11a
237
(K11a
237
)
1
Arc Sequences
7 1 10 9 11 8 2 3 4 6 5
Solving Sequence
2,7
8 1
3,10
4 6 11 5 9
c
7
c
1
c
2
c
3
c
6
c
10
c
5
c
9
c
4
, c
8
, c
11
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
4
u
2
+ 1, u
2
+ a u, u
3
u
2
+ bi
I
u
2
= ha
30
+ 6a
29
+ ··· 178501a + 179509,
7.90647 × 10
98
b + 3.02116 × 10
93
a
29
+ ··· 7.67581 × 10
98
a 4.33584 × 10
97
,
7.90647 × 10
98
u + 1.95599 × 10
93
a
29
+ ··· 6.40705 × 10
97
a + 5.87960 × 10
98
i
I
u
3
= hu
20
+ 3u
19
+ ··· 9u 2, 2u
19
+ 7u
18
+ ··· + b 7, 7u
19
17u
18
+ ··· + 2a + 11i
There are 3 irreducible components with 54 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
u
2
+ 1, u
2
+ a u, u
3
u
2
+ bi
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
8
=
u
u
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
2
1
a
10
=
u
2
+ u
u
3
+ u
2
a
4
=
u
3
2u
2
u + 2
u
3
+ u
2
+ u 2
a
6
=
u
3
u
3
+ u
a
11
=
u
2
+ u + 1
u
3
a
5
=
u
3
u
2
u + 1
u
3
+ u 1
a
9
=
u
u
a
9
=
u
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 = 0.866025 0.500000I
a = 1.36603 1.36603I
b = 0.50000 + 1.86603I
3.28987 2.02988I 6.00000 + 3.46410I
u = 0.866025 + 0.500000I
a = 1.36603 + 1.36603I
b = 0.50000 1.86603I
3.28987 + 2.02988I 6.00000 3.46410I
u = 0.866025 0.500000I
a = 0.366025 + 0.366025I
b = 0.500000 + 0.133975I
3.28987 + 2.02988I 6.00000 3.46410I
u = 0.866025 + 0.500000I
a = 0.366025 0.366025I
b = 0.500000 0.133975I
3.28987 2.02988I 6.00000 + 3.46410I
3
II. I
u
2
=
ha
30
+6a
29
+· · ·178501a +179509, 7.91×10
98
b+3.02×10
93
a
29
+· · ·7.68×
10
98
a4.34×10
97
, 7.91×10
98
u+1.96×10
93
a
29
+· · ·6.41×10
97
a+5.88×10
98
i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
2.47391 × 10
6
a
29
0.0000196618a
28
+ ··· + 0.0810355a 0.743645
a
8
=
2.47391 × 10
6
a
29
+ 0.0000196618a
28
+ ··· 0.0810355a + 0.743645
2.47391 × 10
6
a
29
0.0000196618a
28
+ ··· + 0.0810355a 0.743645
a
1
=
1
1.09771 × 10
6
a
29
+ 3.05323 × 10
6
a
28
+ ··· 0.375833a 0.516603
a
3
=
1.09771 × 10
6
a
29
+ 3.05323 × 10
6
a
28
+ ··· 0.375833a + 0.483397
9.28312 × 10
8
a
29
+ 1.02754 × 10
6
a
28
+ ··· 0.139510a + 1.14958
a
10
=
a
3.82113 × 10
6
a
29
0.0000217288a
28
+ ··· + 0.970827a + 0.0548392
a
4
=
1.89842 × 10
6
a
29
+ 0.0000132058a
28
+ ··· + 0.279129a 0.248265
8.54338 × 10
6
a
29
+ 0.0000469141a
28
+ ··· 0.285527a + 0.266923
a
6
=
1.72109 × 10
6
a
29
+ 4.33312 × 10
6
a
28
+ ··· + 0.243131a 1.20462
4.19501 × 10
6
a
29
0.0000239949a
28
+ ··· 0.162095a + 0.460973
a
11
=
4.69893 × 10
6
a
29
+ 0.0000337818a
28
+ ··· + 0.526474a 0.901147
2.25932 × 10
6
a
29
0.0000165428a
28
+ ··· + 0.263533a + 0.553672
a
5
=
8.49271 × 10
7
a
29
8.80968 × 10
6
a
28
+ ··· + 1.27113a + 0.364823
4.58007 × 10
6
a
29
0.0000302880a
28
+ ··· 1.09869a + 0.742161
a
9
=
8.77998 × 10
6
a
29
+ 0.0000558227a
28
+ ··· 0.544864a + 0.463239
7.89724 × 10
6
a
29
0.0000469046a
28
+ ··· + 0.0431680a + 0.403789
a
9
=
8.77998 × 10
6
a
29
+ 0.0000558227a
28
+ ··· 0.544864a + 0.463239
7.89724 × 10
6
a
29
0.0000469046a
28
+ ··· + 0.0431680a + 0.403789
(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 = 0.204399 + 0.532644I
a = 1.67764 0.81186I
b = 0.239948 + 0.474144I
2.93870 1.66084I 6.48958 + 3.96405I
u = 0.204399 0.532644I
a = 1.67764 + 0.81186I
b = 0.239948 0.474144I
2.93870 + 1.66084I 6.48958 3.96405I
u = 0.778519 0.756850I
a = 1.59408 0.00875I
b = 1.72035 + 0.55832I
6.82325 1.50523I 3.84867 + 2.74048I
u = 0.778519 + 0.756850I
a = 1.59408 + 0.00875I
b = 1.72035 0.55832I
6.82325 + 1.50523I 3.84867 2.74048I
u = 0.778519 + 0.756850I
a = 1.49450 0.73575I
b = 1.23440 1.21329I
6.82325 + 1.50523I 3.84867 2.74048I
u = 0.778519 0.756850I
a = 1.49450 + 0.73575I
b = 1.23440 + 1.21329I
6.82325 1.50523I 3.84867 + 2.74048I
u = 0.690784 0.795701I
a = 1.41221 1.34544I
b = 1.97578 0.62398I
5.27292 3.51852I 6.28698 + 2.59027I
u = 0.690784 + 0.795701I
a = 1.41221 + 1.34544I
b = 1.97578 + 0.62398I
5.27292 + 3.51852I 6.28698 2.59027I
u = 1.006637 0.715109I
a = 0.74331 2.21473I
b = 1.97271 + 1.76769I
4.31617 + 9.21780I 8.14540 7.39135I
u = 1.006637 + 0.715109I
a = 0.74331 + 2.21473I
b = 1.97271 1.76769I
4.31617 9.21780I 8.14540 + 7.39135I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.035188 0.117787I
a = 0.534326 0.761139I
b = 0.739803 + 0.262149I
0.91830 3.60340I 14.1637 + 4.4767I
u = 1.035188 + 0.117787I
a = 0.534326 + 0.761139I
b = 0.739803 0.262149I
0.91830 + 3.60340I 14.1637 4.4767I
u = 0.928223 + 0.554966I
a = 0.493032 1.065331I
b = 0.294935 + 1.104221I
1.46912 2.07402I 11.82822 + 2.67122I
u = 0.928223 0.554966I
a = 0.493032 + 1.065331I
b = 0.294935 1.104221I
1.46912 + 2.07402I 11.82822 2.67122I
u = 0.946375 0.717051I
a = 0.39928 1.57114I
b = 1.47413 + 1.93693I
6.30676 4.09199I 4.95573 + 3.15094I
u = 0.946375 + 0.717051I
a = 0.39928 + 1.57114I
b = 1.47413 1.93693I
6.30676 + 4.09199I 4.95573 3.15094I
u = 0.946375 0.717051I
a = 0.00440 2.05002I
b = 0.74872 + 1.77319I
6.30676 4.09199I 4.95573 + 3.15094I
u = 0.946375 + 0.717051I
a = 0.00440 + 2.05002I
b = 0.74872 1.77319I
6.30676 + 4.09199I 4.95573 3.15094I
u = 1.006637 + 0.715109I
a = 0.47335 2.09230I
b = 2.33201 + 1.69789I
4.31617 9.21780I 8.14540 + 7.39135I
u = 1.006637 0.715109I
a = 0.47335 + 2.09230I
b = 2.33201 1.69789I
4.31617 + 9.21780I 8.14540 7.39135I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.035188 0.117787I
a = 0.677075 0.330278I
b = 0.463476 + 0.850858I
0.91830 3.60340I 14.1637 + 4.4767I
u = 1.035188 + 0.117787I
a = 0.677075 + 0.330278I
b = 0.463476 0.850858I
0.91830 + 3.60340I 14.1637 4.4767I
u = 0.928223 0.554966I
a = 0.758022 0.736401I
b = 0.133579 + 1.262481I
1.46912 + 2.07402I 11.82822 2.67122I
u = 0.928223 + 0.554966I
a = 0.758022 + 0.736401I
b = 0.133579 1.262481I
1.46912 2.07402I 11.82822 + 2.67122I
u = 0.860077
a = 0.83254 1.43098I
b = 0.71605 + 1.23075I
1.86559 10.5634
u = 0.860077
a = 0.83254 + 1.43098I
b = 0.71605 1.23075I
1.86559 10.5634
u = 0.204399 + 0.532644I
a = 0.926592 0.094912I
b = 0.089525 1.059529I
2.93870 1.66084I 6.48958 + 3.96405I
u = 0.204399 0.532644I
a = 0.926592 + 0.094912I
b = 0.089525 + 1.059529I
2.93870 + 1.66084I 6.48958 3.96405I
u = 0.690784 + 0.795701I
a = 1.67640 1.02771I
b = 2.04609 0.19429I
5.27292 + 3.51852I 6.28698 2.59027I
u = 0.690784 0.795701I
a = 1.67640 + 1.02771I
b = 2.04609 + 0.19429I
5.27292 3.51852I 6.28698 + 2.59027I
7
III. I
u
3
=
hu
20
+3u
19
+· · ·9u2, 2u
19
+7u
18
+· · ·+b7, 7u
19
17u
18
+· · ·+2a+11i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
8
=
u
u
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
10
=
7
2
u
19
+
17
2
u
18
+ ··· 26u
11
2
2u
19
7u
18
+ ··· + 26u + 7
a
4
=
1
2
u
19
+
1
2
u
18
+ ··· u +
1
2
u
19
2u
18
+ ··· + 4u + 1
a
6
=
u
3
u
3
+ u
a
11
=
3
2
u
19
+
7
2
u
18
+ ··· 10u
3
2
u
19
3u
18
+ ··· + 11u + 3
a
5
=
1
2
u
19
3
2
u
18
+ ··· + 6u +
3
2
u
19
2u
18
+ ··· + 6u + 1
a
9
=
u
7
2u
5
+ 2u
3
2u
u
9
+ u
7
u
5
+ u
a
9
=
u
7
2u
5
+ 2u
3
2u
u
9
+ u
7
u
5
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.094699 0.484177I
a = 1.233086 0.654159I
b = 1.03313 + 1.31314I
5.87096 0.28405I 5.48153 + 0.41216I
u = 1.094699 + 0.484177I
a = 1.233086 + 0.654159I
b = 1.03313 1.31314I
5.87096 + 0.28405I 5.48153 0.41216I
u = 1.045459 0.735740I
a = 1.13601 2.09362I
b = 2.72801 + 1.35299I
9.8480 13.6547I 5.09222 + 8.08354I
u = 1.045459 + 0.735740I
a = 1.13601 + 2.09362I
b = 2.72801 1.35299I
9.8480 + 13.6547I 5.09222 8.08354I
u = 0.983982 0.659235I
a = 0.16056 + 1.74533I
b = 1.30857 1.61152I
0.77253 5.59830I 13.4954 + 6.8250I
u = 0.983982 + 0.659235I
a = 0.16056 1.74533I
b = 1.30857 + 1.61152I
0.77253 + 5.59830I 13.4954 6.8250I
u = 0.669875 0.667713I
a = 1.190213 0.674360I
b = 1.247573 0.342984I
0.157185 + 0.414126I 12.01664 2.08787I
u = 0.669875 + 0.667713I
a = 1.190213 + 0.674360I
b = 1.247573 + 0.342984I
0.157185 0.414126I 12.01664 + 2.08787I
u = 0.662840 0.867795I
a = 1.32667 + 1.54703I
b = 2.22187 + 0.12584I
11.02128 + 7.69202I 3.25100 3.40395I
u = 0.662840 + 0.867795I
a = 1.32667 1.54703I
b = 2.22187 0.12584I
11.02128 7.69202I 3.25100 + 3.40395I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.401234
a = 0.870492
b = 0.349271
0.553031 18.0199
u = 0.223918 0.785611I
a = 0.982811 0.810382I
b = 0.416577 + 0.953566I
8.53664 4.27425I 2.38649 + 3.51536I
u = 0.223918 + 0.785611I
a = 0.982811 + 0.810382I
b = 0.416577 0.953566I
8.53664 + 4.27425I 2.38649 3.51536I
u = 0.842451 0.593410I
a = 0.075889 + 0.192728I
b = 0.178299 + 0.117331I
1.45151 + 2.34993I 9.21397 4.74077I
u = 0.842451 + 0.593410I
a = 0.075889 0.192728I
b = 0.178299 0.117331I
1.45151 2.34993I 9.21397 + 4.74077I
u = 0.888477 0.831617I
a = 0.066014 0.353048I
b = 0.352253 0.258777I
15.2534 + 3.0855I 1.70716 2.62885I
u = 0.888477 + 0.831617I
a = 0.066014 + 0.353048I
b = 0.352253 + 0.258777I
15.2534 3.0855I 1.70716 + 2.62885I
u = 1.02515
a = 0.617971
b = 0.633516
4.75848 19.2848
u = 1.137886 0.165885I
a = 0.490772 0.388916I
b = 0.493927 0.523954I
3.92725 + 7.17367I 8.70322 5.73165I
u = 1.137886 + 0.165885I
a = 0.490772 + 0.388916I
b = 0.493927 + 0.523954I
3.92725 7.17367I 8.70322 + 5.73165I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
(u
4
u
2
+ 1)
(1 + 2u 2u
2
2u
3
+ 6u
4
8u
6
+ 2u
7
+ 9u
8
4u
9
6u
10
+ 4u
11
+ 3u
12
2u
13
u
14
+ u
15
)
2
(u
20
+ 3u
19
+ ··· 9u 2)
c
2
(u
2
+ u + 1)
2
(1 + 12u
3
+ 42u
4
+ 96u
5
+ 158u
6
+ 206u
7
+ 221u
8
+ 196u
9
+ 146u
10
+ 90u
11
+ 45u
12
+ 18u
13
+ 5u
14
+ u
15
)
2
(u
20
+ 7u
19
+ ··· + 33u + 4)
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(u
2
+ 1)
2
(u
20
+ 11u
18
+ ··· 3u 1)(u
30
+ u
29
+ ··· + 6u + 1)
c
6
(u
2
u + 1)
2
(1 + 12u
3
+ 42u
4
+ 96u
5
+ 158u
6
+ 206u
7
+ 221u
8
+ 196u
9
+ 146u
10
+ 90u
11
+ 45u
12
+ 18u
13
+ 5u
14
+ u
15
)
2
(u
20
+ 7u
19
+ ··· + 33u + 4)
c
8
u
4
(1 4u + 8u
3
+ 10u
4
2u
5
18u
6
+ 14u
7
3u
8
+ 4u
9
6u
10
+ 8u
11
u
12
u
14
+ u
15
)
2
(u
20
+ 3u
19
+ ··· + 48u 32)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
(y
2
y + 1)
2
(1 + 12y
3
42y
4
+ 96y
5
158y
6
+ 206y
7
221y
8
+ 196y
9
146y
10
+ 90y
11
45y
12
+ 18y
13
5y
14
+ y
15
)
2
(y
20
7y
19
+ ··· 33y + 4)
c
2
, c
6
(y
2
+ y + 1)
2
(1 84y
2
172y
3
+ 98y
4
+ 596y
5
+ 638y
6
+ 118y
7
293y
8
196y
9
+ 102y
10
+ 218y
11
+ 147y
12
+ 54y
13
+ 11y
14
+ y
15
)
2
(y
20
+ 13y
19
+ ··· 561y + 16)
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(y + 1)
4
(y
20
+ 22y
19
+ ··· 5y + 1)(y
30
+ 23y
29
+ ··· 16y + 1)
c
8
y
4
(1 + 16y 84y
2
+ 116y
3
238y
4
+ 568y
5
318y
6
+ 322y
7
133y
8
+ 204y
9
18y
10
+ 74y
11
5y
12
+ 14y
13
y
14
+ y
15
)
2
(y
20
y
19
+ ··· + 3328y + 1024)
12