11n
108
(K11n
108
)
1
Arc Sequences
7 1 8 9 11 2 10 4 6 8 9
Solving Sequence
7,10 2,8
1 3 6 9 11 5 4
c
7
c
1
c
2
c
6
c
9
c
11
c
5
c
4
c
3
, c
8
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
12
+ u
11
+ 4u
10
+ 3u
9
+ 8u
8
+ 5u
7
+ 10u
6
+ 5u
5
+ 7u
4
+ 3u
3
+ 4u
2
+ u + 1,
u
11
+ 3u
9
u
8
+ 5u
7
3u
6
+ 5u
5
5u
4
+ 2u
3
4u
2
+ a + u 2,
u
11
+ u
10
2u
9
+ 4u
8
3u
7
+ 7u
6
3u
5
+ 8u
4
u
3
+ 4u
2
+ b u + 3i
I
u
2
= hu
48
+ 12u
46
+ ··· + u + 1, 2.70726 × 10
41
u
47
8.88694 × 10
39
u
46
+ ··· + 3.53065 × 10
41
a 5.45892 × 10
41
,
1.51698 × 10
41
u
47
2.55741 × 10
40
u
46
+ ··· + 3.53065 × 10
41
b + 3.08990 × 10
41
i
There are 2 irreducible components with 60 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
12
+ u
11
+ · · · + u + 1, u
11
+ 3u
9
+ · · · + a 2, u
11
+ u
10
+ · · · + b + 3i
(i) Arc colorings
a
7
=
0
u
a
10
=
u
11
3u
9
+ u
8
5u
7
+ 3u
6
5u
5
+ 5u
4
2u
3
+ 4u
2
u + 2
u
11
u
10
+ 2u
9
4u
8
+ 3u
7
7u
6
+ 3u
5
8u
4
+ u
3
4u
2
+ u 3
a
2
=
1
0
a
8
=
u
11
+ 2u
10
+ 4u
9
+ 6u
8
+ 7u
7
+ 10u
6
+ 7u
5
+ 10u
4
+ 2u
3
+ 5u
2
+ u + 2
u
11
2u
10
+ ··· 3u 3
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
6
=
u
u
a
9
=
u
11
u
10
4u
9
3u
8
8u
7
4u
6
9u
5
2u
4
5u
3
+ u
2
2u + 1
u
11
+ 3u
9
+ 6u
7
+ 7u
5
u
4
+ 4u
3
u
2
+ 2u 2
a
11
=
2u
11
u
10
6u
9
2u
8
10u
7
3u
6
10u
5
3u
4
4u
3
3u
2
3u
3u
11
+ u
10
+ 9u
9
+ u
8
+ 15u
7
+ 15u
5
2u
4
+ 6u
3
+ 4u 2
a
5
=
u
10
+ u
9
+ 3u
8
+ 3u
7
+ 5u
6
+ 5u
5
+ 5u
4
+ 5u
3
+ 2u
2
+ 2u + 2
u
11
3u
10
+ ··· 5u 4
a
4
=
u
7
+ u
6
+ 3u
5
+ 2u
4
+ 4u
3
+ 3u
2
+ 3u + 2
u
11
u
10
3u
9
2u
8
5u
7
3u
6
6u
5
2u
4
3u
3
u
2
3u 1
a
4
=
u
7
+ u
6
+ 3u
5
+ 2u
4
+ 4u
3
+ 3u
2
+ 3u + 2
u
11
u
10
3u
9
2u
8
5u
7
3u
6
6u
5
2u
4
3u
3
u
2
3u 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 = 0.730983 1.067708I
a = 0.783954 + 0.556807I
b = 1.35855 0.66994I
2.81163 + 4.85898I 13.04273 4.67018I
u = 0.730983 + 1.067708I
a = 0.783954 0.556807I
b = 1.35855 + 0.66994I
2.81163 4.85898I 13.04273 + 4.67018I
u = 0.632129 0.585112I
a = 1.10397 + 1.34380I
b = 0.559707 0.469674I
4.36500 + 0.58143I 14.6367 + 0.1461I
u = 0.632129 + 0.585112I
a = 1.10397 1.34380I
b = 0.559707 + 0.469674I
4.36500 0.58143I 14.6367 0.1461I
u = 0.138854 1.248377I
a = 0.381667 + 0.139275I
b = 0.115022 0.348600I
5.04286 + 3.33069I 6.48064 3.71539I
u = 0.138854 + 1.248377I
a = 0.381667 0.139275I
b = 0.115022 + 0.348600I
5.04286 3.33069I 6.48064 + 3.71539I
u = 0.100519 0.675159I
a = 1.053101 + 0.625200I
b = 2.41399 0.59791I
2.69108 2.27732I 0.79309 + 1.51304I
u = 0.100519 + 0.675159I
a = 1.053101 0.625200I
b = 2.41399 + 0.59791I
2.69108 + 2.27732I 0.79309 1.51304I
u = 0.539832 0.700072I
a = 0.674704 1.237520I
b = 0.287415 0.388642I
7.43656 1.12784I 13.7843 + 5.8074I
u = 0.539832 + 0.700072I
a = 0.674704 + 1.237520I
b = 0.287415 + 0.388642I
7.43656 + 1.12784I 13.7843 5.8074I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.562653 1.041425I
a = 0.608810 0.596943I
b = 1.39817 + 1.49582I
6.28022 3.33267I 14.8487 + 3.1328I
u = 0.562653 + 1.041425I
a = 0.608810 + 0.596943I
b = 1.39817 1.49582I
6.28022 + 3.33267I 14.8487 3.1328I
4
II. I
u
2
=
hu
48
+12u
46
+· · ·+u+1, 2.71×10
41
u
47
8.89×10
39
u
46
+· · ·+3.53×10
41
a
5.46 × 10
41
, 1.52 × 10
41
u
47
2.56 × 10
40
u
46
+ · · · + 3.53 × 10
41
b + 3.09 × 10
41
i
(i) Arc colorings
a
7
=
0
u
a
10
=
0.766788u
47
+ 0.0251708u
46
+ ··· + 0.164071u + 1.54615
0.429659u
47
+ 0.0724346u
46
+ ··· 2.67741u 0.875164
a
2
=
1
0
a
8
=
0.322663u
47
0.602755u
46
+ ··· + 3.13105u 1.73661
0.109188u
47
0.0229087u
46
+ ··· 1.75297u + 0.640189
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
6
=
u
u
a
9
=
1.06472u
47
+ 0.344126u
46
+ ··· + 0.598805u + 1.64376
0.727587u
47
0.246520u
46
+ ··· 3.11215u 0.972770
a
11
=
0.599807u
47
0.0654044u
46
+ ··· + 6.32362u + 4.07675
0.649280u
47
+ 0.125140u
46
+ ··· 4.52987u 0.940151
a
5
=
0.221347u
47
+ 0.967359u
46
+ ··· 9.15818u + 2.61103
0.453752u
47
0.874863u
46
+ ··· + 2.71439u 1.96348
a
4
=
0.884277u
47
+ 0.669229u
46
+ ··· 4.06934u + 1.69715
0.470816u
47
0.159105u
46
+ ··· + 3.40112u 0.619755
a
4
=
0.884277u
47
+ 0.669229u
46
+ ··· 4.06934u + 1.69715
0.470816u
47
0.159105u
46
+ ··· + 3.40112u 0.619755
(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 = 0.941336 0.747392I
a = 0.616176 + 0.497628I
b = 0.207780 + 0 .089537I
3.52678 + 2.80965I 12.72101 4.52739I
u = 0.941336 + 0.747392I
a = 0.616176 0.497628I
b = 0.207780 0.089537I
3.52678 2.80965I 12.72101 + 4.52739I
u = 0.929872
a = 1.41886
b = 0.440153
5.55263 20.1448
u = 0.918656 0.227193I
a = 0.910590 0.833969I
b = 0.1296668 0.0357852I
0.10143 2.03024I 9.27605 + 1.57972I
u = 0.918656 + 0.227193I
a = 0.910590 + 0 .833969I
b = 0.1296668 + 0 .0357852I
0.10143 + 2.03024I 9.27605 1.57972I
u = 0.796507 0.983972I
a = 0.514905 + 0.397431I
b = 1.278404 0.522506I
2.78226 + 3.51974I 11.16150 1.38253I
u = 0.796507 + 0.983972I
a = 0.514905 0.397431I
b = 1.278404 + 0 .522506I
2.78226 3.51974I 11.16150 + 1.38253I
u = 0.604419 1.211909I
a = 0.623406 + 0.760107I
b = 1.25709 0.98472I
2.15454 + 5.46033I 6.95643 10.69062I
u = 0.604419 + 1.211909I
a = 0.623406 0.760107I
b = 1.25709 + 0 .98472I
2.15454 5.46033I 6.95643 + 10.69062I
u = 0.585474 1.204260I
a = 0.934193 0.462853I
b = 1.90619 + 1.08716I
3.03425 + 7.47849I 5.84589 4.96713I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.585474 + 1.204260I
a = 0.934193 + 0 .462853I
b = 1.90619 1.08716I
3.03425 7.47849I 5.84589 + 4.96713I
u = 0.499909 1.039799I
a = 0.791679 0.683852I
b = 1.67535 0.45469I
3.44366 + 6.15745I 7.35660 5.98847I
u = 0.499909 + 1.039799I
a = 0.791679 + 0 .683852I
b = 1.67535 + 0.45469I
3.44366 6.15745I 7.35660 + 5.98847I
u = 0.404207 1.040885I
a = 0.661689 1.055534I
b = 1.57736 + 0 .35227I
4.08127 + 0.33982I 5.63392 1.74002I
u = 0.404207 + 1.040885I
a = 0.661689 + 1.055534I
b = 1.57736 0.35227I
4.08127 0.33982I 5.63392 + 1.74002I
u = 0.398216 0.553392I
a = 0.166915 0.954149I
b = 1.88103 + 0.11548I
1.85237 2.17126I 11.26403 + 0.26114I
u = 0.398216 + 0.553392I
a = 0.166915 + 0 .954149I
b = 1.88103 0.11548I
1.85237 + 2.17126I 11.26403 0.26114I
u = 0.361369
a = 0.918562
b = 0.192100
0.656820 15.2638
u = 0.321176 1.348335I
a = 0.229333 0.501506I
b = 0.519328 + 1 .144245I
5.15741 + 2.22291I 6.72402 + 2.42656I
u = 0.321176 + 1.348335I
a = 0.229333 + 0.501506I
b = 0.519328 1.144245I
5.15741 2.22291I 6.72402 2.42656I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.162479 0.997160I
a = 0.796642 + 0 .082165I
b = 1.297063 + 0.362469I
2.18331 + 1.63548I 3.97796 4.36559I
u = 0.162479 + 0.997160I
a = 0.796642 0.082165I
b = 1.297063 0.362469I
2.18331 1.63548I 3.97796 + 4.36559I
u = 0.106371 0.362688I
a = 2.56164 0.08908I
b = 1.40244 + 0.74659I
2.06429 + 2.66223I 9.42585 6.21325I
u = 0.106371 + 0.362688I
a = 2.56164 + 0 .08908I
b = 1.40244 0.74659I
2.06429 2.66223I 9.42585 + 6.21325I
u = 0.11793 1.41674I
a = 0.334531 + 0.638468I
b = 0.738377 1.160398I
5.42165 + 4.76380I 6.19605 6.93136I
u = 0.11793 + 1 .41674I
a = 0.334531 0.638468I
b = 0.738377 + 1 .160398I
5.42165 4.76380I 6.19605 + 6.93136I
u = 0.333524 1.091233I
a = 0.793898 + 0 .506003I
b = 1.69619 + 0.30553I
5.53120 0.07674I 3.78825 + 1.36866I
u = 0.333524 + 1 .091233I
a = 0.793898 0.506003I
b = 1.69619 0.30553I
5.53120 + 0.07674I 3.78825 1.36866I
u = 0.381747 0.518500I
a = 0.80594 + 1 .69233I
b = 0.781188 + 0 .380254I
7.19540 0.40564I 10.12062 3.68058I
u = 0.381747 + 0 .518500I
a = 0.80594 1.69233I
b = 0.781188 0.380254I
7.19540 + 0.40564I 10.12062 + 3.68058I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.448179 0.746413I
a = 0.85173 1.68189I
b = 0.538015 + 0 .411437I
3.91125 1.34687I 8.42551 + 6.39280I
u = 0.448179 + 0 .746413I
a = 0.85173 + 1.68189I
b = 0.538015 0.411437I
3.91125 + 1.34687I 8.42551 6.39280I
u = 0.459033 0.937886I
a = 0.870938 1.085995I
b = 1.60601 + 1 .37167I
3.26073 2.43030I 10.07781 + 2.10293I
u = 0.459033 + 0 .937886I
a = 0.870938 + 1.085995I
b = 1.60601 1.37167I
3.26073 + 2.43030I 10.07781 2.10293I
u = 0.485670 1.065709I
a = 0.599525 + 0 .414902I
b = 1.34446 1.57301I
5.39976 3.46453I 4.57734 + 4.61522I
u = 0.485670 + 1 .065709I
a = 0.599525 0.414902I
b = 1.34446 + 1.57301I
5.39976 + 3.46453I 4.57734 4.61522I
u = 0.490465 0.282839I
a = 1.05131 1.42733I
b = 1.182052 + 0.116426I
2.19332 + 2.77840I 8.82502 3.26643I
u = 0.490465 + 0 .282839I
a = 1.05131 + 1 .42733I
b = 1.182052 0.116426I
2.19332 2.77840I 8.82502 + 3.26643I
u = 0.520417 1.079784I
a = 0.868092 + 0.833067I
b = 1.71003 0.29825I
4.30258 7.07120I 6.19230 + 6.78102I
u = 0.520417 + 1 .079784I
a = 0.868092 0.833067I
b = 1.71003 + 0 .29825I
4.30258 + 7.07120I 6.19230 6.78102I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.674452 1.194326I
a = 0.981295 + 0 .572996I
b = 2.05005 1.07562I
1.3361 14.6102I 8.09331 + 8.42578I
u = 0.674452 + 1 .194326I
a = 0.981295 0.572996I
b = 2.05005 + 1.07562I
1.3361 + 14.6102I 8.09331 8.42578I
u = 0.706876 0.959813I
a = 1.113878 0.269922I
b = 1.86890 + 0 .51778I
2.97315 6.07189I 11.5551 + 10.8395I
u = 0.706876 + 0 .959813I
a = 1.113878 + 0.269922I
b = 1.86890 0.51778I
2.97315 + 6.07189I 11.5551 10.8395I
u = 0.757890 0.744138I
a = 0.330028 1.128433I
b = 0.022736 + 0.298296I
3.63018 + 0.51155I 11.50496 4.41570I
u = 0.757890 + 0 .744138I
a = 0.330028 + 1.128433I
b = 0.022736 0.298296I
3.63018 0.51155I 11.50496 + 4.41570I
u = 1.008190 0.399133I
a = 0.871224 + 0 .951733I
b = 0.0707079 + 0.0668180I
1.11909 + 8.52957I 10.59618 5.34140I
u = 1.008190 + 0 .399133I
a = 0.871224 0.951733I
b = 0.0707079 0.0668180I
1.11909 8.52957I 10.59618 + 5.34140I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
12
+ u
11
+ ··· + u + 1)(u
48
+ 12u
46
+ ··· + u + 1)
c
2
(u
12
+ 7u
11
+ ··· + 7u + 1)(u
48
+ 24u
47
+ ··· + 13u + 1)
c
3
, c
4
(u
12
5u
10
+ u
9
+ 11u
8
4u
7
16u
6
+ 5u
5
+ 15u
4
2u
3
6u
2
+ 1)
(u
48
+ u
47
+ ··· 16u 11)
c
5
(u
12
+ u
10
3u
9
2u
7
+ 4u
6
u
5
+ 3u
4
+ u
3
4u
2
+ 1)
(u
48
+ 3u
47
+ ··· 14u + 1)
c
6
(u
12
u
11
+ ··· u + 1)(u
48
+ 12u
46
+ ··· + u + 1)
c
7
(u
12
+ 2u
11
+ ··· + 2u + 1)(u
48
+ u
47
+ ··· 268u 119)
c
8
(u
12
5u
10
u
9
+ 11u
8
+ 4u
7
16u
6
5u
5
+ 15u
4
+ 2u
3
6u
2
+ 1)
(u
48
+ u
47
+ ··· 16u 11)
c
9
(u
12
4u
10
+ u
9
+ 3u
8
u
7
+ 4u
6
2u
5
3u
3
+ u
2
+ 1)
(u
48
+ u
47
+ ··· 10u 27)
c
10
(u
12
2u
11
+ ··· 2u + 1)(u
48
+ u
47
+ ··· 268u 119)
c
11
(u
12
+ 2u
11
u
10
2u
9
+ 3u
8
+ u
7
+ u
6
+ 7u
5
+ 2u
4
+ 2u
2
+ 1)
(u
48
+ 5u
47
+ ··· + 22u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
12
+ 7y
11
+ ··· + 7y + 1)(y
48
+ 24y
47
+ ··· + 13y + 1)
c
2
(y
12
+ 3y
11
+ ··· y + 1)(y
48
+ 8y
47
+ ··· 71y + 1)
c
3
, c
4
, c
8
(y
12
10y
11
+ ··· 12y + 1)(y
48
17y
47
+ ··· 2302y + 121)
c
5
(y
12
+ 2y
11
+ ··· 8y + 1)(y
48
+ 35y
47
+ ··· 126y + 1)
c
7
, c
10
(y
12
10y
11
+ ··· + 2y + 1)(y
48
25y
47
+ ··· 291260y + 14161)
c
9
(y
12
8y
11
+ ··· + 2y + 1)(y
48
19y
47
+ ··· 6904y + 729)
c
11
(y
12
6y
11
+ ··· + 4y + 1)(y
48
37y
47
+ ··· + 18y + 1)
12