10
163
(K10n
35
)
1
Arc Sequences
4 5 7 9 10 4 3 1 2 3
Solving Sequence
3,7
4
8,10
1 9 6 5 2
c
3
c
7
c
10
c
8
c
6
c
5
c
2
c
1
, c
4
, c
9
Representation Ideals
I =
4
\
i=1
I
u
i
\
I
v
1
I
u
1
= hb
8
4b
6
+ 2b
5
+ 3b
4
b
3
+ 3b
2
10b + 7, 3b
7
14b
5
b
4
+ 9b
3
+ 2b
2
+ 14b + 13u 29,
b
7
+ b
6
+ b
5
+ 4b
4
6b
3
4b
2
+ 2b + 13a 10i
I
u
2
= hb
8
+ b
7
2b
5
8b
4
7b
3
+ 13b
2
+ 8b + 3,
28b
7
25b
6
17b
5
+ 5b
4
+ 181b
3
+ 72b
2
293b + 203u 1,
44b
7
2b
6
+ 52b
5
+ 99b
4
+ 301b
3
36b
2
941b + 203a + 160i
I
u
3
= hu
6
2u
5
+ 4u
4
4u
3
+ 4u
2
u + 1, u
5
+ u
4
2u
3
+ a 2, u
5
+ 2u
4
4u
3
+ 4u
2
+ b 3u + 1i
I
u
4
= hu
14
+ 5u
13
+ 15u
12
+ 30u
11
+ 47u
10
+ 55u
9
+ 48u
8
+ 22u
7
2u
6
17u
5
15u
4
14u
3
4u
2
+ u + 5,
u
13
+ 5u
12
+ 15u
11
+ 31u
10
+ 50u
9
+ 63u
8
+ 61u
7
+ 42u
6
+ 17u
5
+ u
4
8u
3
9u
2
+ b 8u + 1,
u
13
10u
11
45u
10
108u
9
195u
8
267u
7
283u
6
212u
5
102u
4
20u
3
+ 26u
2
+ 5a + 41u + 41i
I
v
1
= hb + 1, v 1, ai
There are 5 irreducible components with 37 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
= hb
8
4b
6
+ 2b
5
+ 3b
4
b
3
+ 3b
2
10b + 7, 3b
7
+ 13u + · · · + 14b
29, b
7
+ b
6
+ · · · + 13a 10i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
0.230769b
7
+ 1.07692b
5
+ ··· 1.07692b + 2.23077
a
4
=
1
0.230769b
6
+ 0.307692b
5
+ ··· + 0.384615b 0.0769231
a
8
=
0.230769b
7
1.07692b
5
+ ··· + 1.07692b 2.23077
0.230769b
7
+ 1.07692b
5
+ ··· 1.07692b + 2.23077
a
10
=
0.0769231b
7
0.0769231b
6
+ ··· 0.153846b + 0.769231
b
a
1
=
0.0769231b
7
0.0769231b
6
+ ··· 1.15385b + 0.769231
b
a
9
=
0.615385b
7
+ 0.538462b
6
+ ··· + 1.76923b 3.46154
1
a
6
=
0.230769b
7
+ 1.07692b
5
+ ··· 1.07692b + 2.23077
0.307692b
7
+ 0.846154b
6
+ ··· + 1.84615b 0.923077
a
5
=
0.307692b
7
+ 0.384615b
6
+ ··· + 0.0769231b 0.769231
0.153846b
7
+ 0.307692b
6
+ ··· + 1.23077b 0.923077
a
2
=
0.461538b
7
+ 0.461538b
6
+ ··· + 0.923077b 1.61538
0.307692b
7
0.769231b
5
+ ··· + 0.769231b 1.30769
(ii) Obstruction class = 1
(iii) Cusp Shapes =
44
13
b
7
+ 4b
6
136
13
b
5
84
13
b
4
+
80
13
b
3
+
12
13
b
2
+
188
13
b
278
13
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.070696 + 0.758745I
a = 0.878270 + 0.343518I
b = 1.77170 0.19130I
5.03685 2.59539I 13.53952 + 0.91892I
u = 1.070696 0.758745I
a = 0.878270 0.343518I
b = 1.77170 + 0.19130I
5.03685 + 2.59539I 13.53952 0.91892I
u = 0.070696 + 0.758745I
a = 0.87053 + 1.13328I
b = 0.350716 1.044375I
1.74699 2.59539I 1.53952 + 0.91892I
u = 0.070696 0.758745I
a = 0.87053 1.13328I
b = 0.350716 + 1.044375I
1.74699 + 2.59539I 1.53952 0.91892I
u = 0.070696 + 0.758745I
a = 1.32191 0.58540I
b = 0.921412 0.580396I
1.74699 2.59539I 1.53952 + 0.91892I
u = 0.070696 0.758745I
a = 1.32191 + 0.58540I
b = 0.921412 + 0.580396I
1.74699 + 2.59539I 1.53952 0.91892I
u = 1.070696 0.758745I
a = 1.185830 + 0.661666I
b = 1.201002 0.298580I
5.03685 + 2.59539I 13.53952 0.91892I
u = 1.070696 + 0.758745I
a = 1.185830 0.661666I
b = 1.201002 + 0.298580I
5.03685 2.59539I 13.53952 + 0.91892I
3
II. I
u
2
= hb
8
+ b
7
2b
5
8b
4
7b
3
+ 13b
2
+ 8b + 3, 28b
7
+ 203u + · · ·
293b 1, 44b
7
2b
6
+ · · · + 203a + 160i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
0.137931b
7
+ 0.123153b
6
+ ··· + 1.44335b + 0.00492611
a
4
=
1
0.00985222b
7
+ 0.182266b
6
+ ··· 1.74384b 0.581281
a
8
=
0.137931b
7
0.123153b
6
+ ··· 1.44335b 0.00492611
0.137931b
7
+ 0.123153b
6
+ ··· + 1.44335b + 0.00492611
a
10
=
0.216749b
7
+ 0.00985222b
6
+ ··· + 4.63547b 0.788177
b
a
1
=
0.216749b
7
+ 0.00985222b
6
+ ··· + 3.63547b 0.788177
b
a
9
=
0.674877b
7
0.556650b
6
+ ··· 10.4039b 4.32512
1
a
6
=
0.137931b
7
+ 0.123153b
6
+ ··· + 1.44335b + 0.00492611
0.216749b
7
+ 0.438424b
6
+ ··· + 0.778325b + 0.0689655
a
5
=
0.330049b
7
0.605911b
6
+ ··· 2.58128b 4.52709
0.0886700b
7
+ 0.0689655b
6
+ ··· + 1.44828b 0.374384
a
2
=
0.197044b
7
0.0689655b
6
+ ··· + 2.55172b 2.05419
0.152709b
7
+ 0.0394089b
6
+ ··· + 2.54187b + 0.418719
(ii) Obstruction class = 1
(iii) Cusp Shapes =
264
203
b
7
12
203
b
6
+
312
203
b
5
+
1000
203
b
4
+
316
29
b
3
+
596
203
b
2
6052
203
b
2694
203
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.93338 + 1.13249I
a = 0.692709 + 0.652910I
b = 1.47465 0.63084I
3.88602 4.68603I 7.29059 + 10.27938I
u = 0.93338 1.13249I
a = 0.692709 0.652910I
b = 1.47465 + 0.63084I
3.88602 + 4.68603I 7.29059 10.27938I
u = 0.433380 0.525827I
a = 2.14039 1.49390I
b = 0.269251 0.341177I
0.59615 4.68603I 4.70941 + 10.27938I
u = 0.433380 + 0.525827I
a = 2.14039 + 1.49390I
b = 0.269251 + 0.341177I
0.59615 + 4.68603I 4.70941 10.27938I
u = 0.433380 0.525827I
a = 0.637691 0.013525I
b = 0.14207 1.77290I
0.59615 4.68603I 4.70941 + 10.27938I
u = 0.433380 + 0.525827I
a = 0.637691 + 0.013525I
b = 0.14207 + 1.77290I
0.59615 + 4.68603I 4.70941 10.27938I
u = 0.93338 1.13249I
a = 0.970793 + 0.502018I
b = 1.385972 0.175069I
3.88602 + 4.68603I 7.29059 10.27938I
u = 0.93338 + 1.13249I
a = 0.970793 0.502018I
b = 1.385972 + 0.175069I
3.88602 4.68603I 7.29059 + 10.27938I
5
III. I
u
3
= hu
6
2u
5
+ 4u
4
4u
3
+ 4u
2
u + 1, u
5
+ u
4
2u
3
+ a
2, u
5
+ 2u
4
4u
3
+ 4u
2
+ b 3u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
8
=
u
u
a
10
=
u
5
u
4
+ 2u
3
+ 2
u
5
2u
4
+ 4u
3
4u
2
+ 3u 1
a
1
=
u
4
2u
3
+ 4u
2
3u + 3
u
5
2u
4
+ 4u
3
4u
2
+ 3u 1
a
9
=
2u
5
+ 4u
4
7u
3
+ 7u
2
6u
u
5
u
4
+ 2u
3
u
2
+ u + 1
a
6
=
u
u
3
+ u
a
5
=
u
5
+ 2u
4
4u
3
+ 3u
2
2u 1
u
2
u + 1
a
2
=
u
5
u
4
+ u
3
+ u
2
u + 3
u
4
+ 2u
3
3u
2
+ 2u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
5
+ 12u
4
19u
3
+ 23u
2
16u + 6
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.034417 0.580231I
a = 1.93912 + 0.34991I
b = 0.136288 1.137180I
1.27956 3.69612I 0.43558 + 6.39872I
u = 0.034417 + 0.580231I
a = 1.93912 0.34991I
b = 0.136288 + 1.137180I
1.27956 + 3.69612I 0.43558 6.39872I
u = 0.096993 1.308888I
a = 0.425821 0.085567I
b = 0.153300 + 0.549053I
4.36362 + 4.05299I 4.55288 5.52472I
u = 0.096993 + 1.308888I
a = 0.425821 + 0.085567I
b = 0.153300 0.549053I
4.36362 4.05299I 4.55288 + 5.52472I
u = 0.937424 0.916243I
a = 1.013299 0.581830I
b = 1.48299 + 0.38301I
3.99825 + 3.41127I 5.61730 2.91658I
u = 0.937424 + 0.916243I
a = 1.013299 + 0.581830I
b = 1.48299 0.38301I
3.99825 3.41127I 5.61730 + 2.91658I
7
IV.
I
u
4
= hu
14
+5u
13
+· · ·+u+5, u
13
+5u
12
+· · ·+b+1, u
13
10u
11
+· · ·+5a+41i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
8
=
u
u
a
10
=
1
5
u
13
+ 2u
11
+ ···
41
5
u
41
5
u
13
5u
12
+ ··· + 8u 1
a
1
=
4
5
u
13
+ 5u
12
+ ···
81
5
u
36
5
u
13
5u
12
+ ··· + 8u 1
a
9
=
11
5
u
13
+ 8u
12
+ ··· +
61
5
u +
51
5
u
13
u
12
+ ··· 14u + 6
a
6
=
u
u
3
+ u
a
5
=
9
5
u
13
8u
12
+ ··· +
36
5
u +
26
5
u
13
4u
12
+ ··· 6u 9
a
2
=
1
5
u
13
u
12
+ ···
16
5
u
16
5
2u
12
9u
11
+ ··· + 14u + 4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
13
5u
12
+ 2u
11
+ 43u
10
+ 98u
9
+ 192u
8
+ 233u
7
+ 231u
6
+
106u
5
+ 55u
4
27u
3
21u
2
60u 6
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.104556 0.803929I
a = 0.895316 0.749503I
b = 1.59147 0.10810I
6.35421 + 6.00703I 6.42492 3.68584I
u = 1.104556 + 0.803929I
a = 0.895316 + 0.749503I
b = 1.59147 + 0.10810I
6.35421 6.00703I 6.42492 + 3.68584I
u = 0.91106 1.12096I
a = 1.113573 0.631511I
b = 1.72243 + 0.67293I
5.3164 13.2900I 4.73276 + 7.55975I
u = 0.91106 + 1.12096I
a = 1.113573 + 0.631511I
b = 1.72243 0.67293I
5.3164 + 13.2900I 4.73276 7.55975I
u = 0.809699 0.855443I
a = 1.48215 + 0.63750I
b = 1.74544 0.75171I
4.94416 4.48113I 10.56248 + 7.82532I
u = 0.809699 + 0.855443I
a = 1.48215 0.63750I
b = 1.74544 + 0.75171I
4.94416 + 4.48113I 10.56248 7.82532I
u = 0.752287 0.954057I
a = 0.769408 + 1.137546I
b = 1.66410 + 0.12170I
4.62410 1.43381I 9.01327 1.28996I
u = 0.752287 + 0.954057I
a = 0.769408 1.137546I
b = 1.66410 0.12170I
4.62410 + 1.43381I 9.01327 + 1.28996I
u = 0.17524 1.43298I
a = 0.028144 0.275446I
b = 0.389777 + 0.088598I
3.73877 + 3.84212I 7.98139 1.57763I
u = 0.17524 + 1.43298I
a = 0.028144 + 0.275446I
b = 0.389777 0.088598I
3.73877 3.84212I 7.98139 + 1.57763I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.269018 0.823102I
a = 0.678836 0.196935I
b = 0.020522 + 0.611730I
0.79193 + 2.01282I 1.55516 4.15380I
u = 0.269018 + 0.823102I
a = 0.678836 + 0.196935I
b = 0.020522 0.611730I
0.79193 2.01282I 1.55516 + 4.15380I
u = 0.633342 0.004347I
a = 0.435692 0.534977I
b = 0.273616 + 0.340717I
1.38615 + 0.45192I 8.23002 1.56844I
u = 0.633342 + 0.004347I
a = 0.435692 + 0.534977I
b = 0.273616 0.340717I
1.38615 0.45192I 8.23002 + 1.56844I
10
V. I
v
1
= hb + 1, v 1, ai
(i) Arc colorings
a
3
=
1
0
a
7
=
1
0
a
4
=
1
0
a
8
=
1
0
a
10
=
0
1
a
1
=
1
1
a
9
=
0
1
a
6
=
1
0
a
5
=
1
1
a
2
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
11
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
1.64493 6.00000
12
VI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u + 1)(u
6
+ u
5
+ 2u
4
+ u
3
+ u
2
+ 1)
(u
8
2u
7
+ 4u
6
+ 4u
5
+ 3u
4
+ 11u
3
+ 17u
2
+ 12u + 9)
(u
8
+ u
7
+ 2u
6
8u
5
+ 6u
4
3u
3
+ 9u
2
2u + 1)
(u
14
+ 4u
12
+ ··· 2u + 3)
c
2
, c
4
(u 1)(u
6
+ u
4
+ ··· u + 1)(u
8
+ 4u
5
+ ··· + 2u + 1)
(u
8
+ u
7
+ ··· 2u + 3)(u
14
+ u
13
+ ··· + 3u + 1)
c
3
u(u
4
+ u
3
+ u
2
u + 1)
2
(u
4
+ 2u
3
+ 2u
2
+ u + 1)
2
(u
6
2u
5
+ ··· u + 1)(u
14
5u
13
+ ··· u + 5)
c
6
, c
7
u(u
4
+ u
3
+ u
2
u + 1)
2
(u
4
+ 2u
3
+ 2u
2
+ u + 1)
2
(u
6
+ 2u
5
+ ··· + u + 1)(u
14
5u
13
+ ··· u + 5)
c
8
, c
10
(u + 1)(u
6
3u
5
+ 4u
4
5u
3
+ 5u
2
2u + 1)
(u
8
4u
6
+ 2u
5
+ 3u
4
u
3
+ 3u
2
10u + 7)
(u
8
+ u
7
+ ··· + 8u + 3)(u
14
10u
12
+ ··· 2u + 1)
c
9
u(u
4
2u
3
+ 2u
2
u + 1)
2
(u
4
u
3
+ u
2
+ u + 1)
2
(u
6
+ 3u
5
+ ··· u
2
+ 1)(u
14
+ 10u
13
+ ··· + 28u + 5)
13
VII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)(y
6
+ 3y
5
+ 4y
4
+ 5y
3
+ 5y
2
+ 2y + 1)
(y
8
+ 3y
7
+ 32y
6
16y
5
+ 30y
4
+ 71y
3
+ 81y
2
+ 14y + 1)
(y
8
+ 4y
7
+ 38y
6
+ 86y
5
+ 123y
4
43y
3
+ 79y
2
+ 162y + 81)
(y
14
+ 8y
13
+ ··· + 62y + 9)
c
2
, c
4
(y 1)(y
6
+ 2y
5
+ 5y
4
+ 5y
3
+ 4y
2
+ 3y + 1)
(y
8
+ 14y
6
14y
5
+ 27y
4
11y
3
+ 3y
2
2y + 1)
(y
8
+ 3y
7
+ 16y
6
+ 4y
5
+ 34y
4
13y
3
+ 37y
2
10y + 9)
(y
14
5y
13
+ ··· 13y + 1)
c
3
, c
6
, c
7
y(y
4
+ 2y
2
+ 3y + 1)
2
(y
4
+ y
3
+ 5y
2
+ y + 1)
2
(y
6
+ 4y
5
+ ··· + 7y + 1)(y
14
+ 5y
13
+ ··· 41y + 25)
c
8
, c
10
(y 1)(y
6
y
5
4y
4
+ 5y
3
+ 13y
2
+ 6y + 1)
(y
8
8y
7
+ 22y
6
22y
5
+ 3y
4
+ y
3
+ 31y
2
58y + 49)
(y
8
y
7
12y
6
+ 36y
5
+ 26y
4
225y
3
+ 233y
2
+ 14y + 9)
(y
14
20y
13
+ ··· 6y + 1)
c
9
y(y
4
+ 2y
2
+ 3y + 1)
2
(y
4
+ y
3
+ 5y
2
+ y + 1)
2
(y
6
y
5
+ ··· 2y + 1)(y
14
+ 26y
12
+ ··· + 246y + 25)
14