767a 0.412868
a
10
=
1.49153a
45
+ 3.33944a
44
+ ··· + 10.5425a + 0.370366
2.51927a
45
5.29404a
44
+ ··· 7.29469a 3.00918
a
1
=
1.03829a
45
2.36039a
44
+ ··· 4.86853a 0.0696640
2.09693a
45
+ 4.33290a
44
+ ··· + 16.5234a + 2.55756
a
2
=
3.67717a
45
7.79261a
44
+ ··· 31.0499a 1.65692
2.29254a
45
4.75735a
44
+ ··· 23.6301a 1.49594
a
6
=
1.27389a
45
2.75281a
44
+ ··· 3.90256a 1.06031
2.31218a
45
5.11320a
44
+ ··· 8.77109a 1.12997
a
5
=
0.286047a
45
+ 0.374273a
44
+ ··· + 3.85532a + 1.78044
2.84917a
45
6.31901a
44
+ ··· 17.5366a 0.846780
a
5
=
0.286047a
45
+ 0.374273a
44
+ ··· + 3.85532a + 1.78044
2.84917a
45
6.31901a
44
+ ··· 17.5366a 0.846780
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.121237 0.604443I
a = 1.96799 0.50345I
b = 0.816301 0.653869I
0.47190 + 2.34013I 2.62944 2.83732I
u = 0.121237 + 0.604443I
a = 1.96799 + 0.50345I
b = 0.816301 + 0.653869I
0.47190 2.34013I 2.62944 + 2.83732I
u = 0.423739 1.023077I
a = 1.52877 0.23560I
b = 1.38675 1.86005I
2.61521 + 3.21096I 5.70075 2.17483I
u = 0.423739 + 1.023077I
a = 1.52877 + 0.23560I
b = 1.38675 + 1.86005I
2.61521 3.21096I 5.70075 + 2.17483I
u = 0.857444 + 0.223332I
a = 1.192312 0.390626I
b = 0.684976 + 1.146969I
1.26940 3.50227I 6.61882 + 3.38553I
u = 0.857444 0.223332I
a = 1.192312 + 0.390626I
b = 0.684976 1.146969I
1.26940 + 3.50227I 6.61882 3.38553I
u = 1.050365 0.349306I
a = 0.969734 0.171942I
b = 0.327669 + 0.836471I
3.69234 + 0.67223I 9.57904 0.98278I
u = 1.050365 + 0.349306I
a = 0.969734 + 0.171942I
b = 0.327669 0.836471I
3.69234 0.67223I 9.57904 + 0.98278I
u = 1.050365 0.349306I
a = 0.47954 1.34001I
b = 1.28346 0.62225I
3.69234 + 0.67223I 9.57904 0.98278I
u = 1.050365 + 0.349306I
a = 0.47954 + 1.34001I
b = 1.28346 + 0.62225I
3.69234 0.67223I 9.57904 + 0.98278I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.33812
a = 0.36081 1.41869I
b = 0.209247 + 0.080581I
9.53870 9.98619
u = 1.33812
a = 0.36081 + 1.41869I
b = 0.209247 0.080581I
9.53870 9.98619
u = 0.975157 0.564788I
a = 0.346398 0.076301I
b = 0.699100 + 0.357480I
0.57975 3.22642I 2.48526 + 3.26705I
u = 0.975157 + 0.564788I
a = 0.346398 + 0.076301I
b = 0.699100 0.357480I
0.57975 + 3.22642I 2.48526 3.26705I
u = 0.892339 + 0.406575I
a = 0.21203 1.59722I
b = 1.18068 2.10521I
0.68141 1.67196I 4.30301 + 3.03015I
u = 0.892339 0.406575I
a = 0.21203 + 1.59722I
b = 1.18068 + 2.10521I
0.68141 + 1.67196I 4.30301 3.03015I
u = 1.017600 + 0.636625I
a = 0.16347 1.47190I
b = 2.06369 1.98214I
1.67882 8.70149I 0.49306 + 7.84909I
u = 1.017600 0.636625I
a = 0.16347 + 1.47190I
b = 2.06369 + 1.98214I
1.67882 + 8.70149I 0.49306 7.84909I
u = 0.639801 + 0.747481I
a = 0.0864885 0.0620471I
b = 0.767955 0.842473I
2.85626 + 3.41905I 2.17452 2.62575I
u = 0.639801 0.747481I
a = 0.0864885 + 0.0620471I
b = 0.767955 + 0.842473I
2.85626 3.41905I 2.17452 + 2.62575I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.649214 0.610986I
a = 0.082512 0.147353I
b = 0.849522 + 0.608294I
1.56921 1.42863I 0.37479 + 3.46803I
u = 0.649214 + 0.610986I
a = 0.082512 + 0.147353I
b = 0.849522 0.608294I
1.56921 + 1.42863I 0.37479 3.46803I
u = 0.121237 0.604443I
a = 0.113464 0.603092I
b = 0.323845 0.972214I
0.47190 + 2.34013I 2.62944 2.83732I
u = 0.121237 + 0.604443I
a = 0.113464 + 0.603092I
b = 0.323845 + 0.972214I
0.47190 2.34013I 2.62944 + 2.83732I
u = 1.183711 + 0.666071I
a = 0.20297 1.42955I
b = 2.44074 1.52827I
5.02301 + 9.28326I 6.87076 5.60434I
u = 1.183711 0.666071I
a = 0.20297 + 1.42955I
b = 2.44074 + 1.52827I
5.02301 9.28326I 6.87076 + 5.60434I
u = 1.058660 + 0.462903I
a = 0.22764 1.50710I
b = 1.57105 1.60140I
2.96583 + 6.20103I 7.62650 6.52033I
u = 1.058660 0.462903I
a = 0.22764 + 1.50710I
b = 1.57105 + 1.60140I
2.96583 6.20103I 7.62650 + 6.52033I
u = 1.017600 + 0.636625I
a = 0.349146 0.003394I
b = 0.611263 0.298378I
1.67882 8.70149I 0.49306 + 7.84909I
u = 1.017600 0.636625I
a = 0.349146 + 0.003394I
b = 0.611263 + 0.298378I
1.67882 + 8.70149I 0.49306 7.84909I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.183711 0.666071I
a = 0.387200 1.264036I
b = 2.21599 0.24794I
5.02301 9.28326I 6.87076 + 5.60434I
u = 1.183711 + 0.666071I
a = 0.387200 + 1.264036I
b = 2.21599 + 0.24794I
5.02301 + 9.28326I 6.87076 5.60434I
u = 0.975157 0.564788I
a = 0.466122 1.235899I
b = 1.84684 0.75184I
0.57975 3.22642I 2.48526 + 3.26705I
u = 0.975157 + 0.564788I
a = 0.466122 + 1.235899I
b = 1.84684 + 0.75184I
0.57975 + 3.22642I 2.48526 3.26705I
u = 0.857444 0.223332I
a = 0.60019 1.41825I
b = 1.16662 1.29751I
1.26940 + 3.50227I 6.61882 3.38553I
u = 0.857444 + 0.223332I
a = 0.60019 + 1.41825I
b = 1.16662 + 1.29751I
1.26940 3.50227I 6.61882 + 3.38553I
u = 0.649214 0.610986I
a = 0.975387 0.510152I
b = 0.792753 + 0.728471I
1.56921 1.42863I 0.37479 + 3.46803I
u = 0.649214 + 0.610986I
a = 0.975387 + 0.510152I
b = 0.792753 0.728471I
1.56921 + 1.42863I 0.37479 3.46803I
u = 1.058660 0.462903I
a = 1.006285 0.213786I
b = 0.203444 + 1.122089I
2.96583 6.20103I 7.62650 + 6.52033I
u = 1.058660 + 0.462903I
a = 1.006285 + 0.213786I
b = 0.203444 1.122089I
2.96583 + 6.20103I 7.62650 6.52033I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.892339 + 0.406575I
a = 1.300061 0.220918I
b = 0.60583 + 1.51568I
0.68141 1.67196I 4.30301 + 3.03015I
u = 0.892339 0.406575I
a = 1.300061 + 0.220918I
b = 0.60583 1.51568I
0.68141 + 1.67196I 4.30301 3.03015I
u = 0.423739 1.023077I
a = 1.303729 0.482824I
b = 1.95133 + 0.53720I
2.61521 + 3.21096I 5.70075 2.17483I
u = 0.423739 + 1.023077I
a = 1.303729 + 0.482824I
b = 1.95133 0.53720I
2.61521 3.21096I 5.70075 + 2.17483I
u = 0.639801 0.747481I
a = 1.54283 0.03133I
b = 0.37453 1.84292I
2.85626 3.41905I 2.17452 + 2.62575I
u = 0.639801 + 0.747481I
a = 1.54283 + 0.03133I
b = 0.37453 + 1.84292I
2.85626 + 3.41905I 2.17452 2.62575I
10
III. I
v
1
= hv + 1, b
2
+ b + 1, ai
(i) Arc colorings
a
3
=
1
0
a
7
=
1
0
a
8
=
1
0
a
4
=
1
0
a
9
=
1
0
a
11
=
0
b
a
10
=
1
b + 1
a
1
=
b
b
a
2
=
b
b + 1
a
6
=
b
b
a
5
=
0
b
a
5
=
0
b
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
11
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 0.500000 0.866025I
4.05977I 6.92820I
v = 1.00000
a = 0
b = 0.500000 + 0.866025I
4.05977I 6.92820I
12
IV. I
v
2
= hb + v, b
2
+ b + 1, ai
(i) Arc colorings
a
3
=
1
0
a
7
=
b
0
a
8
=
b
0
a
4
=
1
0
a
9
=
b
0
a
11
=
0
b
a
10
=
b
1
a
1
=
1
b
a
2
=
b + 1
b + 1
a
6
=
1
b
a
5
=
b + 1
b
a
5
=
b + 1
b
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
13
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
2
1(vol +
1CS) Cusp shape
v = 0.500000 + 0.866025I
a = 0
b = 0.500000 0.866025I
0 3.00000
v = 0.500000 0.866025I
a = 0
b = 0.500000 + 0.866025I
0 3.00000
14
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
2
+ u + 1)
2
(u
15
+ u
14
+ ··· + 2u 1)(u
46
+ 2u
45
+ ··· + 3u + 1)
c
2
, c
5
, c
11
(u
2
+ u + 1)
2
(u
15
+ 5u
14
+ ··· + 18u
2
1)(u
46
+ 16u
45
+ ··· 7u + 1)
c
3
, c
7
u
4
(u
15
+ 5u
14
+ ··· + 12u + 4)
(2 5u + 4u
2
+ 4u
3
14u
4
+ 7u
5
+ 24u
6
31u
7
20u
8
+ 51u
9
+ 8u
10
62u
11
+ 14u
12
+ 54u
13
32u
14
32u
15
+ 36u
16
+ 9u
17
24u
18
+ 2u
19
+ 10u
20
3u
21
2u
22
+ u
23
)
2
c
4
, c
10
(u
2
u + 1)
2
(u
15
+ u
14
+ ··· + 2u 1)(u
46
+ 2u
45
+ ··· + 3u + 1)
c
8
u
4
(u
15
+ 5u
14
+ ··· + 48u + 16)
(4 + 9u 38u
3
58u
4
+ 91u
5
+ 662u
6
+ 1839u
7
+ 3558u
8
+ 5415u
9
+ 6876u
10
+ 7532u
11
+ 7320u
12
+ 6468u
13
+ 5320u
14
+ 4116u
15
+ 2966u
16
+ 1929u
17
+ 1088u
18
+ 510u
19
+ 190u
20
+ 53u
21
+ 10u
22
+ u
23
)
2
c
9
(u
2
u + 1)
2
(u
15
+ 5u
14
+ ··· + 18u
2
1)(u
46
+ 16u
45
+ ··· 7u + 1)
15
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
6
c
10
(y
2
+ y + 1)
2
(y
15
+ 5y
14
+ ··· + 18y
2
1)(y
46
+ 16y
45
+ ··· 7y + 1)
c
2
, c
5
(y
2
+ y + 1)
2
(y
15
+ 13y
14
+ ··· + 36y 1)
(y
46
+ 28y
45
+ ··· 31y + 1)
c
3
, c
7
y
4
(y
15
5y
14
+ ··· + 48y 16)
(4 + 9y 38y
3
+ 58y
4
+ 91y
5
662y
6
+ 1839y
7
3558y
8
+ 5415y
9
6876y
10
+ 7532y
11
7320y
12
+ 6468y
13
5320y
14
+ 4116y
15
2966y
16
+ 1929y
17
1088y
18
+ 510y
19
190y
20
+ 53y
21
10y
22
+ y
23
)
2
c
8
y
4
(y
15
+ 3y
14
+ ··· 1024y 256)(y
46
+ 12y
45
+ ··· 2592y + 256)
c
9
, c
11
(y
2
+ y + 1)
2
(y
15
+ 13y
14
+ ··· + 36y 1)
(y
46
+ 28y
45
+ ··· 31y + 1)
16