11n
137
(K11n
137
)
1
Arc Sequences
8 1 9 8 11 10 2 4 6 7 9
Solving Sequence
6,9
10 7 11
1,4
3 2 5 8
c
9
c
6
c
10
c
11
c
3
c
2
c
5
c
8
c
1
, c
4
, c
7
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
6
6u
5
+ 16u
4
24u
3
+ 23u
2
14u + 5, u
5
5u
4
+ 10u
3
10u
2
+ b + 6u 2,
3u
5
13u
4
+ 23u
3
17u
2
+ 5a + 4u + 3i
I
u
2
= hu
16
+ 2u
15
+ ··· 10u + 1,
1.73365 × 10
15
u
15
4.00311 × 10
15
u
14
+ ··· + 6.06510 × 10
14
b + 6.17175 × 10
15
,
2431814551962578u
15
+ 5014581106198658u
14
+ ··· + 606509508209333a 16828491642260893i
I
u
3
= hu
18
u
17
+ ··· 3598u + 589,
1.09339 × 10
32
u
17
1.03004 × 10
31
u
16
+ ··· + 2.96925 × 10
32
b + 5.87333 × 10
34
,
2.31921 × 10
34
u
17
1.62455 × 10
33
u
16
+ ··· + 1.74889 × 10
35
a + 1.33772 × 10
37
i
There are 3 irreducible components with 40 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
6
6u
5
+ 16u
4
24u
3
+ 23u
2
14u + 5, u
5
5u
4
+ 10u
3
10u
2
+ b + 6u 2, 3u
5
13u
4
+ 23u
3
17u
2
+ 5a + 4u + 3i
(i) Arc colorings
a
6
=
0
u
a
9
=
3
5
u
5
+
13
5
u
4
+ ···
4
5
u
3
5
u
5
+ 5u
4
10u
3
+ 10u
2
6u + 2
a
10
=
3
5
u
5
+
13
5
u
4
+ ···
4
5
u
3
5
u
3
4u
2
+ 5u 3
a
7
=
1
5
u
5
6
5
u
4
+ ···
7
5
u +
6
5
u
5
4u
4
+ 6u
3
3u
2
+ 1
a
11
=
1
5
u
5
+
6
5
u
4
+ ···
23
5
u +
9
5
1
a
1
=
2
5
u
5
+
12
5
u
4
+ ···
31
5
u +
8
5
u 2
a
4
=
1
0
a
3
=
1
5
u
5
+
6
5
u
4
+ ···
18
5
u +
9
5
1
a
2
=
3
5
u
5
+
18
5
u
4
+ ···
49
5
u +
17
5
u 3
a
5
=
1
5
u
5
6
5
u
4
+ ··· +
18
5
u
4
5
1
a
8
=
2
5
u
5
12
5
u
4
+ ··· +
26
5
u
13
5
u
5
+ 5u
4
10u
3
+ 10u
2
6u + 2
a
8
=
2
5
u
5
12
5
u
4
+ ··· +
26
5
u
13
5
u
5
+ 5u
4
10u
3
+ 10u
2
6u + 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 = 0.255138 0.877439I
a = 0.17339 1.43930I
b = 1.00000I
0.26574 + 2.82812I 3.50976 2.97945I
u = 0.255138 + 0.877439I
a = 0.17339 + 1.43930I
b = 1.00000I
0.26574 2.82812I 3.50976 + 2.97945I
u = 1.000000 0.754878I
a = 0.274015 + 0.362993I
b = 1.00000I
4.40332 3.01951
u = 1.000000 + 0.754878I
a = 0.274015 0.362993I
b = 1.00000I
4.40332 3.01951
u = 1.74486 0.87744I
a = 0.647408 + 0.202297I
b = 1.00000I
0.26574 2.82812I 3.50976 + 2.97945I
u = 1.74486 + 0.87744I
a = 0.647408 0.202297I
b = 1.00000I
0.26574 + 2.82812I 3.50976 2.97945I
3
II. I
u
2
=
hu
16
+2u
15
+· · ·10u+1, 1.73×10
15
u
15
4.00×10
15
u
14
+· · ·+6.07×10
14
b+
6.17 × 10
15
, 2.43 × 10
15
u
15
+ 5.01 × 10
15
u
14
+ · · · + 6.07 × 10
14
a 1.68 × 10
16
i
(i) Arc colorings
a
6
=
0
u
a
9
=
4.00952u
15
8.26793u
14
+ ··· 126.612u + 27.7465
2.85841u
15
+ 6.60025u
14
+ ··· + 65.1455u 10.1759
a
10
=
4.00952u
15
8.26793u
14
+ ··· 126.612u + 27.7465
2.73115u
15
+ 6.27741u
14
+ ··· + 63.6248u 10.4247
a
7
=
8.99316u
15
20.3369u
14
+ ··· 210.155u + 35.1290
1.28618u
15
+ 2.77910u
14
+ ··· + 35.1160u 4.08796
a
11
=
1.97679u
15
4.31360u
14
+ ··· 49.5922u + 15.1411
3.61529u
15
+ 7.99868u
14
+ ··· + 90.9157u 16.5704
a
1
=
12.3027u
15
27.3952u
14
+ ··· 316.275u + 59.4225
3.71417u
15
+ 8.49770u
14
+ ··· + 84.1430u 11.9117
a
4
=
1
0
a
3
=
1.73738u
15
+ 4.18410u
14
+ ··· + 34.5508u + 3.22948
3.61529u
15
+ 7.99868u
14
+ ··· + 90.9157u 16.5704
a
2
=
10.4247u
15
23.5806u
14
+ ··· 259.910u + 40.6225
0.0988785u
15
+ 0.499021u
14
+ ··· 6.77273u + 4.65877
a
5
=
1.87792u
15
+ 3.81458u
14
+ ··· + 56.3649u 17.7999
3.61529u
15
7.99868u
14
+ ··· 90.9157u + 16.5704
a
8
=
6.86793u
15
14.8682u
14
+ ··· 191.758u + 37.9223
2.85841u
15
+ 6.60025u
14
+ ··· + 65.1455u 10.1759
a
8
=
6.86793u
15
14.8682u
14
+ ··· 191.758u + 37.9223
2.85841u
15
+ 6.60025u
14
+ ··· + 65.1455u 10.1759
(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 = 3.13152 0.31808I
a = 0.473846 + 0.041423I
b = 0.82217 + 1.15830I
8.6109 + 12.3641I 9.35094 7.14528I
u = 3.13152 + 0.31808I
a = 0.473846 0.041423I
b = 0.82217 1.15830I
8.6109 12.3641I 9.35094 + 7.14528I
u = 1.97730
a = 0.724290
b = 0.888414
6.54271 14.4518
u = 1.38608 0.45928I
a = 0.416903 0.397672I
b = 0.797243 1.086111I
2.59863 + 8.63192I 5.90792 7.27043I
u = 1.38608 + 0.45928I
a = 0.416903 + 0.397672I
b = 0.797243 + 1.086111I
2.59863 8.63192I 5.90792 + 7.27043I
u = 0.239470
a = 1.67118
b = 0.423356
0.656537 15.0457
u = 0.216557 0.046304I
a = 2.13510 + 3.15314I
b = 0.826528 0.979522I
3.36394 4.13872I 7.73528 + 1.97260I
u = 0.216557 + 0.046304I
a = 2.13510 3.15314I
b = 0.826528 + 0.979522I
3.36394 + 4.13872I 7.73528 1.97260I
u = 0.467500 0.213512I
a = 2.03947 1.48957I
b = 0.232716 0.644221I
1.24387 + 2.05073I 9.21244 + 1.11358I
u = 0.467500 + 0.213512I
a = 2.03947 + 1.48957I
b = 0.232716 + 0.644221I
1.24387 2.05073I 9.21244 1.11358I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.857887 0.159297I
a = 0.163729 0.801451I
b = 0.379775 0.677130I
2.70658 + 1.45405I 3.73411 4.71917I
u = 0.857887 + 0.159297I
a = 0.163729 + 0.801451I
b = 0.379775 + 0.677130I
2.70658 1.45405I 3.73411 + 4.71917I
u = 1.231701 0.623882I
a = 0.690658 + 0.586962I
b = 0.494247 + 0.784033I
0.99780 + 5.12268I 7.85223 7.82309I
u = 1.231701 + 0.623882I
a = 0.690658 0.586962I
b = 0.494247 0.784033I
0.99780 5.12268I 7.85223 + 7.82309I
u = 1.85234 1.37345I
a = 0.480404 0.449368I
b = 0.966111 + 0.941274I
10.25569 1.47993I 11.45831 + 1.74331I
u = 1.85234 + 1.37345I
a = 0.480404 + 0.449368I
b = 0.966111 0.941274I
10.25569 + 1.47993I 11.45831 1.74331I
6
III. I
u
3
= hu
18
u
17
+ · · · 3598u + 589, 1.09 × 10
32
u
17
1.03 ×
10
31
u
16
+ · · · + 2.97 × 10
32
b + 5.87 × 10
34
, 2.32 × 10
34
u
17
1.62 ×
10
33
u
16
+ · · · + 1.75 × 10
35
a + 1.34 × 10
37
i
(i) Arc colorings
a
6
=
0
u
a
9
=
0.132611u
17
+ 0.00928906u
16
+ ··· + 387.873u 76.4899
0.368238u
17
+ 0.0346903u
16
+ ··· + 1025.48u 197.805
a
10
=
0.132611u
17
+ 0.00928906u
16
+ ··· + 387.873u 76.4899
0.213184u
17
+ 0.0191754u
16
+ ··· + 593.028u 114.226
a
7
=
0.0565071u
17
+ 0.00633907u
16
+ ··· + 149.592u 29.2688
0.0105699u
17
+ 0.00149785u
16
+ ··· + 30.8052u 4.95031
a
11
=
0.137014u
17
+ 0.00852612u
16
+ ··· + 403.219u 81.0893
0.109001u
17
+ 0.0112857u
16
+ ··· + 295.446u 55.7058
a
1
=
0.134708u
17
+ 0.00583225u
16
+ ··· + 403.872u 78.2342
1
a
4
=
1
0
a
3
=
0.0712508u
17
+ 0.00713066u
16
+ ··· + 190.044u 32.7172
0.0200358u
17
0.00511484u
16
+ ··· 42.0777u + 7.20075
a
2
=
0.119299u
17
+ 0.00948594u
16
+ ··· + 339.894u 67.3015
0.0889654u
17
+ 0.00617084u
16
+ ··· + 253.368u 48.5050
a
5
=
0.0912866u
17
0.0122455u
16
+ ··· 232.122u + 41.9180
0.0200358u
17
+ 0.00511484u
16
+ ··· + 42.0777u 7.20075
a
8
=
0.235627u
17
0.0254012u
16
+ ··· 637.603u + 121.316
0.368238u
17
+ 0.0346903u
16
+ ··· + 1025.48u 197.805
a
8
=
0.235627u
17
0.0254012u
16
+ ··· 637.603u + 121.316
0.368238u
17
+ 0.0346903u
16
+ ··· + 1025.48u 197.805
(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 = 2.93425 0.48080I
a = 0.507686 + 0.003340I
b = 0.945590 0.965095I
10.17134 + 5.50049I 11.48937 2.97298I
u = 2.93425 + 0.48080I
a = 0.507686 0.003340I
b = 0.945590 + 0.965095I
10.17134 5.50049I 11.48937 + 2.97298I
u = 1.99207 1.47309I
a = 0.486905 0.278515I
b = 0.505863 + 0.476260I
1.87293 + 3.41073I 9.88238 4.39642I
u = 1.99207 + 1.47309I
a = 0.486905 + 0.278515I
b = 0.505863 0.476260I
1.87293 3.41073I 9.88238 + 4.39642I
u = 1.096551 0.344363I
a = 0.574173 + 0.427943I
b = 0.881705 + 0.851729I
3.77376 + 2.21388I 8.24115 3.04598I
u = 1.096551 + 0.344363I
a = 0.574173 0.427943I
b = 0.881705 0.851729I
3.77376 2.21388I 8.24115 + 3.04598I
u = 0.047733 1.069326I
a = 0.435880 + 0.196481I
b = 0.257033 0.703723I
3.25448 + 1.10969I 4.55374 6.23947I
u = 0.047733 + 1.069326I
a = 0.435880 0.196481I
b = 0.257033 + 0.703723I
3.25448 1.10969I 4.55374 + 6.23947I
u = 0.318103 0.220761I
a = 0.04108 2.12527I
b = 0.937576 + 0.708026I
3.77376 + 2.21388I 8.24115 3.04598I
u = 0.318103 + 0.220761I
a = 0.04108 + 2.12527I
b = 0.937576 0.708026I
3.77376 2.21388I 8.24115 + 3.04598I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.088453 0.004751I
a = 0.213971 0.418670I
b = 0.033137 1.191065I
3.25448 1.10969I 4.55374 + 6.23947I
u = 1.088453 + 0.004751I
a = 0.213971 + 0.418670I
b = 0.033137 + 1.191065I
3.25448 + 1.10969I 4.55374 6.23947I
u = 1.258469 0.377137I
a = 1.041864 0.183155I
b = 0.076831 + 1.264199I
1.87293 3.41073I 9.88238 + 4.39642I
u = 1.258469 + 0.377137I
a = 1.041864 + 0.183155I
b = 0.076831 1.264199I
1.87293 + 3.41073I 9.88238 4.39642I
u = 1.76892 1.40654I
a = 0.451971 + 0.491832I
b = 1.067290 0.668745I
10.17134 + 5.50049I 11.48937 2.97298I
u = 1.76892 + 1.40654I
a = 0.451971 0.491832I
b = 1.067290 + 0.668745I
10.17134 5.50049I 11.48937 + 2.97298I
u = 2.13666 0.11985I
a = 0.597448 + 0.033512I
b = 0.295309 + 1.123219I
0.453072 5.66670
u = 2.13666 + 0.11985I
a = 0.597448 0.033512I
b = 0.295309 1.123219I
0.453072 5.66670
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
3
(u
16
+ 2u
14
+ ··· 2u 1)(u
18
+ u
17
+ ··· + 8u + 5)
c
2
(u + 1)
6
(u
16
+ 4u
15
+ ··· 14u
2
+ 1)(u
18
+ 7u
17
+ ··· + 136u + 25)
c
5
(u
6
+ u
4
+ 2u
2
+ 1)
(u
9
3u
8
+ 2u
7
+ 5u
6
u
5
13u
4
+ 10u
3
+ 2u
2
+ u 3)
2
(u
16
+ 9u
15
+ ··· + 31u + 22)
c
6
, c
9
, c
10
(u
6
3u
4
+ 2u
2
+ 1)
(u
9
u
8
4u
7
+ 3u
6
+ 5u
5
u
4
2u
3
2u
2
+ u 1)
2
(u
16
+ 3u
15
+ ··· 3u 2)
c
11
(u
3
u
2
+ 1)
2
(1 + u + 2u
2
+ 6u
3
+ 7u
4
+ 11u
5
+ 5u
6
+ 6u
7
+ u
8
+ u
9
)
2
(u
16
+ 3u
15
+ ··· + 41u 8)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
7
, c
8
(y + 1)
6
(y
16
+ 4y
15
+ ··· 14y
2
+ 1)(y
18
+ 7y
17
+ ··· + 136y + 25)
c
2
(y 1)
6
(y
16
+ 24y
15
+ ··· 28y + 1)(y
18
+ 7y
17
+ ··· + 5004y + 625)
c
5
(y
3
+ y
2
+ 2y + 1)
2
(9 + 13y 62y
2
+ 180y
3
223y
4
+ 185y
5
87y
6
+ 32y
7
5y
8
+ y
9
)
2
(y
16
3y
15
+ ··· 3557y + 484)
c
6
, c
9
, c
10
(y
3
3y
2
+ 2y + 1)
2
(y
9
9y
8
+ 32y
7
55y
6
+ 45y
5
19y
4
+ 16y
3
10y
2
3y 1)
2
(y
16
15y
15
+ ··· 21y + 4)
c
11
(y
3
y
2
+ 2y 1)
2
(y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 121y
5
+ 73y
4
+ 20y
3
6y
2
3y 1)
2
(y
16
+ 9y
15
+ ··· 6561y + 64)
11