10
156
(K10n
32
)
1
Arc Sequences
7 5 8 10 8 1 4 5 3 2
Solving Sequence
4,7 2,8
1 3 6 10 5 9
c
7
c
1
c
3
c
6
c
10
c
4
c
8
c
2
, c
5
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
12
+ 3u
11
+ 4u
10
+ 5u
9
2u
8
+ 7u
7
+ 19u
6
3u
5
+ 12u
4
+ 12u
3
+ 10u + 11,
358u
11
1274u
10
+ ··· + 57905b 4904,
140508u
11
870808u
10
+ ··· + 10828235a 10200130i
I
u
2
= hu
5
2u
3
+ 3u
2
2u + 1, u
4
3u
2
+ a + 2u, u
4
+ 3u
2
+ b 2u + 1i
I
u
3
= hu
10
+ 2u
9
+ 6u
8
+ 7u
7
+ 21u
6
+ 22u
5
+ 34u
4
+ 17u
3
+ 13u
2
+ u + 1,
53u
9
119u
8
292u
7
328u
6
926u
5
1062u
4
1154u
3
110u
2
+ 225a + 201u + 493,
53u
9
+ 119u
8
+ 292u
7
+ 328u
6
+ 926u
5
+ 1062u
4
+ 1154u
3
+ 110u
2
+ 225b 201u 268i
There are 3 irreducible components with 27 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
+ 3u
11
+ · · · + 10u + 11, 358u
11
1274u
10
+ · · · + 57905b
4904, 1.41 × 10
5
u
11
8.71 × 10
5
u
10
+ · · · + 1.08 × 10
7
a 1.02 × 10
7
i
(i) Arc colorings
a
4
=
0
u
a
7
=
0.0129761u
11
+ 0.0804201u
10
+ ··· + 0.219219u + 0.941994
0.00618254u
11
+ 0.0220016u
10
+ ··· 0.370210u + 0.0846904
a
2
=
1
0
a
8
=
0.0129761u
11
+ 0.0804201u
10
+ ··· + 0.219219u + 0.941994
0.0839692u
11
0.240256u
10
+ ··· + 0.187446u + 0.541101
a
1
=
0.0243656u
11
0.0950617u
10
+ ··· + 0.655784u + 0.482490
0.0672135u
11
+ 0.159693u
10
+ ··· 0.558501u + 0.227891
a
3
=
0.0100016u
11
0.144111u
10
+ ··· 0.373939u 0.0138264
0.00320809u
11
0.0416890u
10
+ ··· 0.524930u 0.987142
a
6
=
0.00116889u
11
+ 0.00671476u
10
+ ··· 0.569594u + 0.536619
0.0610310u
11
+ 0.181694u
10
+ ··· 0.928711u + 1.31258
a
10
=
0.0428480u
11
+ 0.0646309u
10
+ ··· + 0.0972831u + 0.710381
0.0672135u
11
+ 0.159693u
10
+ ··· 0.558501u + 0.227891
a
5
=
0.0592845u
11
+ 0.206820u
10
+ ··· 0.946361u 0.290927
0.00320809u
11
0.0416890u
10
+ ··· 0.524930u + 0.0128578
a
9
=
0.0674429u
11
+ 0.00229844u
10
+ ··· + 0.633027u + 0.0160250
0.126446u
11
0.278249u
10
+ ··· + 1.18712u + 0.939466
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1432
57905
u
11
+
5096
57905
u
10
+
2256
57905
u
9
34896
57905
u
8
55008
57905
u
7
161656
57905
u
6
+
68516
57905
u
5
+
99424
57905
u
4
257532
57905
u
3
+
90136
57905
u
2
85748
57905
u +
135426
57905
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 2.28602
a = 0.109456
b = 0.754878
2.83439 1.01951
u = 1.47545
a = 0.794497
b = 0.754878
2.83439 1.01951
u = 0.729624 0.294565I
a = 0.533817 0.434611I
b = 0.877439 + 0.744862I
0.92371 2.82812I 5.50976 + 2.97945I
u = 0.729624 + 0.294565I
a = 0.533817 + 0.434611I
b = 0.877439 0.744862I
0.92371 + 2.82812I 5.50976 2.97945I
u = 0.62468 1.86786I
a = 0.403492 0.628311I
b = 0.877439 + 0.744862I
6.97197 2.82812I 5.50976 + 2.97945I
u = 0.62468 + 1.86786I
a = 0.403492 + 0.628311I
b = 0.877439 0.744862I
6.97197 + 2.82812I 5.50976 2.97945I
u = 0.078364 0.958073I
a = 1.11218 + 1.23414I
b = 0.877439 0.744862I
6.97197 + 2.82812I 5.50976 2.97945I
u = 0.078364 + 0.958073I
a = 1.11218 1.23414I
b = 0.877439 + 0.744862I
6.97197 2.82812I 5.50976 + 2.97945I
u = 0.718377 0.787894I
a = 0.846985 0.995749I
b = 0.754878
5.06130 1.01951
u = 0.718377 + 0.787894I
a = 0.846985 + 0.995749I
b = 0.754878
5.06130 1.01951
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.938298 0.642073I
a = 0.201517 + 0.893456I
b = 0.877439 0.744862I
0.92371 + 2.82812I 5.50976 2.97945I
u = 0.938298 + 0.642073I
a = 0.201517 0.893456I
b = 0.877439 + 0.744862I
0.92371 2.82812I 5.50976 + 2.97945I
4
II. I
u
2
= hu
5
2u
3
+ 3u
2
2u + 1, u
4
3u
2
+ a + 2u, u
4
+ 3u
2
+ b 2u + 1i
(i) Arc colorings
a
4
=
0
u
a
7
=
u
4
+ 3u
2
2u
u
4
3u
2
+ 2u 1
a
2
=
1
0
a
8
=
u
4
+ 3u
2
2u
u
3
u
2
+ u 1
a
1
=
2u
4
+ u
3
4u
2
+ 4u
u
4
u
3
+ u
2
2u
a
3
=
u
4
+ u
3
+ 3u
2
5u + 3
u 1
a
6
=
u
3
u
2
+ 2u 1
2u
4
+ 2u
3
3u
2
+ 2u 1
a
10
=
u
4
3u
2
+ 2u
u
4
u
3
+ u
2
2u
a
5
=
u
4
u
3
3u
2
+ 5u 3
u
4
+ u
3
2u
2
+ u
a
9
=
0
u
3
u
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7u
4
+ 4u
3
9u
2
+ 14u 6
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 2.02298
a = 0.424848
b = 0.575152
3.55538 12.9676
u = 0.193660 0.705367I
a = 1.90443 + 0.33976I
b = 0.904429 0.339760I
4.86920 + 1.42206I 0.68335 4.57040I
u = 0.193660 + 0.705367I
a = 1.90443 0.33976I
b = 0.904429 + 0.339760I
4.86920 1.42206I 0.68335 + 4.57040I
u = 0.817831 0.505011I
a = 0.116853 0.784420I
b = 1.116853 + 0.784420I
1.84330 + 3.45949I 2.16713 7.95950I
u = 0.817831 + 0.505011I
a = 0.116853 + 0.784420I
b = 1.116853 0.784420I
1.84330 3.45949I 2.16713 + 7.95950I
6
III. I
u
3
= hu
10
+ 2u
9
+ · · · + u + 1, 53u
9
119u
8
+ · · · + 225a +
493, 53u
9
+ 119u
8
+ · · · + 225b 268i
(i) Arc colorings
a
4
=
0
u
a
7
=
0.235556u
9
+ 0.528889u
8
+ ··· 0.893333u 2.19111
0.235556u
9
0.528889u
8
+ ··· + 0.893333u + 1.19111
a
2
=
1
0
a
8
=
0.235556u
9
+ 0.528889u
8
+ ··· 0.893333u 2.19111
0.00444444u
9
+ 0.00888889u
8
+ ··· + 1.18667u + 1.24889
a
1
=
0.0977778u
9
+ 0.128889u
8
+ ··· + 4.57333u + 4.40889
0.137778u
9
0.657778u
8
+ ··· 3.68000u 2.21778
a
3
=
0.266667u
9
0.733333u
8
+ ··· 5.93333u 0.333333
0.457778u
9
+ 0.617778u
8
+ ··· + 4.04000u 0.102222
a
6
=
0.368889u
9
+ 1.39556u
8
+ ··· + 10.5733u + 3.47556
0.466667u
9
1.26667u
8
+ ··· 6u 1.06667
a
10
=
0.235556u
9
0.528889u
8
+ ··· + 0.893333u + 2.19111
0.137778u
9
0.657778u
8
+ ··· 3.68000u 2.21778
a
5
=
4
15
u
9
+
11
15
u
8
+ ··· +
89
15
u +
1
3
0.657778u
9
1.15111u
8
+ ··· 3.10667u + 0.368889
a
9
=
6
5
u
9
+
7
3
u
8
+ ··· +
82
15
u
37
15
0.0488889u
9
0.0977778u
8
+ ··· 0.0533333u + 2.26222
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
122
75
u
9
+
136
75
u
8
+
553
75
u
7
+
287
75
u
6
+
1994
75
u
5
+
668
75
u
4
+
807
25
u
3
38
5
u
2
+
541
75
u
152
75
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.20055 1.54735I
a = 0.143866 + 0.796701I
b = 1.143866 0.796701I
6.19490 11.16339I 4.37125 + 6.32339I
u = 1.20055 + 1.54735I
a = 0.143866 0.796701I
b = 1.143866 + 0.796701I
6.19490 + 11.16339I 4.37125 6.32339I
u = 0.443805 1.105327I
a = 0.359398 1.080444I
b = 0.640602 + 1.080444I
7.82103 4.41044I 6.40190 + 3.03613I
u = 0.443805 + 1.105327I
a = 0.359398 + 1.080444I
b = 0.640602 1.080444I
7.82103 + 4.41044I 6.40190 3.03613I
u = 0.282508 0.750438I
a = 0.624833 + 0.605238I
b = 0.375167 0.605238I
1.185417 + 0.648518I 7.38806 2.73057I
u = 0.282508 + 0.750438I
a = 0.624833 0.605238I
b = 0.375167 + 0.605238I
1.185417 0.648518I 7.38806 + 2.73057I
u = 0.025281 0.303928I
a = 2.25258 + 0.40809I
b = 1.252581 0.408094I
1.96302 + 2.37863I 1.27520 1.22709I
u = 0.025281 + 0.303928I
a = 2.25258 0.40809I
b = 1.252581 + 0.408094I
1.96302 2.37863I 1.27520 + 1.22709I
u = 0.90158 1.50334I
a = 0.092946 0.536743I
b = 1.092946 + 0.536743I
0.90131 + 5.21099I 4.11400 8.12783I
u = 0.90158 + 1.50334I
a = 0.092946 + 0.536743I
b = 1.092946 0.536743I
0.90131 5.21099I 4.11400 + 8.12783I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
3
u
2
+ 1)
4
(u
5
+ u
4
u
3
2u
2
+ u + 1)
(u
10
+ 4u
9
+ 6u
8
12u
6
15u
5
+ u
4
+ 21u
3
+ 25u
2
+ 14u + 4)
c
2
(u
2
+ u 1)
6
(u
5
+ u
4
2u
3
u
2
+ u + 1)
(u
10
6u
9
+ 14u
8
18u
7
+ 22u
6
31u
5
+ 26u
4
7u
3
+ 4u
2
12u + 8)
c
3
(u
5
+ 2u
3
+ u
2
+ 1)(u
10
u
7
+ 5u
6
+ u
3
+ u
2
+ u + 1)
(u
12
+ u
11
+ 2u
10
+ 3u
9
+ 6u
8
+ 3u
7
+ 9u
6
+ 5u
5
2u
4
8u
2
+ 6u 1)
c
4
, c
7
(u
5
+ 2u
3
u
2
1)(u
10
u
7
+ 5u
6
+ u
3
+ u
2
+ u + 1)
(u
12
+ u
11
+ 2u
10
+ 3u
9
+ 6u
8
+ 3u
7
+ 9u
6
+ 5u
5
2u
4
8u
2
+ 6u 1)
c
5
, c
9
(u
5
+ u
3
+ 2u
2
+ 1)
(u
10
+ 2u
9
7u
8
18u
7
+ 9u
6
+ 46u
5
+ 25u
4
13u
3
10u
2
+ u + 1)
(u
12
+ u
11
+ ··· 46u 19)
c
6
(u
3
u
2
+ 1)
4
(u
5
u
4
u
3
+ 2u
2
+ u 1)
(u
10
+ 4u
9
+ 6u
8
12u
6
15u
5
+ u
4
+ 21u
3
+ 25u
2
+ 14u + 4)
c
8
(u
5
+ u
3
2u
2
1)
(u
10
+ 2u
9
7u
8
18u
7
+ 9u
6
+ 46u
5
+ 25u
4
13u
3
10u
2
+ u + 1)
(u
12
+ u
11
+ ··· 46u 19)
c
10
(u
3
+ u
2
+ 2u + 1)
4
(u
5
3u
4
+ 7u
3
8u
2
+ 5u 1)
(u
10
+ 4u
9
+ ··· 4u + 16)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
3
y
2
+ 2y 1)
4
(y
5
3y
4
+ 7y
3
8y
2
+ 5y 1)
(y
10
4y
9
+ ··· + 4y + 16)
c
2
(y
2
3y + 1)
6
(y
5
5y
4
+ 8y
3
7y
2
+ 3y 1)
(y
10
8y
9
+ ··· 80y + 64)
c
3
, c
4
, c
7
(y
5
+ 4y
4
+ 4y
3
y
2
2y 1)
(y
10
+ 10y
8
y
7
+ 27y
6
+ 4y
5
+ 12y
4
+ 9y
3
y
2
+ y + 1)
(y
12
+ 3y
11
+ ··· 20y + 1)
c
5
, c
8
, c
9
(y
5
+ 2y
4
+ ··· 4y 1)(y
10
18y
9
+ ··· 21y + 1)
(y
12
9y
11
+ ··· + 240y + 361)
c
10
(y
3
+ 3y
2
+ 2y 1)
4
(y
5
+ 5y
4
+ 11y
3
+ 9y 1)
(y
10
+ 8y
9
+ ··· + 1424y + 256)
10