11n
136
(K11n
136
)
1
Arc Sequences
6 1 9 7 10 2 4 1 5 9 4
Solving Sequence
4,9 1,3
2 8 7 5 6 11 10
c
3
c
2
c
8
c
7
c
4
c
6
c
11
c
10
c
1
, c
5
, c
9
Representation Ideals
I =
5
\
i=1
I
u
i
I
u
1
= hu 1, a, b + 1i
I
u
2
= hu
6
u
5
u
4
+ 2u
3
3u
2
+ 2u 1, b + u, u
4
+ 2u
2
+ a u + 1i
I
u
3
= hu
6
3u
5
+ 2u
4
+ u
3
3u
2
+ 2u 1, b u + 1, u
5
+ 3u
4
u
3
3u
2
+ a + 3u 1i
I
u
4
= hu
12
+ 2u
11
7u
10
13u
9
+ 22u
8
+ 32u
7
37u
6
30u
5
+ 31u
4
+ 6u
3
6u
2
3u + 1, b + u,
166u
11
+ 369u
10
+ ··· + 169a 288i
I
u
5
= hu
30
+ 2u
29
+ ··· 66u 79,
5.14017 × 10
40
u
29
4.96074 × 10
40
u
28
+ ··· + 4.56302 × 10
41
b + 6.36971 × 10
42
,
2.43559 × 10
42
u
29
7.16169 × 10
42
u
28
+ ··· + 3.60479 × 10
43
a 1.96694 × 10
44
i
There are 5 irreducible components with 55 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 1, a, b + 1i
(i) Arc colorings
a
4
=
0
1
a
9
=
0
1
a
1
=
1
0
a
3
=
0
1
a
2
=
1
1
a
8
=
1
1
a
7
=
1
0
a
5
=
1
1
a
6
=
0
1
a
11
=
1
1
a
10
=
1
2
a
10
=
1
2
(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
b = 1.00000
4.93480 18.0000
3
II. I
u
2
= hu
6
u
5
u
4
+ 2u
3
3u
2
+ 2u 1, b + u, u
4
+ 2u
2
+ a u + 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
u
4
2u
2
+ u 1
u
a
1
=
1
0
a
3
=
u 1
u
5
+ u
4
+ u
3
2u
2
+ 3u 1
a
2
=
u
5
+ u
4
+ u
3
2u
2
+ 4u 2
u
5
+ u
4
+ u
3
2u
2
+ 3u 1
a
8
=
u
4
2u
2
1
u
a
7
=
u
4
2u
2
1
u
5
+ u
4
+ 2u
3
2u
2
+ u 1
a
5
=
2u
5
2u
4
3u
3
+ 4u
2
5u + 3
2u
5
u
4
3u
3
+ 3u
2
3u + 1
a
6
=
2u
5
4u
3
+ u
2
2u
u
4
2u
2
+ u 1
a
11
=
1
u
2
a
10
=
u
5
2u
3
+ 1
u
2
u + 1
a
10
=
u
5
2u
3
+ 1
u
2
u + 1
(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 = 1.58749
a = 1.27668
b = 1.58749
10.0050 16.0412
u = 0.056351 0.865615I
a = 1.095746 0.524927I
b = 0.056351 + 0.865615I
0.69572 + 5.66603I 8.99565 5.65371I
u = 0.056351 + 0.865615I
a = 1.095746 + 0.524927I
b = 0.056351 0.865615I
0.69572 5.66603I 8.99565 + 5.65371I
u = 0.520377 0.559444I
a = 0.732480 + 0.654168I
b = 0.520377 + 0.559444I
3.09094 + 3.67876I 6.55000 7.14850I
u = 0.520377 + 0.559444I
a = 0.732480 0.654168I
b = 0.520377 0.559444I
3.09094 3.67876I 6.55000 + 7.14850I
u = 1.43404
a = 0.550153
b = 1.43404
4.14809 6.86752
5
III.
I
u
3
= hu
6
3u
5
+2u
4
+u
3
3u
2
+2u1, bu+1, u
5
+3u
4
u
3
3u
2
+a+3u1i
(i) Arc colorings
a
4
=
0
u
a
9
=
u
5
3u
4
+ u
3
+ 3u
2
3u + 1
u 1
a
1
=
1
0
a
3
=
u 1
u
5
+ 3u
4
2u
3
u
2
+ 3u 1
a
2
=
u
5
+ 3u
4
2u
3
u
2
+ 4u 2
u
5
+ 3u
4
2u
3
u
2
+ 3u 1
a
8
=
u
5
3u
4
+ u
3
+ 3u
2
2u
u 1
a
7
=
u
5
3u
4
+ u
3
+ 3u
2
2u
u
5
2u
4
u
3
+ 2u
2
1
a
5
=
2u
5
6u
4
+ 3u
3
+ 4u
2
6u + 3
2u
5
5u
4
+ u
3
+ 4u
2
4u + 1
a
6
=
2u
4
+ 4u
3
+ u
2
3u + 2
u
5
3u
4
+ u
3
+ 3u
2
3u + 1
a
11
=
1
u
2
a
10
=
u
5
2u
4
u
3
+ 2u
2
u + 1
u
2
+ u
a
10
=
u
5
2u
4
u
3
+ 2u
2
u + 1
u
2
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.04852
a = 1.39776
b = 2.04852
7.71260 26.4539
u = 0.210979 0.638344I
a = 1.010648 + 0.698701I
b = 0.789021 0.638344I
2.21137 + 1.58317I 8.77306 + 1.69425I
u = 0.210979 + 0.638344I
a = 1.010648 0.698701I
b = 0.789021 + 0.638344I
2.21137 1.58317I 8.77306 1.69425I
u = 0.789021 0.638344I
a = 0.669484 0.462841I
b = 0.210979 0.638344I
2.21137 1.58317I 8.77306 1.69425I
u = 0.789021 + 0.638344I
a = 0.669484 + 0.462841I
b = 0.210979 + 0.638344I
2.21137 + 1.58317I 8.77306 + 1.69425I
u = 2.04852
a = 0.715431
b = 1.04852
7.71260 26.4539
7
IV.
I
u
4
= hu
12
+ 2u
11
+ · · · 3u + 1, b + u, 166u
11
+ 369u
10
+ · · · + 169a 288i
(i) Arc colorings
a
4
=
0
u
a
9
=
0.982249u
11
2.18343u
10
+ ··· 0.710059u + 1.70414
u
a
1
=
1
0
a
3
=
0.840237u
11
1.65089u
10
+ ··· 4.39053u + 2.33728
0.633136u
11
+ 1.12426u
10
+ ··· + 0.674556u 0.218935
a
2
=
0.207101u
11
0.526627u
10
+ ··· 3.71598u + 2.11834
0.633136u
11
+ 1.12426u
10
+ ··· + 0.674556u 0.218935
a
8
=
0.982249u
11
2.18343u
10
+ ··· 1.71006u + 1.70414
u
a
7
=
0.982249u
11
2.18343u
10
+ ··· 1.71006u + 1.70414
0.633136u
11
+ 1.12426u
10
+ ··· 1.32544u 0.218935
a
5
=
0.426036u
11
0.597633u
10
+ ··· + 5.04142u 1.89941
0.171598u
11
0.106509u
10
+ ··· + 0.136095u + 0.473373
a
6
=
0.112426u
11
+ 0.171598u
10
+ ··· 1.49704u + 0.792899
0.0414201u
11
+ 0.0946746u
10
+ ··· 1.34320u + 0.0236686
a
11
=
1
u
2
a
10
=
0.248521u
11
0.431953u
10
+ ··· 1.05917u + 1.14201
0.142012u
11
0.467456u
10
+ ··· 1.68047u + 0.633136
a
10
=
0.248521u
11
0.431953u
10
+ ··· 1.05917u + 1.14201
0.142012u
11
0.467456u
10
+ ··· 1.68047u + 0.633136
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.77190 0.53892I
a = 1.076764 0.037721I
b = 1.77190 + 0.53892I
7.6014 13.7948I 13.8218 + 7.4992I
u = 1.77190 + 0.53892I
a = 1.076764 + 0.037721I
b = 1.77190 0.53892I
7.6014 + 13.7948I 13.8218 7.4992I
u = 1.72036 0.03367I
a = 0.923801 + 0.441615I
b = 1.72036 + 0.03367I
9.15791 + 5.54846I 14.9776 4.7158I
u = 1.72036 + 0.03367I
a = 0.923801 0.441615I
b = 1.72036 0.03367I
9.15791 5.54846I 14.9776 + 4.7158I
u = 0.385507 0.310495I
a = 0.89688 + 2.08681I
b = 0.385507 + 0.310495I
2.70102 + 2.45198I 9.00502 1.91716I
u = 0.385507 + 0.310495I
a = 0.89688 2.08681I
b = 0.385507 0.310495I
2.70102 2.45198I 9.00502 + 1.91716I
u = 0.264871
a = 1.36172
b = 0.264871
0.612207 16.2730
u = 0.678073 0.310386I
a = 0.86009 2.09158I
b = 0.678073 + 0.310386I
1.13692 4.86316I 15.3188 + 3.9545I
u = 0.678073 + 0.310386I
a = 0.86009 + 2.09158I
b = 0.678073 0.310386I
1.13692 + 4.86316I 15.3188 3.9545I
u = 1.14566
a = 0.194228
b = 1.14566
4.93861 18.1860
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.49443 0.38475I
a = 0.965619 + 0.097254I
b = 1.49443 + 0.38475I
1.76862 + 2.36514I 8.64736 0.93899I
u = 1.49443 + 0.38475I
a = 0.965619 0.097254I
b = 1.49443 0.38475I
1.76862 2.36514I 8.64736 + 0.93899I
10
V. I
u
5
=
hu
30
+2u
29
+· · ·66u79, 5.14×10
40
u
29
4.96×10
40
u
28
+· · ·+4.56×10
41
b+
6.37× 10
42
, 2.44×10
42
u
29
7.16 ×10
42
u
28
+· · · +3.60 ×10
43
a 1.97× 10
44
i
(i) Arc colorings
a
4
=
0
u
a
9
=
0.0675654u
29
+ 0.198672u
28
+ ··· + 17.0406u + 5.45647
0.112648u
29
+ 0.108716u
28
+ ··· + 22.9117u 13.9594
a
1
=
1
0
a
3
=
0.164982u
29
+ 0.393738u
28
+ ··· + 44.6296u + 9.56552
0.256101u
29
0.612130u
28
+ ··· 60.7640u 21.0783
a
2
=
0.0911189u
29
0.218392u
28
+ ··· 16.1344u 11.5128
0.256101u
29
0.612130u
28
+ ··· 60.7640u 21.0783
a
8
=
0.180214u
29
+ 0.307388u
28
+ ··· + 39.9524u 8.50294
0.112648u
29
+ 0.108716u
28
+ ··· + 22.9117u 13.9594
a
7
=
0.180214u
29
+ 0.307388u
28
+ ··· + 39.9524u 8.50294
0.0554301u
29
+ 0.0308805u
28
+ ··· + 12.1755u 9.76926
a
5
=
0.189781u
29
+ 0.545825u
28
+ ··· + 64.8850u + 35.9629
0.0153590u
29
+ 0.0911915u
28
+ ··· + 22.7845u + 11.3153
a
6
=
0.140549u
29
+ 0.168848u
28
+ ··· + 11.5189u 20.3136
0.000758985u
29
0.0658007u
28
+ ··· 13.6852u 16.3414
a
11
=
1
u
2
a
10
=
0.203041u
29
+ 0.606896u
28
+ ··· + 71.6076u + 32.1207
0.142608u
29
+ 0.246809u
28
+ ··· + 31.3399u + 3.40162
a
10
=
0.203041u
29
+ 0.606896u
28
+ ··· + 71.6076u + 32.1207
0.142608u
29
+ 0.246809u
28
+ ··· + 31.3399u + 3.40162
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
11
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.79388 0.09420I
a = 1.108314 0.024265I
b = 1.45017 0.51636I
10.17636 + 2.57627I 15.0709 4.0254I
u = 1.79388 + 0.09420I
a = 1.108314 + 0.024265I
b = 1.45017 + 0.51636I
10.17636 2.57627I 15.0709 + 4.0254I
u = 1.65186 0.31915I
a = 1.063152 + 0.008618I
b = 1.66298 0.62132I
4.38929 7.65996I 11.60171 + 4.83891I
u = 1.65186 + 0.31915I
a = 1.063152 0.008618I
b = 1.66298 + 0.62132I
4.38929 + 7.65996I 11.60171 4.83891I
u = 1.46931 0.07774I
a = 1.004278 0.274861I
b = 1.82057 0.02441I
6.55037 + 0.76607I 13.52677 0.03940I
u = 1.46931 + 0.07774I
a = 1.004278 + 0.274861I
b = 1.82057 + 0.02441I
6.55037 0.76607I 13.52677 + 0.03940I
u = 1.45017 0.51636I
a = 1.183107 + 0.523275I
b = 1.79388 0.09420I
10.17636 2.57627I 15.0709 + 4.0254I
u = 1.45017 + 0.51636I
a = 1.183107 0.523275I
b = 1.79388 + 0.09420I
10.17636 + 2.57627I 15.0709 4.0254I
u = 0.990302
a = 1.62073
b = 1.93654
7.32542 4.72892
u = 0.674810 0.174597I
a = 1.076875 + 0.877302I
b = 0.627325 + 0.826408I
1.85339 + 2.65754I 12.13634 3.34510I
12
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.674810 + 0.174597I
a = 1.076875 0.877302I
b = 0.627325 0.826408I
1.85339 2.65754I 12.13634 + 3.34510I
u = 0.572998 0.112628I
a = 0.052794 + 0.870427I
b = 0.186932 + 0.933368I
2.10570 + 2.66884I 8.49589 5.19452I
u = 0.572998 + 0.112628I
a = 0.052794 0.870427I
b = 0.186932 0.933368I
2.10570 2.66884I 8.49589 + 5.19452I
u = 0.186932 0.933368I
a = 0.533860 0.034297I
b = 0.572998 + 0.112628I
2.10570 + 2.66884I 8.49589 5.19452I
u = 0.186932 + 0.933368I
a = 0.533860 + 0.034297I
b = 0.572998 0.112628I
2.10570 2.66884I 8.49589 + 5.19452I
u = 0.23544 1.52005I
a = 0.134071 0.431702I
b = 0.581176 + 0.044236I
0.89474 + 6.09921I 15.4033 6.7831I
u = 0.23544 + 1.52005I
a = 0.134071 + 0.431702I
b = 0.581176 0.044236I
0.89474 6.09921I 15.4033 + 6.7831I
u = 0.314929 1.087779I
a = 0.356643 + 0.537841I
b = 0.710560 0.015696I
0.330230 0.679087I 12.40066 + 0.76832I
u = 0.314929 + 1.087779I
a = 0.356643 0.537841I
b = 0.710560 + 0.015696I
0.330230 + 0.679087I 12.40066 0.76832I
u = 0.581176 0.044236I
a = 1.028829 0.603856I
b = 0.23544 + 1.52005I
0.89474 + 6.09921I 15.4033 6.7831I
13
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.581176 + 0.044236I
a = 1.028829 + 0.603856I
b = 0.23544 1.52005I
0.89474 6.09921I 15.4033 + 6.7831I
u = 0.627325 0.826408I
a = 0.264607 0.894844I
b = 0.674810 + 0.174597I
1.85339 + 2.65754I 12.13634 3.34510I
u = 0.627325 + 0.826408I
a = 0.264607 + 0.894844I
b = 0.674810 0.174597I
1.85339 2.65754I 12.13634 + 3.34510I
u = 0.710560 0.015696I
a = 0.974165 + 0.329118I
b = 0.314929 1.087779I
0.330230 + 0.679087I 12.40066 0.76832I
u = 0.710560 + 0.015696I
a = 0.974165 0.329118I
b = 0.314929 + 1.087779I
0.330230 0.679087I 12.40066 + 0.76832I
u = 1.66298 0.62132I
a = 0.994935 0.159133I
b = 1.65186 0.31915I
4.38929 + 7.65996I 11.60171 4.83891I
u = 1.66298 + 0.62132I
a = 0.994935 + 0.159133I
b = 1.65186 + 0.31915I
4.38929 7.65996I 11.60171 + 4.83891I
u = 1.82057 0.02441I
a = 0.819704 0.189938I
b = 1.46931 0.07774I
6.55037 0.76607I 13.52677 + 0.03940I
u = 1.82057 + 0.02441I
a = 0.819704 + 0.189938I
b = 1.46931 + 0.07774I
6.55037 + 0.76607I 13.52677 0.03940I
u = 1.93654
a = 0.828805
b = 0.990302
7.32542 4.72892
14
VI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u + 1)(u
6
2u
4
+ 2u
2
+ u 1)(u
6
2u
4
u
3
+ 2u
2
1)
(u
12
u
11
+ ··· + 2u 1)(u
30
+ u
29
+ ··· + 2u
2
+ 1)
c
2
, c
10
(u + 1)(u
6
+ 4u
5
+ 8u
4
+ 10u
3
+ 8u
2
+ 5u + 1)
(u
6
+ 4u
5
+ ··· + 4u + 1)(u
12
+ 7u
11
+ ··· + 8u + 1)
(u
30
+ 17u
29
+ ··· 4u + 1)
c
3
u(u
3
u
2
1)
2
(u
6
+ 2u
5
+ u
4
+ 3u
3
+ 2u
2
+ u + 1)
(u
12
+ 11u
11
+ ··· + 88u + 8)
(3 + 8u 11u
2
+ 13u
3
+ 4u
4
62u
5
+ 78u
6
+ 34u
7
169u
8
+ 148u
9
9u
10
61u
11
+ 31u
12
+ 2u
13
5u
14
+ u
15
)
2
c
4
(u + 1)(u
3
+ u + 1)
2
(u
6
u
5
+ 2u
4
3u
3
+ u
2
2u + 1)
(u
12
+ 7u
11
+ ··· 4u 8)
(1 + 5u 9u
2
+ 13u
3
13u
4
+ 19u
5
21u
6
+ 27u
7
26u
8
+ 24u
9
20u
10
+ 16u
11
11u
12
+ 7u
13
3u
14
+ u
15
)
2
c
6
, c
9
(u + 1)(u
6
2u
4
+ 2u
2
u 1)(u
6
2u
4
+ u
3
+ 2u
2
1)
(u
12
u
11
+ ··· + 2u 1)(u
30
+ u
29
+ ··· + 2u
2
+ 1)
c
7
(u + 1)(u
3
+ u 1)
2
(u
6
+ u
5
+ 2u
4
+ 3u
3
+ u
2
+ 2u + 1)
(u
12
+ 7u
11
+ ··· 4u 8)
(1 + 5u 9u
2
+ 13u
3
13u
4
+ 19u
5
21u
6
+ 27u
7
26u
8
+ 24u
9
20u
10
+ 16u
11
11u
12
+ 7u
13
3u
14
+ u
15
)
2
c
8
, c
11
(u 1)(u
6
3u
5
+ 2u
4
+ u
3
3u
2
+ 2u 1)
(u
6
u
5
+ ··· + 2u 1)(u
12
+ 2u
11
+ ··· 3u + 1)
(u
30
+ 2u
29
+ ··· 66u 79)
15
VII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y 1)(y
6
4y
5
+ 8y
4
11y
3
+ 8y
2
4y + 1)
(y
6
4y
5
+ ··· 5y + 1)(y
12
7y
11
+ ··· 8y + 1)
(y
30
17y
29
+ ··· + 4y + 1)
c
2
(y 1)(y
6
10y
3
20y
2
9y + 1)(y
6
8y
4
23y
3
8y
2
+ 1)
(y
12
+ y
11
+ ··· 32y + 1)(y
30
5y
29
+ ··· 112y + 1)
c
3
y(y
3
y
2
2y 1)
2
(y
6
2y
5
7y
4
7y
3
+ 3y + 1)
(y
12
7y
11
+ ··· 1760y + 64)(y
30
42y
29
+ ··· + 36y + 81)
c
4
, c
7
(y 1)(y
3
+ 2y
2
+ y 1)
2
(y
6
+ 3y
5
7y
3
7y
2
2y + 1)
(y
12
+ 5y
11
+ ··· 656y + 64)
(1 + 7y + 23y
2
+ 83y
3
+ 165y
4
+ 249y
5
+ 311y
6
+ 311y
7
+ 240y
8
+ 152y
9
+ 86y
10
+ 50y
11
+ 31y
12
+ 15y
13
+ 5y
14
+ y
15
)
2
c
5
, c
9
(y 1)(y
6
4y
5
+ 8y
4
11y
3
+ 8y
2
4y + 1)
(y
6
4y
5
+ ··· 5y + 1)(y
12
7y
11
+ ··· 8y + 1)
(y
30
17y
29
+ ··· + 4y + 1)
c
8
, c
11
(y 1)(y
6
5y
5
+ 4y
4
3y
3
+ y
2
+ 2y + 1)
(y
6
3y
5
+ ··· + 2y + 1)(y
12
18y
11
+ ··· 21y + 1)
(y
30
30y
29
+ ··· 120802y + 6241)
c
10
(y 1)(y
6
10y
3
20y
2
9y + 1)(y
6
8y
4
23y
3
8y
2
+ 1)
(y
12
+ y
11
+ ··· 32y + 1)(y
30
5y
29
+ ··· 112y + 1)
16