10
50
(K10a
82
)
1
Arc Sequences
6 10 7 8 9 1 5 4 2 3
Solving Sequence
2,10 3,6
1 7 4 9 5 8
c
2
c
1
c
6
c
3
c
9
c
5
c
8
c
4
, c
7
, c
10
Representation Ideals
I = I
u
1
\
I
v
1
I
u
1
= hu
29
+ u
28
+ ··· 4u 8, 2.35533 × 10
21
u
28
1.08295 × 10
21
u
27
+ ··· + 3.21297 × 10
22
b + 4.99760 × 10
22
,
1.40279 × 10
22
u
28
1.34020 × 10
22
u
27
+ ··· + 1.28519 × 10
23
a + 3.78490 × 10
21
i
I
v
1
= hb + 1, v
3
v
2
+ 1, ai
There are 2 irreducible components with 32 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
29
+u
28
+· · ·4u 8, 2.36×10
21
u
28
1.08× 10
21
u
27
+· · ·+3.21 ×10
22
b +
5.00× 10
22
, 1.40× 10
22
u
28
1.34 × 10
22
u
27
+· · · +1.29 × 10
23
a + 3.78 × 10
21
i
(i) Arc colorings
a
2
=
1
0
a
10
=
0.109151u
28
+ 0.104281u
27
+ ··· + 1.05028u 0.0294502
0.0733071u
28
+ 0.0337056u
27
+ ··· + 2.66211u 1.55545
a
3
=
0.109151u
28
+ 0.104281u
27
+ ··· + 1.05028u 0.0294502
0.0977462u
28
0.105212u
27
+ ··· 1.80838u + 1.51649
a
6
=
0
u
a
1
=
1
u
2
a
7
=
u
u
3
+ u
a
4
=
0.0742290u
28
0.0130112u
27
+ ··· + 2.33206u 0.385222
0.0403995u
28
0.0272617u
27
+ ··· 1.13546u + 0.501755
a
9
=
0.182458u
28
+ 0.137986u
27
+ ··· + 3.71239u 1.58490
0.0733071u
28
+ 0.0337056u
27
+ ··· + 2.66211u 1.55545
a
5
=
0.0801372u
28
+ 0.0142674u
27
+ ··· 1.32021u + 0.654627
0.0123357u
28
0.0384861u
27
+ ··· + 1.53266u + 0.0912353
a
8
=
0.129825u
28
+ 0.142058u
27
+ ··· + 2.93148u 0.254056
0.0232146u
28
+ 0.0398460u
27
+ ··· + 2.23303u 0.796703
(ii) Obstruction class = 1
(iii) Cusp Shapes =
9329261216550331417115
64259342851074266550324
u
28
+
21591074691190881271517
64259342851074266550324
u
27
+ ···
133668533129886119618135
10709890475179044425054
u
29447886342587988197125
16064835712768566637581
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.298970 0.143296I
a = 0.405123 0.011225I
b = 1.46649 0.06834I
3.48935 4.29283I 8.53955 + 3.19264I
u = 1.298970 + 0.143296I
a = 0.405123 + 0.011225I
b = 1.46649 + 0.06834I
3.48935 + 4.29283I 8.53955 3.19264I
u = 0.63881 1.44580I
a = 1.39978 0.79345I
b = 1.54068 0.30648I
7.67865 + 11.19889I 9.19156 6.17598I
u = 0.63881 + 1.44580I
a = 1.39978 + 0.79345I
b = 1.54068 + 0.30648I
7.67865 11.19889I 9.19156 + 6.17598I
u = 0.538894 0.689414I
a = 0.718190 + 0.375236I
b = 0.093803 + 0.571484I
4.46963 + 2.10537I 0.57633 3.98592I
u = 0.538894 + 0.689414I
a = 0.718190 0.375236I
b = 0.093803 0.571484I
4.46963 2.10537I 0.57633 + 3.98592I
u = 0.525371
a = 1.43709
b = 0.304151
1.50367 5.88400
u = 0.479164
a = 0.476311
b = 1.09947
2.07267 2.13092
u = 0.45548 1.52023I
a = 1.52448 0.57596I
b = 1.57403 0.21687I
9.09072 + 1.97634I 10.56391 0.15391I
u = 0.45548 + 1.52023I
a = 1.52448 + 0.57596I
b = 1.57403 + 0.21687I
9.09072 1.97634I 10.56391 + 0.15391I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.215027 1.248976I
a = 0.502392 + 0.258146I
b = 0.574714 + 0.809142I
5.26114 + 2.70743I 11.83350 3.32702I
u = 0.215027 + 1.248976I
a = 0.502392 0.258146I
b = 0.574714 0.809142I
5.26114 2.70743I 11.83350 + 3.32702I
u = 0.112616 1.303260I
a = 2.10664 0.24142I
b = 1.46854 0.05369I
6.34917 + 2.02688I 11.64196 3.46616I
u = 0.112616 + 1.303260I
a = 2.10664 + 0.24142I
b = 1.46854 + 0.05369I
6.34917 2.02688I 11.64196 + 3.46616I
u = 0.063501 1.233244I
a = 0.496703 0.223803I
b = 0.673518 0.754049I
1.62082 + 1.51334I 8.49380 0.41799I
u = 0.063501 + 1.233244I
a = 0.496703 + 0.223803I
b = 0.673518 + 0.754049I
1.62082 1.51334I 8.49380 + 0.41799I
u = 0.175226 0.644435I
a = 0.714606 0.159961I
b = 0.332600 0.298296I
0.389560 0.938777I 6.80996 + 7.32576I
u = 0.175226 + 0.644435I
a = 0.714606 + 0.159961I
b = 0.332600 + 0.298296I
0.389560 + 0.938777I 6.80996 7.32576I
u = 0.257766 1.113062I
a = 2.39745 + 0.78028I
b = 1.377159 + 0.122752I
0.14603 4.37313I 7.64888 + 4.01970I
u = 0.257766 + 1.113062I
a = 2.39745 0.78028I
b = 1.377159 0.122752I
0.14603 + 4.37313I 7.64888 4.01970I
4
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.338315 1.255878I
a = 0.503754 0.288020I
b = 0.496046 0.855361I
1.04610 6.94187I 7.09973 + 6.05967I
u = 0.338315 + 1.255878I
a = 0.503754 + 0.288020I
b = 0.496046 + 0.855361I
1.04610 + 6.94187I 7.09973 6.05967I
u = 0.56484 1.49174I
a = 1.44633 + 0.69483I
b = 1.56175 + 0.26987I
12.27836 6.66801I 13.30046 + 3.89200I
u = 0.56484 + 1.49174I
a = 1.44633 0.69483I
b = 1.56175 0.26987I
12.27836 + 6.66801I 13.30046 3.89200I
u = 0.603790 0.612719I
a = 0.465663 + 0.062524I
b = 1.109445 + 0.283231I
1.43725 + 1.10103I 6.03106 + 0.28755I
u = 0.603790 + 0.612719I
a = 0.465663 0.062524I
b = 1.109445 0.283231I
1.43725 1.10103I 6.03106 0.28755I
u = 0.607413 0.112242I
a = 1.40896 0.45087I
b = 0.356186 0.206024I
2.53302 + 3.25312I 0.46847 3.58405I
u = 0.607413 + 0.112242I
a = 1.40896 + 0.45087I
b = 0.356186 + 0.206024I
2.53302 3.25312I 0.46847 + 3.58405I
u = 1.30242
a = 0.405165
b = 1.46813
7.43008 12.5867
5
II. I
v
1
= hb + 1, v
3
v
2
+ 1, ai
(i) Arc colorings
a
2
=
1
0
a
10
=
0
1
a
3
=
1
1
a
6
=
v
0
a
1
=
1
0
a
7
=
v
0
a
4
=
v
2
+ 1
1
a
9
=
1
1
a
5
=
0
v
a
8
=
v
v
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = v
2
+ 5v 11
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 0.754878
a = 0
b = 1.00000
2.75839 15.3442
v = 0.877439 0.744862I
a = 0
b = 1.00000
1.37919 + 2.82812I 6.82789 2.41717I
v = 0.877439 + 0.744862I
a = 0
b = 1.00000
1.37919 2.82812I 6.82789 + 2.41717I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
u
3
(u
29
+ u
28
+ ··· 4u 8)
c
2
, c
9
, c
10
(u 1)
3
(u
29
+ 4u
28
+ ··· + 2u + 1)
c
3
(u
3
+ u
2
1)(u
29
+ 2u
28
+ ··· 15u 9)
c
4
, c
8
(u
3
+ u
2
+ 2u + 1)(u
29
+ 2u
28
+ ··· + u + 1)
c
5
(u
3
u
2
+ 1)(u
29
+ 2u
28
+ ··· 15u 9)
c
7
(u
3
u
2
+ 2u 1)(u
29
+ 2u
28
+ ··· + u + 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
y
3
(y
29
+ 21y
28
+ ··· + 144y 64)
c
2
, c
9
(y 1)
3
(y
29
30y
28
+ ··· + 18y 1)
c
3
(y
3
y
2
+ 2y 1)(y
29
24y
28
+ ··· + 621y 81)
c
4
, c
8
(1y 0.32471796)(1.00000y
2
+ 3.324718y + 3.07960)
(1y
29
+ 24.000000y
28
+ ··· + 13.0000000y 1.00000000)
c
5
+ 1.0000000(1y
3
y
2
+ 2.0000000y 1.00000000)
(1y
29
24.000000y
28
+ ··· + 621.00000y 81.000000)
c
7
(y
3
+ 3y
2
+ 2y 1)(y
29
+ 24y
28
+ ··· + 13y 1)
c
10
1.0000000(1y 1.00000000)
3
(1y
29
30.000000y
28
+ ··· + 18.0000000y 1.00000000)
9