10
100
(K10a
104
)
1
Arc Sequences
5 9 6 1 8 10 4 2 3 7
Solving Sequence
5,8 2,6
9 3 10 1 4 7
c
5
c
8
c
2
c
9
c
1
c
4
c
7
c
3
, c
6
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
24
u
23
+ ··· + 8u + 1, 8.57301 × 10
15
u
23
+ 2.04876 × 10
16
u
22
+ ··· + 3.83883 × 10
16
b 6.92888 × 10
16
,
4.13329 × 10
16
u
23
8.54600 × 10
16
u
22
+ ··· + 3.83883 × 10
16
a + 3.99828 × 10
17
i
I
u
2
= hu
6
u
5
3u
4
+ 3u
3
+ 3u
2
3u 1, u
4
+ 2u
2
+ b 1, u
5
u
4
3u
3
+ 3u
2
+ a + 2u 3i
I
u
3
= hu
14
+ u
13
6u
12
6u
11
+ 13u
10
+ 12u
9
8u
8
5u
7
8u
6
11u
5
+ 7u
4
+ 8u
3
+ 3u
2
+ 3u + 1,
12u
13
8u
12
+ ··· + 17a + 31, 16u
13
+ 12u
12
+ ··· + 17b + 30i
There are 3 irreducible components with 44 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
24
u
23
+· · ·+8u +1, 8.57×10
15
u
23
+2.05× 10
16
u
22
+· · ·+3.84 ×10
16
b
6.93 × 10
16
, 4.13 × 10
16
u
23
8.55 × 10
16
u
22
+ · · · + 3.84 × 10
16
a + 4.00 × 10
17
i
(i) Arc colorings
a
5
=
0
u
a
8
=
1.07671u
23
+ 2.22620u
22
+ ··· 7.86626u 10.4153
0.223324u
23
0.533695u
22
+ ··· + 1.17106u + 1.80495
a
2
=
1
0
a
6
=
0.220536u
23
1.34045u
22
+ ··· 86.9825u + 0.285801
1.62084u
23
3.38766u
22
+ ··· + 25.3369u + 1.44363
a
9
=
0.853382u
23
+ 1.69250u
22
+ ··· 6.69520u 8.61040
0.223324u
23
0.533695u
22
+ ··· + 1.17106u + 1.80495
a
3
=
1.32970u
23
+ 2.95305u
22
+ ··· 13.2088u + 11.7767
0.113927u
23
0.111421u
22
+ ··· + 4.48424u 2.01115
a
10
=
0.450634u
23
1.74007u
22
+ ··· 26.5168u + 10.7002
0.907116u
23
1.94557u
22
+ ··· + 14.2160u 0.619577
a
1
=
1
u
2
a
4
=
u
u
3
+ u
a
7
=
0.0250570u
23
0.0719604u
22
+ ··· 0.353618u 9.37638
0.591737u
23
1.35814u
22
+ ··· + 2.57884u + 2.01249
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
14046277262204888
38388304749160487
u
23
59679128012064636
38388304749160487
u
22
+ ···+
21408546407184612
3489845886287317
u +
443200526757334190
38388304749160487
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.43177 0.51875I
a = 0.175026 0.495004I
b = 0.483672
5.67365 + 2.02988I 15.4168 3.4641I
u = 1.43177 + 0.51875I
a = 0.175026 + 0.495004I
b = 0.483672
5.67365 2.02988I 15.4168 + 3.4641I
u = 1.304131 0.360578I
a = 0.37557 1.58654I
b = 1.52087 + 0.16310I
8.63038 + 6.62201I 10.58114 6.66892I
u = 1.304131 + 0.360578I
a = 0.37557 + 1.58654I
b = 1.52087 0.16310I
8.63038 6.62201I 10.58114 + 6.66892I
u = 1.292850 0.036946I
a = 1.52058 + 1.19913I
b = 1.53904
12.59489 2.02988I 14.2695 + 3.4641I
u = 1.292850 + 0.036946I
a = 1.52058 1.19913I
b = 1.53904
12.59489 + 2.02988I 14.2695 3.4641I
u = 1.121551 0.030906I
a = 0.084202 + 0.693521I
b = 0.493180 0.575288I
1.97456 0.05747I 6.57572 0.22068I
u = 1.121551 + 0.030906I
a = 0.084202 0.693521I
b = 0.493180 + 0.575288I
1.97456 + 0.05747I 6.57572 + 0.22068I
u = 0.191754 0.976178I
a = 0.507198 + 0.744269I
b = 0.493180 0.575288I
1.97456 + 4.00229I 6.57572 7.14888I
u = 0.191754 + 0.976178I
a = 0.507198 0.744269I
b = 0.493180 + 0.575288I
1.97456 4.00229I 6.57572 + 7.14888I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.0601819 0.1121888I
a = 9.28795 0.04286I
b = 1.52087 + 0.16310I
8.63038 + 2.56224I 10.58114 + 0.25928I
u = 0.0601819 + 0.1121888I
a = 9.28795 + 0.04286I
b = 1.52087 0.16310I
8.63038 2.56224I 10.58114 0.25928I
u = 0.016927 0.466082I
a = 1.16818 + 1.28257I
b = 0.493180 0.575288I
1.97456 0.05747I 6.57572 0.22068I
u = 0.016927 + 0.466082I
a = 1.16818 1.28257I
b = 0.493180 + 0.575288I
1.97456 + 0.05747I 6.57572 + 0.22068I
u = 0.403158 1.352799I
a = 0.555797 + 0.481353I
b = 1.52087 0.16310I
8.63038 6.62201I 10.58114 + 6.66892I
u = 0.403158 + 1.352799I
a = 0.555797 0.481353I
b = 1.52087 + 0.16310I
8.63038 + 6.62201I 10.58114 6.66892I
u = 1.191398 0.238722I
a = 0.347235 1.108912I
b = 0.493180 + 0.575288I
1.97456 4.00229I 6.57572 + 7.14888I
u = 1.191398 + 0.238722I
a = 0.347235 + 1.108912I
b = 0.493180 0.575288I
1.97456 + 4.00229I 6.57572 7.14888I
u = 1.225192 0.160956I
a = 0.989670 0.751764I
b = 0.483672
5.67365 2.02988I 15.4168 + 3.4641I
u = 1.225192 + 0.160956I
a = 0.989670 + 0.751764I
b = 0.483672
5.67365 + 2.02988I 15.4168 3.4641I
4
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.36996 0.39634I
a = 0.223996 + 0.786508I
b = 1.52087 0.16310I
8.63038 2.56224I 10.58114 0.25928I
u = 1.36996 + 0.39634I
a = 0.223996 0.786508I
b = 1.52087 + 0.16310I
8.63038 + 2.56224I 10.58114 + 0.25928I
u = 1.72946 0.71928I
a = 0.125869 0.571753I
b = 1.53904
12.59489 2.02988I 14.2695 + 3.4641I
u = 1.72946 + 0.71928I
a = 0.125869 + 0.571753I
b = 1.53904
12.59489 + 2.02988I 14.2695 3.4641I
5
II. I
u
2
= hu
6
u
5
3u
4
+ 3u
3
+ 3u
2
3u 1, u
4
+ 2u
2
+ b 1, u
5
u
4
3u
3
+ 3u
2
+ a + 2u 3i
(i) Arc colorings
a
5
=
0
u
a
8
=
u
5
+ u
4
+ 3u
3
3u
2
2u + 3
u
4
2u
2
+ 1
a
2
=
1
0
a
6
=
u
5
+ u
4
+ 2u
3
3u
2
u + 3
u
5
+ 3u
3
2u
a
9
=
u
5
+ 2u
4
+ 3u
3
5u
2
2u + 4
u
4
2u
2
+ 1
a
3
=
u
4
+ 3u
2
u 3
u 1
a
10
=
u
5
+ u
4
+ 2u
3
3u
2
u + 2
u
5
+ 2u
3
u
a
1
=
1
u
2
a
4
=
u
u
3
+ u
a
7
=
u
5
+ 2u
4
+ 2u
3
5u
2
+ 4
u
4
2u
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
5
3u
4
12u
3
+ 8u
2
+ 12u + 2
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.216294 0.265932I
a = 0.431634 0.437492I
b = 0.251489 + 0.528716I
4.59420 + 1.63935I 6.79257 0.07886I
u = 1.216294 + 0.265932I
a = 0.431634 + 0.437492I
b = 0.251489 0.528716I
4.59420 1.63935I 6.79257 + 0.07886I
u = 0.281838
a = 3.26630
b = 0.847445
0.285060 0.503989
u = 1.105921 0.548003I
a = 0.379950 0.907735I
b = 1.46321 + 0.18726I
9.23208 4.33255I 12.59516 + 4.05038I
u = 1.105921 + 0.548003I
a = 0.379950 + 0.907735I
b = 1.46321 0.18726I
9.23208 + 4.33255I 12.59516 4.05038I
u = 1.50258
a = 0.837064
b = 1.58196
12.1109 12.7285
7
III. I
u
3
= hu
14
+ u
13
+ · · · + 3u + 1, 12u
13
8u
12
+ · · · + 17a + 31, 16u
13
+
12u
12
+ · · · + 17b + 30i
(i) Arc colorings
a
5
=
0
u
a
8
=
0.705882u
13
+ 0.470588u
12
+ ··· 1.17647u 1.82353
0.941176u
13
0.705882u
12
+ ··· 2.23529u 1.76471
a
2
=
1
0
a
6
=
0.705882u
13
+ 0.470588u
12
+ ··· 2.17647u 1.82353
0.294118u
13
+ 0.529412u
12
+ ··· 1.82353u 1.17647
a
9
=
1.64706u
13
0.235294u
12
+ ··· 3.41176u 3.58824
0.941176u
13
0.705882u
12
+ ··· 2.23529u 1.76471
a
3
=
1.29412u
13
+ 0.470588u
12
+ ··· + 1.82353u + 2.17647
0.117647u
13
0.411765u
12
+ ··· + 1.52941u + 0.470588
a
10
=
0.588235u
13
+ 0.0588235u
12
+ ··· 1.64706u 2.35294
0.588235u
13
+ 0.941176u
12
+ ··· 1.35294u 0.647059
a
1
=
1
u
2
a
4
=
u
u
3
+ u
a
7
=
1.35294u
13
0.764706u
12
+ ··· 1.58824u 2.41176
0.647059u
13
1.23529u
12
+ ··· + 0.588235u 0.588235
(ii) Obstruction class = 1
(iii) Cusp Shapes =
39
17
u
13
+
25
17
u
12
270
17
u
11
209
17
u
10
+
674
17
u
9
+
580
17
u
8
542
17
u
7
33u
6
363
17
u
5
283
17
u
4
+
535
17
u
3
+
650
17
u
2
+
116
17
u +
190
17
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.53034 0.47327I
a = 0.329980 + 1.104498I
b = 1.62071 0.25886I
14.7349 + 12.8109I 11.37066 6.14968I
u = 1.53034 + 0.47327I
a = 0.329980 1.104498I
b = 1.62071 + 0.25886I
14.7349 12.8109I 11.37066 + 6.14968I
u = 1.278751 0.193074I
a = 0.029088 + 0.703768I
b = 0.493094 1.098776I
5.88215 + 2.52726I 13.28929 3.43101I
u = 1.278751 + 0.193074I
a = 0.029088 0.703768I
b = 0.493094 + 1.098776I
5.88215 2.52726I 13.28929 + 3.43101I
u = 0.662974
a = 1.81684
b = 1.04049
0.339162 13.0624
u = 0.417149
a = 0.305601
b = 0.622591
0.865875 12.3761
u = 0.033402 0.876284I
a = 1.179864 0.535106I
b = 1.45633 + 0.05562I
4.42897 2.24150I 6.33861 + 3.08717I
u = 0.033402 + 0.876284I
a = 1.179864 + 0.535106I
b = 1.45633 0.05562I
4.42897 + 2.24150I 6.33861 3.08717I
u = 0.104768 0.538091I
a = 0.93892 1.17310I
b = 0.303715 + 0.334799I
1.31044 + 0.99980I 2.51765 3.01751I
u = 0.104768 + 0.538091I
a = 0.93892 + 1.17310I
b = 0.303715 0.334799I
1.31044 0.99980I 2.51765 + 3.01751I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.332439 0.033792I
a = 0.435875 0.807892I
b = 1.67750 + 0.35344I
13.16539 3.07431I 13.56108 + 2.64554I
u = 1.332439 + 0.033792I
a = 0.435875 + 0.807892I
b = 1.67750 0.35344I
13.16539 + 3.07431I 13.56108 2.64554I
u = 1.37854 0.38980I
a = 0.240207 + 0.893211I
b = 0.748785 0.823629I
6.90977 8.77559I 9.73876 + 7.09449I
u = 1.37854 + 0.38980I
a = 0.240207 0.893211I
b = 0.748785 + 0.823629I
6.90977 + 8.77559I 9.73876 7.09449I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
6
u
5
+ ··· 3u 1)(u
14
+ u
13
+ ··· + 3u + 1)
(u
24
u
23
+ ··· + 8u + 1)
c
2
(1 + u + 2u
2
+ 2u
3
3u
4
u
5
+ u
6
)
4
(u
6
+ u
5
+ ··· + u + 1)
(u
14
+ 6u
13
+ ··· + 2u 4)
c
3
, c
5
(u
6
+ u
5
+ ··· u 1)(u
14
u
13
+ ··· 5u 1)
(u
24
+ 7u
23
+ ··· + 372u + 73)
c
4
, c
10
(u
6
+ u
5
+ ··· + 3u 1)(u
14
+ u
13
+ ··· + 3u + 1)
(u
24
u
23
+ ··· + 8u + 1)
c
7
(u
2
u + 1)
12
(u
6
u
5
+ u
4
+ u
3
u
2
+ u 1)
(u
14
+ 14u
13
+ ··· 288u 64)
c
8
, c
9
(1 + u + 2u
2
+ 2u
3
3u
4
u
5
+ u
6
)
4
(u
6
u
5
+ ··· u + 1)
(u
14
+ 6u
13
+ ··· + 2u 4)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
6
7y
5
+ ··· 15y + 1)(y
14
13y
13
+ ··· 3y + 1)
(y
24
21y
23
+ ··· + 72y + 1)
c
2
, c
8
, c
9
(y
6
7y
5
+ 17y
4
16y
3
+ 6y
2
5y + 1)
4
(y
6
7y
5
+ ··· + y + 1)(y
14
14y
13
+ ··· 60y + 16)
c
3
(1.58196 + 1.00000y)(0.847445 + 1.00000y)(1.00000y
2
+ 0.5029776y + 0.3427873)(1.00000y
2
+ 2.926423y + 2.17605)
(1y
14
+ 7.0000000y
13
+ ··· 59.000000y + 1.00000000)
(1y
24
+ 11.0000000y
23
+ ··· + 52000.000y + 5329.0000)
c
5
(y
6
+ y
5
+ ··· + y + 1)(y
14
+ 7y
13
+ ··· 59y + 1)
(y
24
+ 11y
23
+ ··· + 52000y + 5329)
c
6
1.0000000
(1y
6
7.0000000y
5
+ ··· 15.000000y + 1.0000000)
(1y
14
13.000000y
13
+ ··· 3.0000000y + 1.0000000)
(1y
24
21.000000y
23
+ ··· + 72.000000y + 1.0000000)
c
7
(y
2
+ y + 1)
12
(y
6
+ y
5
+ y
4
3y
3
3y
2
+ y + 1)
(y
14
+ 2y
13
+ ··· 41984y + 4096)
c
10
1.00000000
(1y
6
7.0000000y
5
+ ··· 15.0000000y + 1.00000000)
(1y
14
13.0000000y
13
+ ··· 3.0000000y + 1.00000000)
(1y
24
21.0000000y
23
+ ··· + 72.000000y + 1.00000000)
12