11n
172
(K11n
172
)
1
Arc Sequences
6 9 1 10 2 4 11 2 7 5 9
Solving Sequence
1,6 2,9
3 4 5 8 11 7 10
c
1
c
2
c
3
c
5
c
8
c
11
c
7
c
10
c
4
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
16
u
15
+ ··· 2u 1, 61u
15
11u
14
+ ··· + 44b 137, 211u
15
99u
14
+ ··· + 44a 229i
I
u
2
= hu
40
+ 2u
39
+ ··· 14u 1,
2.48987 × 10
61
u
39
4.82938 × 10
61
u
38
+ ··· + 1.01371 × 10
62
b + 7.79734 × 10
61
,
2.85349 × 10
62
u
39
+ 5.48698 × 10
62
u
38
+ ··· + 1.01371 × 10
62
a 2.27568 × 10
63
i
There are 2 irreducible components with 56 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
16
u
15
+ · · · 2u 1, 61u
15
11u
14
+ · · · + 44b 137, 211u
15
99u
14
+ · · · + 44a 229i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
9
=
4.79545u
15
+ 2.25000u
14
+ ··· + 20.3636u + 5.20455
1.38636u
15
+ 0.250000u
14
+ ··· + 5.59091u + 3.11364
a
3
=
1.43182u
15
+ 0.500000u
14
+ ··· 13.2045u 2.81818
1.13636u
15
+ 2.25000u
14
+ ··· + 0.340909u 0.386364
a
4
=
2.56818u
15
1.75000u
14
+ ··· 13.5455u 2.43182
1.13636u
15
+ 2.25000u
14
+ ··· + 0.340909u 0.386364
a
5
=
u
u
3
+ u
a
8
=
6.61364u
15
+ 4.50000u
14
+ ··· + 24.6591u + 4.63636
2.31818u
15
+ 1.50000u
14
+ ··· + 6.54545u + 2.68182
a
11
=
3
4
u
15
+ 3u
14
+ ···
21
4
u
7
2
1.13636u
15
1.50000u
14
+ ··· 4.09091u 1.86364
a
7
=
6.31818u
15
+ 8u
14
+ ··· + 13.0455u 2.81818
1.09091u
15
+ 0.500000u
14
+ ··· + 1.22727u 0.590909
a
10
=
0.863636u
15
+ 3u
14
+ ··· 2.59091u 2.36364
1.61364u
15
1.75000u
14
+ ··· 6.40909u 2.88636
a
10
=
0.863636u
15
+ 3u
14
+ ··· 2.59091u 2.36364
1.61364u
15
1.75000u
14
+ ··· 6.40909u 2.88636
(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 = 1.08888
a = 1.65520
b = 1.14875
3.38602 19.6054
u = 0.379929 0.431781I
a = 1.21215 0.78064I
b = 0.10398 + 1.46365I
5.56667 + 2.95635I 4.74772 2.75237I
u = 0.379929 + 0.431781I
a = 1.21215 + 0.78064I
b = 0.10398 1.46365I
5.56667 2.95635I 4.74772 + 2.75237I
u = 0.299377 1.065241I
a = 0.498460 0.668912I
b = 0.795627 0.500550I
0.62726 + 1.86405I 2.38240 3.76150I
u = 0.299377 + 1.065241I
a = 0.498460 + 0.668912I
b = 0.795627 + 0.500550I
0.62726 1.86405I 2.38240 + 3.76150I
u = 0.190166 0.829536I
a = 1.53060 + 0.24921I
b = 0.602481 + 0.142061I
0.363930 + 0.209601I 0.372974 + 0.607008I
u = 0.190166 + 0.829536I
a = 1.53060 0.24921I
b = 0.602481 0.142061I
0.363930 0.209601I 0.372974 0.607008I
u = 0.050757 0.668728I
a = 2.51851 1.76824I
b = 0.486835 0.436096I
1.13425 + 5.09162I 5.21325 6.81011I
u = 0.050757 + 0.668728I
a = 2.51851 + 1.76824I
b = 0.486835 + 0.436096I
1.13425 5.09162I 5.21325 + 6.81011I
u = 0.19518 1.54614I
a = 0.24009 1.67107I
b = 0.13897 1.81400I
12.18101 1.94801I 0.634861 0.077298I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.19518 + 1.54614I
a = 0.24009 + 1.67107I
b = 0.13897 + 1.81400I
12.18101 + 1.94801I 0.634861 + 0.077298I
u = 0.361947 1.340667I
a = 0.190390 + 0.008678I
b = 0.736073 + 0.114852I
4.61886 6.15067I 1.67402 + 4.95826I
u = 0.361947 + 1.340667I
a = 0.190390 0.008678I
b = 0.736073 0.114852I
4.61886 + 6.15067I 1.67402 4.95826I
u = 0.597072 0.748618I
a = 0.419496 0.041748I
b = 0.087841 + 1.127950I
8.25349 1.91163I 5.00806 + 3.84199I
u = 0.597072 + 0.748618I
a = 0.419496 + 0.041748I
b = 0.087841 1.127950I
8.25349 + 1.91163I 5.00806 3.84199I
u = 1.62094
a = 0.996790
b = 1.18892
1.43155 5.25645
4
II. I
u
2
=
hu
40
+2u
39
+· · ·14u1, 2.49×10
61
u
39
4.83×10
61
u
38
+· · ·+1.01×10
62
b+
7.80 × 10
61
, 2.85 × 10
62
u
39
+ 5.49 × 10
62
u
38
+ · · · + 1.01 × 10
62
a 2.28 × 10
63
i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
9
=
2.81489u
39
5.41275u
38
+ ··· + 216.961u + 22.4489
0.245619u
39
+ 0.476405u
38
+ ··· 11.7481u 0.769185
a
3
=
2.24338u
39
+ 4.44433u
38
+ ··· 166.087u 20.5892
0.0737940u
39
0.141157u
38
+ ··· + 8.96434u + 0.546333
a
4
=
2.31718u
39
+ 4.58549u
38
+ ··· 175.051u 21.1355
0.0737940u
39
0.141157u
38
+ ··· + 8.96434u + 0.546333
a
5
=
u
u
3
+ u
a
8
=
3.06419u
39
5.89858u
38
+ ··· + 228.486u + 23.0011
0.257433u
39
+ 0.501984u
38
+ ··· 11.6776u 0.781961
a
11
=
1.64745u
39
3.29893u
38
+ ··· + 121.757u + 19.2119
0.0580217u
39
+ 0.116218u
38
+ ··· 7.26051u 0.542306
a
7
=
3.53017u
39
6.64205u
38
+ ··· + 246.267u + 13.8065
0.0482060u
39
+ 0.0516457u
38
+ ··· 8.18351u + 0.0416260
a
10
=
1.62591u
39
3.25910u
38
+ ··· + 120.037u + 19.2014
0.0334484u
39
+ 0.0685099u
38
+ ··· 5.51644u 0.528595
a
10
=
1.62591u
39
3.25910u
38
+ ··· + 120.037u + 19.2014
0.0334484u
39
+ 0.0685099u
38
+ ··· 5.51644u 0.528595
(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 = 1.75447 0.27463I
a = 0.0220510 + 0.0395348I
b = 0.10898 + 2.12585I
10.09044 + 5.18311I 2.70240 3.54435I
u = 1.75447 + 0.27463I
a = 0.0220510 0.0395348I
b = 0.10898 2.12585I
10.09044 5.18311I 2.70240 + 3.54435I
u = 1.21545
a = 1.31118
b = 0.649335
2.31916 3.45588
u = 0.84317 1.82666I
a = 0.593658 0.965993I
b = 0.61921 2.17342I
14.8294 + 4.4956I 3.27609 2.70545I
u = 0.84317 + 1.82666I
a = 0.593658 + 0.965993I
b = 0.61921 + 2.17342I
14.8294 4.4956I 3.27609 + 2.70545I
u = 0.63008 1.69717I
a = 0.609510 + 1.233262I
b = 0.55981 + 2.10243I
16.3560 + 13.3915I 2.10100 5.83549I
u = 0.63008 + 1.69717I
a = 0.609510 1.233262I
b = 0.55981 2.10243I
16.3560 13.3915I 2.10100 + 5.83549I
u = 0.381500 0.672173I
a = 0.286606 0.290964I
b = 0.248699 0.413490I
0.03644 + 1.50292I 1.25089 6.14683I
u = 0.381500 + 0.672173I
a = 0.286606 + 0.290964I
b = 0.248699 + 0.413490I
0.03644 1.50292I 1.25089 + 6.14683I
u = 0.325667 1.141167I
a = 0.798264 + 0.284615I
b = 1.025039 + 0.573847I
1.21242 + 1.07873I 3.63948 + 1.17826I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.325667 + 1.141167I
a = 0.798264 0.284615I
b = 1.025039 0.573847I
1.21242 1.07873I 3.63948 1.17826I
u = 0.246524 1.044101I
a = 0.211827 0.946805I
b = 0.266752 0.501923I
1.63884 + 2.52981I 3.46896 4.78238I
u = 0.246524 + 1.044101I
a = 0.211827 + 0.946805I
b = 0.266752 + 0.501923I
1.63884 2.52981I 3.46896 + 4.78238I
u = 0.197863 0.060558I
a = 2.88076 3.04382I
b = 0.577307 0.064688I
1.31470 + 0.60771I 8.49421 3.04989I
u = 0.197863 + 0.060558I
a = 2.88076 + 3.04382I
b = 0.577307 + 0.064688I
1.31470 0.60771I 8.49421 + 3.04989I
u = 0.10749 1.61398I
a = 0.339161 0.635536I
b = 0.942745 0.706252I
6.22320 3.43752I 2.13285 + 2.29436I
u = 0.10749 + 1.61398I
a = 0.339161 + 0.635536I
b = 0.942745 + 0.706252I
6.22320 + 3.43752I 2.13285 2.29436I
u = 0.0965200 0.0759702I
a = 2.97374 13.61605I
b = 0.588296 + 0.344976I
0.55843 + 5.04023I 8.92316 6.11668I
u = 0.0965200 + 0.0759702I
a = 2.97374 + 13.61605I
b = 0.588296 0.344976I
0.55843 5.04023I 8.92316 + 6.11668I
u = 0.05196 1.65626I
a = 0.220045 + 0.567547I
b = 1.53393 + 0.95218I
6.89381 + 4.31374I 3.00921 2.52408I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.05196 + 1.65626I
a = 0.220045 0.567547I
b = 1.53393 0.95218I
6.89381 4.31374I 3.00921 + 2.52408I
u = 0.13173 1.46303I
a = 0.06034 1.93217I
b = 0.14005 1.63978I
12.97916 2.42998I 9.06445 + 4.17297I
u = 0.13173 + 1.46303I
a = 0.06034 + 1.93217I
b = 0.14005 + 1.63978I
12.97916 + 2.42998I 9.06445 4.17297I
u = 0.14644 1.58425I
a = 0.29330 + 1.55586I
b = 0.49563 + 2.13728I
14.3940 1.3454I 5.37205 + 0.15160I
u = 0.14644 + 1.58425I
a = 0.29330 1.55586I
b = 0.49563 2.13728I
14.3940 + 1.3454I 5.37205 0.15160I
u = 0.27705 1.55024I
a = 0.36039 + 1.64659I
b = 0.46245 + 2.16251I
10.55226 6.76566I 1.06220 + 4.68199I
u = 0.27705 + 1.55024I
a = 0.36039 1.64659I
b = 0.46245 2.16251I
10.55226 + 6.76566I 1.06220 4.68199I
u = 0.28024 1.51688I
a = 0.47715 1.61323I
b = 0.31655 1.86696I
9.55118 0.96633I 0.232502 0.266577I
u = 0.28024 + 1.51688I
a = 0.47715 + 1.61323I
b = 0.31655 + 1.86696I
9.55118 + 0.96633I 0.232502 + 0.266577I
u = 0.345255 0.125454I
a = 1.73592 + 1.36825I
b = 0.178266 + 1.332875I
8.09181 + 0.67197I 3.63743 + 1.66692I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.345255 + 0.125454I
a = 1.73592 1.36825I
b = 0.178266 1.332875I
8.09181 0.67197I 3.63743 1.66692I
u = 0.440412 1.198771I
a = 0.196802 0.687503I
b = 0.485884 0.400355I
6.21063 2.44288I 1.57019 + 3.53786I
u = 0.440412 + 1.198771I
a = 0.196802 + 0.687503I
b = 0.485884 + 0.400355I
6.21063 + 2.44288I 1.57019 3.53786I
u = 0.458127 1.196180I
a = 0.644252 + 0.048307I
b = 0.149211 + 0.533355I
5.71740 6.20191I 6.05433 + 5.08780I
u = 0.458127 + 1.196180I
a = 0.644252 0.048307I
b = 0.149211 0.533355I
5.71740 + 6.20191I 6.05433 5.08780I
u = 0.742397 0.144685I
a = 0.110723 0.345539I
b = 0.00282 + 1.58047I
4.59373 2.98825I 4.81250 + 3.01552I
u = 0.742397 + 0.144685I
a = 0.110723 + 0.345539I
b = 0.00282 1.58047I
4.59373 + 2.98825I 4.81250 3.01552I
u = 0.768195 0.090718I
a = 0.230836 + 0.267949I
b = 0.389053 0.579808I
2.47058 + 1.70720I 0.230221 0.591796I
u = 0.768195 + 0.090718I
a = 0.230836 0.267949I
b = 0.389053 + 0.579808I
2.47058 1.70720I 0.230221 + 0.591796I
u = 1.09842
a = 1.59864
b = 1.32148
3.09539 9.16156
9
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
16
u
15
+ ··· 2u 1)(u
40
+ 2u
39
+ ··· 14u 1)
c
2
(u
16
+ 5u
14
+ ··· + 10u
2
4)(u
40
+ u
39
+ ··· 492u 892)
c
3
(u
16
+ 8u
15
+ ··· + 12u + 8)(u
40
+ 5u
39
+ ··· + 248u + 88)
c
4
(u
16
6u
14
+ ··· 14u
2
+ 4)(u
40
+ u
39
+ ··· + 12u + 4)
c
5
(u
16
+ u
15
+ ··· + 2u 1)(u
40
+ 2u
39
+ ··· 14u 1)
c
6
(u
16
+ 2u
15
+ ··· + 11u + 1)(u
40
+ 5u
39
+ ··· 457u + 29)
c
7
(u
16
2u
15
+ ··· + 13u + 1)(u
40
+ 3u
39
+ ··· + 26575u + 7349)
c
8
(u
16
+ 5u
14
+ ··· + 10u
2
4)(u
40
+ u
39
+ ··· 492u 892)
c
9
(u
16
6u
15
+ ··· 2u + 1)(u
40
+ 5u
39
+ ··· + 96u + 11)
c
10
(u
16
6u
14
+ ··· 14u
2
+ 4)(u
40
+ u
39
+ ··· + 12u + 4)
c
11
(u
16
+ 4u
14
+ ··· + 4u
2
1)(u
40
+ u
39
+ ··· 1168u 424)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
16
+ 9y
15
+ ··· + 10y + 1)(y
40
+ 38y
39
+ ··· 78y + 1)
c
2
, c
8
(y
16
+ 10y
15
+ ··· 80y + 16)
(y
40
+ 63y
39
+ ··· + 6026912y + 795664)
c
3
(y
16
20y
15
+ ··· 1168y + 64)
(y
40
59y
39
+ ··· 199840y + 7744)
c
4
, c
10
(y
16
12y
15
+ ··· 112y + 16)(y
40
35y
39
+ ··· + 1888y + 16)
c
6
(y
16
+ 4y
14
+ ··· 91y + 1)(y
40
15y
39
+ ··· 262383y + 841)
c
7
(y
16
+ 10y
15
+ ··· 69y + 1)
(y
40
+ 47y
39
+ ··· + 923130863y + 54007801)
c
9
(y
16
2y
15
+ ··· 2y + 1)(y
40
+ 7y
39
+ ··· 218y + 121)
c
11
(y
16
+ 8y
15
+ ··· 8y + 1)(y
40
+ 57y
39
+ ··· + 1150944y + 179776)
11