10
119
(K10a
85
)
1
Arc Sequences
9 6 1 2 8 10 3 5 4 7
Solving Sequence
2,9 1,5
4 10 3 8 6 7
c
1
c
4
c
9
c
3
c
8
c
5
c
7
c
2
, c
6
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
10
+ 5u
9
+ 14u
8
+ 24u
7
+ 29u
6
+ 24u
5
+ 11u
4
u
3
2u
2
+ u + 1,
41u
9
152u
8
353u
7
428u
6
371u
5
73u
4
+ 248u
3
+ 361u
2
+ 67a + 135u + 77,
63u
9
+ 245u
8
+ 580u
7
+ 741u
6
+ 691u
5
+ 305u
4
100u
3
272u
2
+ 67b + 72u + 50i
I
u
2
= hu
60
+ 4u
58
+ ··· 28u + 3,
2.92641 × 10
187
u
59
8.35049 × 10
186
u
58
+ ··· + 6.28202 × 10
187
b + 1.91667 × 10
188
,
5.56717 × 10
187
u
59
+ 1.54419 × 10
187
u
58
+ ··· + 1.88461 × 10
188
a 5.98410 × 10
188
i
There are 2 irreducible components with 70 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
10
+ 5u
9
+ · · · + u + 1, 41u
9
152u
8
+ · · · + 67a + 77, 63u
9
+
245u
8
+ · · · + 67b + 50i
(i) Arc colorings
a
2
=
1
0
a
9
=
0.611940u
9
+ 2.26866u
8
+ ··· 2.01493u 1.14925
0.940299u
9
3.65672u
8
+ ··· 1.07463u 0.746269
a
1
=
1.59701u
9
+ 6.43284u
8
+ ··· + 2.25373u + 3.53731
2.59701u
9
11.4328u
8
+ ··· 0.253731u 3.53731
a
5
=
0
u
a
4
=
u
u
a
10
=
0.671642u
9
+ 2.61194u
8
+ ··· 2.08955u 0.895522
u
9
4u
8
10u
7
14u
6
15u
5
9u
4
2u
3
+ 3u
2
u 1
a
3
=
0.0447761u
9
0.492537u
8
+ ··· + 3.19403u + 1.94030
1.04478u
9
4.50746u
8
+ ··· 0.194030u 0.940299
a
8
=
0.611940u
9
+ 2.26866u
8
+ ··· 2.01493u 1.14925
0.283582u
9
0.880597u
8
+ ··· 0.895522u + 0.0447761
a
6
=
1.59701u
9
+ 7.43284u
8
+ ··· + 1.25373u + 0.537313
0.716418u
9
3.11940u
8
+ ··· + 0.895522u 1.04478
a
7
=
0.910448u
9
+ 3.98507u
8
+ ··· 1.38806u 1.88060
0.164179u
9
0.194030u
8
+ ··· 1.04478u + 0.552239
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 14u
9
+ 61u
8
+ 157u
7
+ 236u
6
+ 256u
5
+ 174u
4
+ 45u
3
37u
2
+ 3u + 19
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.14707 1.48128I
a = 0.609691 0.292888I
b = 0.702457 + 0.789416I
2.50173 + 4.70796I 2.30544 6.58589I
u = 1.14707 + 1.48128I
a = 0.609691 + 0.292888I
b = 0.702457 0.789416I
2.50173 4.70796I 2.30544 + 6.58589I
u = 1.006319 0.639149I
a = 0.141697 + 0.417011I
b = 0.305504 + 0.686816I
0.60938 + 1.82644I 5.24506 2.77183I
u = 1.006319 + 0.639149I
a = 0.141697 0.417011I
b = 0.305504 0.686816I
0.60938 1.82644I 5.24506 + 2.77183I
u = 0.666955 0.329661I
a = 1.074642 0.447632I
b = 0.93198 + 1.28311I
0.80863 + 2.83685I 3.98230 13.24479I
u = 0.666955 + 0.329661I
a = 1.074642 + 0.447632I
b = 0.93198 1.28311I
0.80863 2.83685I 3.98230 + 13.24479I
u = 0.069226 1.285133I
a = 0.881151 0.152583I
b = 1.100169 0.460898I
5.27004 + 6.36836I 3.95341 6.63467I
u = 0.069226 + 1.285133I
a = 0.881151 + 0.152583I
b = 1.100169 + 0.460898I
5.27004 6.36836I 3.95341 + 6.63467I
u = 0.389573 0.258635I
a = 2.55512 + 1.96682I
b = 0.339768 0.572266I
5.16077 + 2.93340I 2.03148 3.30765I
u = 0.389573 + 0.258635I
a = 2.55512 1.96682I
b = 0.339768 + 0.572266I
5.16077 2.93340I 2.03148 + 3.30765I
3
II. I
u
2
= hu
60
+ 4u
58
+ · · · 28u + 3, 2.93 × 10
187
u
59
8.35 × 10
186
u
58
+
· · · + 6.28 × 10
187
b + 1.92 × 10
188
, 5.57 × 10
187
u
59
+ 1.54 × 10
187
u
58
+ · · · +
1.88 × 10
188
a 5.98 × 10
188
i
(i) Arc colorings
a
2
=
1
0
a
9
=
0.295402u
59
0.0819373u
58
+ ··· 2.29883u + 3.17525
0.465839u
59
+ 0.132927u
58
+ ··· + 20.7915u 3.05104
a
1
=
0.132338u
59
0.117631u
58
+ ··· + 10.6408u 3.11418
0.0127953u
59
+ 0.260649u
58
+ ··· 30.8475u + 5.43814
a
5
=
0
u
a
4
=
u
u
a
10
=
0.331765u
59
0.0994100u
58
+ ··· 3.21522u + 3.32822
0.502201u
59
+ 0.150400u
58
+ ··· + 21.7079u 3.20400
a
3
=
1.61798u
59
+ 0.226769u
58
+ ··· + 73.0966u 7.40978
0.727922u
59
+ 0.0979081u
58
+ ··· 61.9678u + 7.58270
a
8
=
0.295402u
59
0.0819373u
58
+ ··· 2.29883u + 3.17525
0.512381u
59
+ 0.179225u
58
+ ··· + 19.3834u 2.80522
a
6
=
0.151223u
59
+ 0.103858u
58
+ ··· + 15.9126u 2.83148
0.0824824u
59
0.118027u
58
+ ··· + 3.04563u 0.708587
a
7
=
0.830298u
59
+ 0.205190u
58
+ ··· 72.6925u + 8.44583
0.103653u
59
0.237454u
58
+ ··· + 20.3707u 3.23913
(ii) Obstruction class = 1
(iii) Cusp Shapes = 1.43659u
59
+ 0.426058u
58
+ ··· 246.489u + 36.8057
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.238385 0.475118I
a = 0.325818 + 0.733060I
b = 0.41049 1.55431I
1.53778 + 2.56920I 1.40739 3.97397I
u = 1.238385 + 0.475118I
a = 0.325818 0.733060I
b = 0.41049 + 1.55431I
1.53778 2.56920I 1.40739 + 3.97397I
u = 1.16552 1.29014I
a = 0.557684 + 0.308461I
b = 0.693224 1.050902I
1.42019 + 3.81670I 2.39392 1.19874I
u = 1.16552 + 1.29014I
a = 0.557684 0.308461I
b = 0.693224 + 1.050902I
1.42019 3.81670I 2.39392 + 1.19874I
u = 1.01214 1.22490I
a = 0.750680 0.364385I
b = 0.695461 + 0.539687I
2.23892 + 3.83057I 1.35038 + 0.97651I
u = 1.01214 + 1.22490I
a = 0.750680 + 0.364385I
b = 0.695461 0.539687I
2.23892 3.83057I 1.35038 0.97651I
u = 0.990107 0.775650I
a = 0.363869 0.232534I
b = 0.689474 0.479498I
1.01358 + 3.09322I 5.20661 7.95930I
u = 0.990107 + 0.775650I
a = 0.363869 + 0.232534I
b = 0.689474 + 0.479498I
1.01358 3.09322I 5.20661 + 7.95930I
u = 0.98126 1.54901I
a = 0.629038 0.124952I
b = 0.628595 + 0.975527I
2.67312 + 6.89147I 0.54802 9.35333I
u = 0.98126 + 1.54901I
a = 0.629038 + 0.124952I
b = 0.628595 0.975527I
2.67312 6.89147I 0.54802 + 9.35333I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.97062 1.39698I
a = 0.709468 + 0.305181I
b = 1.068754 0.684485I
3.13949 + 6.16052I 0.86534 8.28596I
u = 0.97062 + 1.39698I
a = 0.709468 0.305181I
b = 1.068754 + 0.684485I
3.13949 6.16052I 0.86534 + 8.28596I
u = 0.950374 0.230216I
a = 0.689315 + 0.180915I
b = 0.513079 + 0.172371I
1.77833 + 0.12398I 7.12326 + 1.43443I
u = 0.950374 + 0.230216I
a = 0.689315 0.180915I
b = 0.513079 0.172371I
1.77833 0.12398I 7.12326 1.43443I
u = 0.880526 0.425648I
a = 0.273162 0.703429I
b = 0.560842 0.757300I
0.285692 + 0.756946I 1.15945 + 1.79896I
u = 0.880526 + 0.425648I
a = 0.273162 + 0.703429I
b = 0.560842 + 0.757300I
0.285692 0.756946I 1.15945 1.79896I
u = 0.796474 0.786684I
a = 0.258331 + 0.743451I
b = 0.326012 + 0.849608I
0.44456 + 2.32036I 1.97518 4.64341I
u = 0.796474 + 0.786684I
a = 0.258331 0.743451I
b = 0.326012 0.849608I
0.44456 2.32036I 1.97518 + 4.64341I
u = 0.621503 0.751780I
a = 1.44905 0.67817I
b = 0.163932 + 0.584162I
3.26872 + 3.24397I 0.45716 6.47664I
u = 0.621503 + 0.751780I
a = 1.44905 + 0.67817I
b = 0.163932 0.584162I
3.26872 3.24397I 0.45716 + 6.47664I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.539561 0.979683I
a = 0.983442 0.483042I
b = 1.31460 + 1.41628I
7.31597 + 5.19158I 7.13412 5.50514I
u = 0.539561 + 0.979683I
a = 0.983442 + 0.483042I
b = 1.31460 1.41628I
7.31597 5.19158I 7.13412 + 5.50514I
u = 0.307590 0.697419I
a = 1.179949 0.437400I
b = 0.584375 0.606604I
1.69174 + 2.07365I 0.35018 3.75765I
u = 0.307590 + 0.697419I
a = 1.179949 + 0.437400I
b = 0.584375 + 0.606604I
1.69174 2.07365I 0.35018 + 3.75765I
u = 0.299852 1.192800I
a = 1.204775 0.173112I
b = 0.181867 0.911330I
8.17971 + 6.88009I 6.32444 5.98403I
u = 0.299852 + 1.192800I
a = 1.204775 + 0.173112I
b = 0.181867 + 0.911330I
8.17971 6.88009I 6.32444 + 5.98403I
u = 0.223189 0.473560I
a = 0.407288 0.415644I
b = 0.588695 0.710675I
0.08914 + 1.52136I 0.92288 3.10853I
u = 0.223189 + 0.473560I
a = 0.407288 + 0.415644I
b = 0.588695 + 0.710675I
0.08914 1.52136I 0.92288 + 3.10853I
u = 0.000901 0.445406I
a = 3.69660 1.84265I
b = 0.227140 0.595597I
5.53864 3.01539I 19.4332 + 8.4307I
u = 0.000901 + 0.445406I
a = 3.69660 + 1.84265I
b = 0.227140 + 0.595597I
5.53864 + 3.01539I 19.4332 8.4307I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.161175 0.349105I
a = 3.30875 1.31611I
b = 0.203979 + 1.309977I
6.64861 0.13946I 6.32934 + 0.07570I
u = 0.161175 + 0.349105I
a = 3.30875 + 1.31611I
b = 0.203979 1.309977I
6.64861 + 0.13946I 6.32934 0.07570I
u = 0.222891 0.173390I
a = 3.03718 1.65389I
b = 1.181954 + 0.206918I
0.48057 1.96275I 6.28895 + 2.92214I
u = 0.222891 + 0.173390I
a = 3.03718 + 1.65389I
b = 1.181954 0.206918I
0.48057 + 1.96275I 6.28895 2.92214I
u = 0.223247 0.733150I
a = 0.57721 1.32238I
b = 0.478818 0.735448I
4.15031 + 0.75171I 5.55583 + 1.52775I
u = 0.223247 + 0.733150I
a = 0.57721 + 1.32238I
b = 0.478818 + 0.735448I
4.15031 0.75171I 5.55583 1.52775I
u = 0.224592 1.103138I
a = 0.0614888 0.0773880I
b = 0.747558 + 0.713910I
3.57991 + 4.96662I 0.78194 5.62106I
u = 0.224592 + 1.103138I
a = 0.0614888 + 0.0773880I
b = 0.747558 0.713910I
3.57991 4.96662I 0.78194 + 5.62106I
u = 0.308920 0.321579I
a = 2.25812 + 1.22542I
b = 1.92160 + 0.42821I
4.17954 7.17743I 1.10548 + 10.24381I
u = 0.308920 + 0.321579I
a = 2.25812 1.22542I
b = 1.92160 0.42821I
4.17954 + 7.17743I 1.10548 10.24381I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.636544 0.230444I
a = 1.27435 0.81298I
b = 1.03529 + 1.13558I
0.77022 2.47923I 0.39058 8.23551I
u = 0.636544 + 0.230444I
a = 1.27435 + 0.81298I
b = 1.03529 1.13558I
0.77022 + 2.47923I 0.39058 + 8.23551I
u = 0.723525 0.554514I
a = 1.152803 + 0.612199I
b = 0.868791 + 0.810031I
1.66896 4.09599I 2.80131 + 9.03279I
u = 0.723525 + 0.554514I
a = 1.152803 0.612199I
b = 0.868791 0.810031I
1.66896 + 4.09599I 2.80131 9.03279I
u = 0.783244 0.988185I
a = 1.092514 + 0.378136I
b = 0.86821 1.35253I
8.85403 4.69065I 5.81787 + 4.01531I
u = 0.783244 + 0.988185I
a = 1.092514 0.378136I
b = 0.86821 + 1.35253I
8.85403 + 4.69065I 5.81787 4.01531I
u = 0.83604 1.24262I
a = 0.647217 + 0.481424I
b = 0.136133 0.745299I
3.96912 + 1.82451I 2.76104 2.92871I
u = 0.83604 + 1.24262I
a = 0.647217 0.481424I
b = 0.136133 + 0.745299I
3.96912 1.82451I 2.76104 + 2.92871I
u = 0.860595 0.757350I
a = 0.846219 0.544161I
b = 0.835203 0.947302I
1.92721 10.19582I 0.72576 + 8.25674I
u = 0.860595 + 0.757350I
a = 0.846219 + 0.544161I
b = 0.835203 + 0.947302I
1.92721 + 10.19582I 0.72576 8.25674I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.866958 0.207225I
a = 0.571049 0.451805I
b = 0.936750 + 0.956743I
1.77283 3.70444I 2.15456 + 4.83060I
u = 0.866958 + 0.207225I
a = 0.571049 + 0.451805I
b = 0.936750 0.956743I
1.77283 + 3.70444I 2.15456 4.83060I
u = 1.10129 1.06163I
a = 0.881434 0.438247I
b = 0.88904 + 1.33193I
3.32542 9.91193I 0.14733 + 6.82962I
u = 1.10129 + 1.06163I
a = 0.881434 + 0.438247I
b = 0.88904 1.33193I
3.32542 + 9.91193I 0.14733 6.82962I
u = 1.14876 1.24339I
a = 0.818110 + 0.354584I
b = 0.92101 1.32266I
6.9242 15.9387I 2.21564 + 8.66212I
u = 1.14876 + 1.24339I
a = 0.818110 0.354584I
b = 0.92101 + 1.32266I
6.9242 + 15.9387I 2.21564 8.66212I
u = 1.31875 1.79980I
a = 0.405736 0.355761I
b = 0.016053 + 0.631044I
6.90749 + 6.01393I 7.81339 6.22163I
u = 1.31875 + 1.79980I
a = 0.405736 + 0.355761I
b = 0.016053 0.631044I
6.90749 6.01393I 7.81339 + 6.22163I
u = 1.55967 0.51228I
a = 0.194064 0.780344I
b = 0.089967 + 0.920793I
6.87032 1.61985I 7.52046 + 1.01232I
u = 1.55967 + 0.51228I
a = 0.194064 + 0.780344I
b = 0.089967 0.920793I
6.87032 + 1.61985I 7.52046 1.01232I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
10
u
9
+ ··· + 2u
2
+ 1)(u
60
+ 6u
59
+ ··· + 543u + 79)
c
2
(u
10
+ 2u
8
+ 4u
7
+ 2u
6
+ 3u
5
+ 6u
4
+ 4u
3
+ u
2
+ u + 1)
(u
60
+ u
59
+ ··· + 314u + 71)
c
3
(u
10
+ 5u
9
+ ··· + 5u + 1)(u
60
+ 4u
59
+ ··· 90u + 31)
c
4
(u
10
5u
9
+ 14u
8
24u
7
+ 29u
6
24u
5
+ 11u
4
+ u
3
2u
2
u + 1)
(u
60
+ 4u
58
+ ··· + 28u + 3)
c
5
(u
10
+ u
9
+ 5u
8
+ 6u
7
+ 12u
6
+ 13u
5
+ 15u
4
+ 12u
3
+ 9u
2
+ 4u + 1)
(u
60
+ 4u
59
+ ··· + 23u + 1)
c
6
(u
10
u
9
2u
8
+ 4u
7
5u
5
+ 5u
4
+ 2u
3
4u
2
+ 1)
(u
60
+ 2u
59
+ ··· + 21u + 13)
c
7
(u
10
+ 3u
7
+ 2u
6
+ 2u
5
+ 2u
4
+ 3u
3
+ 4u
2
+ u + 1)
(u
60
+ u
59
+ ··· + 1880u + 667)
c
8
(u
10
u
9
+ 5u
8
6u
7
+ 12u
6
13u
5
+ 15u
4
12u
3
+ 9u
2
4u + 1)
(u
60
+ 4u
59
+ ··· + 23u + 1)
c
9
(u
10
+ 2u
8
+ ··· u + 1)(u
60
+ u
59
+ ··· 22u + 1)
c
10
(u
10
+ u
9
2u
8
4u
7
+ 5u
5
+ 5u
4
2u
3
4u
2
+ 1)
(u
60
+ 2u
59
+ ··· + 21u + 13)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y
10
+ y
9
+ 7y
8
y
7
+ 12y
6
+ 6y
5
+ 18y
4
+ 16y
3
+ 12y
2
+ 4y + 1)
(y
60
+ 14y
59
+ ··· + 122113y + 6241)
c
2
(y
10
+ 4y
9
+ 8y
8
+ 4y
7
+ 6y
6
11y
5
+ 12y
4
6y
3
+ 5y
2
+ y + 1)
(y
60
+ 13y
59
+ ··· + 80466y + 5041)
c
3
(y
10
+ 3y
9
+ 2y
8
8y
7
35y
6
+ 138y
4
+ 105y
3
+ 27y
2
+ 5y + 1)
(y
60
16y
59
+ ··· 65698y + 961)
c
4
(y
10
+ 3y
9
+ 14y
8
+ 18y
7
+ 3y
6
+ 46y
5
+ 33y
4
35y
3
+ 28y
2
5y + 1)
(y
60
+ 8y
59
+ ··· + 152y + 9)
c
5
, c
8
(y
10
+ 9y
9
+ ··· + 2y + 1)(y
60
+ 46y
59
+ ··· 77y + 1)
c
6
, c
10
(y
10
5y
9
+ ··· 8y + 1)(y
60
36y
59
+ ··· 2183y + 169)
c
7
(y
10
+ 4y
8
5y
7
12y
5
+ 2y
4
+ 7y
3
+ 14y
2
+ 7y + 1)
(y
60
+ 21y
59
+ ··· + 11382388y + 444889)
c
9
(y
10
+ 4y
9
+ 12y
8
+ 16y
7
+ 18y
6
+ 6y
5
+ 12y
4
y
3
+ 7y
2
+ y + 1)
(y
60
+ 5y
59
+ ··· 22y + 1)
12