10
117
(K10a
99
)
1
Arc Sequences
8 5 10 9 1 3 4 2 7 6
Solving Sequence
3,10 4,6
7 8 1 5 2 9
c
3
c
6
c
7
c
10
c
5
c
2
c
9
c
1
, c
4
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
9
u
8
+ 4u
7
+ u
6
+ u
5
+ 8u
4
+ 3u
2
+ 7u + 1,
9u
8
+ 10u
7
41u
6
+ 10u
5
34u
4
36u
3
+ 14u
2
+ 20b 33u 16,
27u
8
+ 30u
7
113u
6
10u
5
32u
4
208u
3
+ 32u
2
+ 10a 89u 168i
I
u
2
= hu
60
+ 5u
58
+ ··· 9u + 1, 2.07481 × 10
162
u
59
+ 1.62572 × 10
162
u
58
+ ··· + 2.44194 × 10
160
b + 1.07278 × 10
162
,
3.11578 × 10
162
u
59
+ 1.13230 × 10
162
u
58
+ ··· + 1.22097 × 10
160
a 5.18954 × 10
162
i
There are 2 irreducible components with 69 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
9
u
8
+ 4u
7
+ u
6
+ u
5
+ 8u
4
+ 3u
2
+ 7u + 1, 9u
8
+ 10u
7
+ · · · +
20b 16, 27u
8
+ 30u
7
+ · · · + 10a 168i
(i) Arc colorings
a
3
=
1
0
a
10
=
27
10
u
8
3u
7
+ ··· +
89
10
u +
84
5
9
20
u
8
1
2
u
7
+ ··· +
33
20
u +
4
5
a
4
=
1.15000u
8
+ 1.25000u
7
+ ··· 5.05000u 7.85000
0.0500000u
8
0.450000u
6
+ ··· 2.85000u 0.700000
a
6
=
0
u
a
7
=
u
u
a
8
=
1.55000u
8
1.75000u
7
+ ··· + 4.85000u + 7.95000
11
20
u
8
1
2
u
7
+ ··· +
7
20
u +
1
5
a
1
=
27
10
u
8
3u
7
+ ··· +
89
10
u +
84
5
1
4
u
8
+
5
4
u
6
+ ··· +
9
4
u +
1
2
a
5
=
6u
8
7u
7
+ 25u
6
+ 2u
5
+ 5u
4
+ 47u
3
8u
2
+ 18u + 39
1
10
u
8
1
4
u
7
+ ···
4
5
u +
3
20
a
2
=
17
10
u
8
7
4
u
7
+ ··· +
22
5
u +
201
20
9
20
u
8
1
4
u
7
+ ··· +
3
20
u +
1
20
a
9
=
2.70000u
8
3.25000u
7
+ ··· + 9.40000u + 16.5500
9
20
u
8
3
4
u
7
+ ··· +
43
20
u +
11
20
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
96
5
u
8
+
89
4
u
7
803
10
u
6
23
4
u
5
197
10
u
4
1483
10
u
3
+
116
5
u
2
669
10
u
2331
20
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.911604 0.103130I
a = 0.346143 + 0.236844I
b = 0.055585 + 1.071073I
3.57395 + 1.78451I 10.19091 0.99326I
u = 0.911604 + 0.103130I
a = 0.346143 0.236844I
b = 0.055585 1.071073I
3.57395 1.78451I 10.19091 + 0.99326I
u = 0.153592
a = 15.2840
b = 0.524555
0.416370 105.202
u = 0.161935 1.373032I
a = 0.812399 + 0.143691I
b = 1.040637 0.285855I
0.72688 6.70635I 3.06894 + 7.87674I
u = 0.161935 + 1.373032I
a = 0.812399 0.143691I
b = 1.040637 + 0.285855I
0.72688 + 6.70635I 3.06894 7.87674I
u = 0.53282 1.62052I
a = 0.667140 + 0.065384I
b = 0.244831 + 0.626842I
4.15988 1.15529I 3.41918 + 3.86401I
u = 0.53282 + 1.62052I
a = 0.667140 0.065384I
b = 0.244831 0.626842I
4.15988 + 1.15529I 3.41918 3.86401I
u = 0.793645 0.872272I
a = 0.941131 + 0.574780I
b = 1.11367 + 1.37911I
3.22264 4.71392I 2.42181 + 4.00779I
u = 0.793645 + 0.872272I
a = 0.941131 0.574780I
b = 1.11367 1.37911I
3.22264 + 4.71392I 2.42181 4.00779I
3
II. I
u
2
=
hu
60
+5u
58
+· · ·9u+1, 2.07×10
162
u
59
+1.63×10
162
u
58
+· · ·+2.44×10
160
b+
1.07×10
162
, 3.12×10
162
u
59
+1.13×10
162
u
58
+· · ·+1.22×10
160
a5.19×10
162
i
(i) Arc colorings
a
3
=
1
0
a
10
=
255.189u
59
92.7381u
58
+ ··· 3646.39u + 425.035
84.9656u
59
66.5751u
58
+ ··· 224.018u 43.9316
a
4
=
34.6302u
59
+ 0.548999u
58
+ ··· 1007.20u + 175.024
45.2599u
59
7.84990u
58
+ ··· 999.711u + 150.594
a
6
=
0
u
a
7
=
u
u
a
8
=
125.168u
59
+ 33.9794u
58
+ ··· + 2395.22u 338.061
86.3935u
59
+ 31.8053u
58
+ ··· + 1365.56u 165.434
a
1
=
255.189u
59
92.7381u
58
+ ··· 3646.39u + 425.035
69.6829u
59
73.4913u
58
+ ··· + 355.436u 136.670
a
5
=
14.0463u
59
+ 9.41040u
58
+ ··· 32.8635u + 25.4338
2.39712u
59
+ 4.14434u
58
+ ··· 207.295u + 44.0406
a
2
=
130.734u
59
31.4426u
58
+ ··· 2579.21u + 368.152
83.1300u
59
29.6977u
58
+ ··· 1324.62u + 160.529
a
9
=
269.967u
59
107.999u
58
+ ··· 3581.15u + 398.872
99.7439u
59
81.8360u
58
+ ··· 158.775u 70.0946
(ii) Obstruction class = 1
(iii) Cusp Shapes = 70.9758u
59
81.5130u
58
+ ··· + 434.666u 184.833
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.327123 0.173804I
a = 0.019142 + 0.750014I
b = 0.07505 + 1.52982I
2.01747 2.78040I 6.20209 + 2.60384I
u = 1.327123 + 0.173804I
a = 0.019142 0.750014I
b = 0.07505 1.52982I
2.01747 + 2.78040I 6.20209 2.60384I
u = 1.23945 1.72511I
a = 0.386708 + 0.384033I
b = 0.008423 + 0.554656I
0.66314 6.63784I 0.93799 + 7.96550I
u = 1.23945 + 1.72511I
a = 0.386708 0.384033I
b = 0.008423 0.554656I
0.66314 + 6.63784I 0.93799 7.96550I
u = 1.19326 1.23882I
a = 0.737435 0.338220I
b = 0.82689 1.22977I
3.51043 + 9.84939I 1.68857 6.86281I
u = 1.19326 + 1.23882I
a = 0.737435 + 0.338220I
b = 0.82689 + 1.22977I
3.51043 9.84939I 1.68857 + 6.86281I
u = 1.171170 0.662981I
a = 0.413796 + 0.335905I
b = 0.654495 0.222235I
5.22239 + 2.37989I 14.0342 3.0404I
u = 1.171170 + 0.662981I
a = 0.413796 0.335905I
b = 0.654495 + 0.222235I
5.22239 2.37989I 14.0342 + 3.0404I
u = 1.16052 1.17459I
a = 0.855833 + 0.367253I
b = 0.96616 + 1.29625I
0.3709 + 16.1254I 4.05494 8.96831I
u = 1.16052 + 1.17459I
a = 0.855833 0.367253I
b = 0.96616 1.29625I
0.3709 16.1254I 4.05494 + 8.96831I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.96089 1.31598I
a = 0.664551 + 0.085232I
b = 0.824128 + 0.877490I
3.14073 + 5.21448I 7.85957 5.74859I
u = 0.96089 + 1.31598I
a = 0.664551 0.085232I
b = 0.824128 0.877490I
3.14073 5.21448I 7.85957 + 5.74859I
u = 0.870982 0.095094I
a = 0.721633 + 0.694495I
b = 0.93774 + 1.26399I
3.61034 + 3.01624I 10.20173 6.52927I
u = 0.870982 + 0.095094I
a = 0.721633 0.694495I
b = 0.93774 1.26399I
3.61034 3.01624I 10.20173 + 6.52927I
u = 0.847394 0.210567I
a = 0.702593 + 0.065112I
b = 0.758675 + 0.611167I
1.45806 + 0.40357I 7.54100 1.19625I
u = 0.847394 + 0.210567I
a = 0.702593 0.065112I
b = 0.758675 0.611167I
1.45806 0.40357I 7.54100 + 1.19625I
u = 0.81149 1.93630I
a = 0.475078 0.173694I
b = 0.282250 0.670079I
4.32803 + 0.39404I 0.42763 + 3.09033I
u = 0.81149 + 1.93630I
a = 0.475078 + 0.173694I
b = 0.282250 + 0.670079I
4.32803 0.39404I 0.42763 3.09033I
u = 0.514151
a = 1.27475
b = 0.906827
1.17963 9.59748
u = 0.441193 1.061616I
a = 1.339016 0.042189I
b = 0.286214 + 0.854751I
1.94749 + 7.99248I 0.63462 7.50100I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.441193 + 1.061616I
a = 1.339016 + 0.042189I
b = 0.286214 0.854751I
1.94749 7.99248I 0.63462 + 7.50100I
u = 0.244986 0.494041I
a = 1.78295 1.55598I
b = 0.520353 + 0.534999I
1.62697 0.89360I 2.00328 3.90798I
u = 0.244986 + 0.494041I
a = 1.78295 + 1.55598I
b = 0.520353 0.534999I
1.62697 + 0.89360I 2.00328 + 3.90798I
u = 0.184074
a = 14.7227
b = 0.469315
0.389771 203.392
u = 0.173386 1.263771I
a = 0.954624 + 0.242109I
b = 0.208530 0.777299I
4.80506 + 1.60983I 3.59396 7.39228I
u = 0.173386 + 1.263771I
a = 0.954624 0.242109I
b = 0.208530 + 0.777299I
4.80506 1.60983I 3.59396 + 7.39228I
u = 0.031984 0.666688I
a = 0.397815 + 0.640557I
b = 0.538025 0.635820I
1.41050 + 1.53960I 1.08247 1.71398I
u = 0.031984 + 0.666688I
a = 0.397815 0.640557I
b = 0.538025 + 0.635820I
1.41050 1.53960I 1.08247 + 1.71398I
u = 0.005931 1.204396I
a = 0.198017 + 0.116361I
b = 0.791211 + 0.664753I
1.99327 + 4.92703I 6.26354 5.97246I
u = 0.005931 + 1.204396I
a = 0.198017 0.116361I
b = 0.791211 0.664753I
1.99327 4.92703I 6.26354 + 5.97246I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.314536 0.264463I
a = 2.39818 1.29354I
b = 1.81425 + 0.48435I
2.02262 7.30145I 9.29133 + 9.27741I
u = 0.314536 + 0.264463I
a = 2.39818 + 1.29354I
b = 1.81425 0.48435I
2.02262 + 7.30145I 9.29133 9.27741I
u = 0.352953 0.262356I
a = 1.80614 + 1.55531I
b = 1.242095 0.227160I
0.86533 2.68701I 3.94597 + 4.66832I
u = 0.352953 + 0.262356I
a = 1.80614 1.55531I
b = 1.242095 + 0.227160I
0.86533 + 2.68701I 3.94597 4.66832I
u = 0.385422 0.306091I
a = 1.25101 0.71324I
b = 0.775420 + 1.016884I
2.81417 + 0.85474I 6.39097 + 3.00432I
u = 0.385422 + 0.306091I
a = 1.25101 + 0.71324I
b = 0.775420 1.016884I
2.81417 0.85474I 6.39097 3.00432I
u = 0.479030 0.178046I
a = 1.02519 + 2.38226I
b = 0.419068 + 0.994844I
3.24917 2.13718I 0.66405 + 3.46334I
u = 0.479030 + 0.178046I
a = 1.02519 2.38226I
b = 0.419068 0.994844I
3.24917 + 2.13718I 0.66405 3.46334I
u = 0.540790 0.509640I
a = 1.05126 1.14986I
b = 0.112870 0.986911I
3.10319 + 2.00739I 0.07017 3.73576I
u = 0.540790 + 0.509640I
a = 1.05126 + 1.14986I
b = 0.112870 + 0.986911I
3.10319 2.00739I 0.07017 + 3.73576I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.656591 0.103431I
a = 0.13250 + 2.72776I
b = 0.047283 + 0.408643I
0.647397 + 0.825379I 13.47557 + 3.84496I
u = 0.656591 + 0.103431I
a = 0.13250 2.72776I
b = 0.047283 0.408643I
0.647397 0.825379I 13.47557 3.84496I
u = 0.694021 0.906910I
a = 0.753750 0.501984I
b = 1.02141 1.47167I
2.63315 3.07819I 2.31600 + 1.81054I
u = 0.694021 + 0.906910I
a = 0.753750 + 0.501984I
b = 1.02141 + 1.47167I
2.63315 + 3.07819I 2.31600 1.81054I
u = 0.728116 1.050760I
a = 0.909156 + 0.512571I
b = 1.23044 + 1.25774I
2.92948 5.60961I 4.14975 + 9.97823I
u = 0.728116 + 1.050760I
a = 0.909156 0.512571I
b = 1.23044 1.25774I
2.92948 + 5.60961I 4.14975 9.97823I
u = 0.819658 0.613048I
a = 0.404891 0.964109I
b = 0.521047 + 0.599991I
4.56427 + 0.06184I 9.79376 + 2.89093I
u = 0.819658 + 0.613048I
a = 0.404891 + 0.964109I
b = 0.521047 0.599991I
4.56427 0.06184I 9.79376 2.89093I
u = 0.843186 0.654005I
a = 0.652685 + 0.499980I
b = 0.693182 0.924086I
0.60848 4.96181I 4.51323 + 6.29782I
u = 0.843186 + 0.654005I
a = 0.652685 0.499980I
b = 0.693182 + 0.924086I
0.60848 + 4.96181I 4.51323 6.29782I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.887791 0.659627I
a = 0.981211 0.593925I
b = 0.911068 + 0.857587I
4.45627 9.81230I 7.61602 + 8.31170I
u = 0.887791 + 0.659627I
a = 0.981211 + 0.593925I
b = 0.911068 0.857587I
4.45627 + 9.81230I 7.61602 8.31170I
u = 0.891724 0.853255I
a = 1.092303 0.397025I
b = 0.92727 1.34544I
3.18307 7.41862I 1.90065 + 8.64551I
u = 0.891724 + 0.853255I
a = 1.092303 + 0.397025I
b = 0.92727 + 1.34544I
3.18307 + 7.41862I 1.90065 8.64551I
u = 0.92399 1.46999I
a = 0.706064 0.209188I
b = 0.993764 0.432758I
1.75235 6.26871I 10.64174 + 5.67281I
u = 0.92399 + 1.46999I
a = 0.706064 + 0.209188I
b = 0.993764 + 0.432758I
1.75235 + 6.26871I 10.64174 5.67281I
u = 0.929673 0.916332I
a = 0.952809 + 0.572168I
b = 0.81360 + 1.26582I
4.28857 3.85212I 0.611567 + 0.357094I
u = 0.929673 + 0.916332I
a = 0.952809 0.572168I
b = 0.81360 1.26582I
4.28857 + 3.85212I 0.611567 0.357094I
u = 1.36952 0.80230I
a = 0.475150 + 0.678348I
b = 0.373335 + 0.846791I
3.50470 2.65606I 0.26583 + 6.15253I
u = 1.36952 + 0.80230I
a = 0.475150 0.678348I
b = 0.373335 0.846791I
3.50470 + 2.65606I 0.26583 6.15253I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
9
+ u
8
u
7
u
6
u
5
u
4
+ 4u
3
+ 2u
2
2u 1)
(u
60
16u
58
+ ··· + 24u + 19)
c
2
(u
9
+ 4u
8
+ 10u
7
+ 20u
6
+ 34u
5
+ 42u
4
+ 30u
3
+ 9u
2
1)
(u
60
+ u
59
+ ··· + 252u + 29)
c
3
(u
9
+ u
7
4u
6
+ 4u
5
6u
4
+ 6u
3
2u
2
+ 2u 1)
(u
60
+ 3u
59
+ ··· + 100u 47)
c
4
(u
9
2u
8
u
7
3u
6
3u
5
5u
3
4u
2
u 1)
(u
60
+ u
59
+ ··· 295u 37)
c
5
(u
9
2u
8
+ 3u
7
7u
6
+ u
5
10u
4
4u
3
6u
2
4u 1)
(u
60
+ u
59
+ ··· 328u 49)
c
6
(u
9
+ u
8
+ ··· + 7u 1)(u
60
+ 5u
58
+ ··· + 9u + 1)
c
7
(u
9
u
8
+ 3u
7
+ u
6
+ u
5
+ 3u
4
+ 3u
3
+ 5u
2
+ 1)
(u
60
+ 2u
59
+ ··· + 74u 19)
c
8
(u
9
u
8
u
7
+ u
6
u
5
+ u
4
+ 4u
3
2u
2
2u + 1)
(u
60
16u
58
+ ··· + 24u + 19)
c
9
(u
9
+ 4u
8
+ ··· + u
3
+ 1)(u
60
+ 3u
59
+ ··· 16u 1)
c
10
(u
9
+ 2u
8
+ 3u
7
+ 7u
6
+ u
5
+ 10u
4
4u
3
+ 6u
2
4u + 1)
(u
60
+ u
59
+ ··· 328u 49)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y
9
3y
8
+ y
7
+ 11y
6
17y
5
+ y
4
+ 22y
3
22y
2
+ 8y 1)
(y
60
32y
59
+ ··· 1602y + 361)
c
2
(y
9
+ 4y
8
+ 8y
7
+ 4y
6
+ 4y
5
76y
4
+ 184y
3
+ 3y
2
+ 18y 1)
(y
60
5y
59
+ ··· 61300y + 841)
c
3
(y
9
+ 2y
8
+ 9y
7
+ 4y
6
16y
5
+ 20y
3
+ 8y
2
1)
(y
60
+ 13y
59
+ ··· + 47810y + 2209)
c
4
(y
9
6y
8
17y
7
13y
6
+ y
5
+ 4y
4
+ 25y
3
6y
2
7y 1)
(y
60
+ 9y
59
+ ··· 29527y + 1369)
c
5
, c
10
(y
9
+ 2y
8
17y
7
91y
6
195y
5
220y
4
126y
3
24y
2
+ 4y 1)
(y
60
+ 37y
59
+ ··· + 42258y + 2401)
c
6
(y
9
+ 7y
8
+ 20y
7
+ 23y
6
+ 5y
5
12y
4
36y
3
25y
2
+ 43y 1)
(y
60
+ 10y
59
+ ··· 45y + 1)
c
7
(y
9
+ 5y
8
+ 13y
7
+ 17y
6
+ 23y
5
11y
4
23y
3
31y
2
10y 1)
(y
60
+ 42y
58
+ ··· 7604y + 361)
c
8
(y
9
3y
8
+ y
7
+ 11y
6
17y
5
+ y
4
+ 22y
3
22y
2
+ 8y 1)
(y
60
32y
59
+ ··· 1602y + 361)
c
9
(y
9
6y
8
+ 5y
7
3y
6
+ 2y
5
8y
4
5y
3
4y
2
1)
(y
60
+ y
59
+ ··· 10y + 1)
12