10
131
(K10n
19
)
1
Arc Sequences
4 8 5 2 8 10 1 2 7 6
Solving Sequence
1,4 2,6
10 7 8 3 5 9
c
1
c
10
c
6
c
7
c
2
c
5
c
9
c
3
, c
4
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
3
+ b
2
+ 2b + 1, u 1, b + a + 1i
I
u
2
= hu
18
4u
17
+ ··· + 3u 1, u
17
3u
16
+ ··· + 4a + 5, u
17
3u
16
+ ··· + 4b + 1i
There are 2 irreducible components with 21 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
3
+ b
2
+ 2b + 1, u 1, b + a + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
6
=
b 1
b
a
10
=
b
2
+ b + 1
b
2
a
7
=
2b
2
2b 2
b
2
b 1
a
8
=
b
2
b 1
b
2
b 1
a
3
=
1
1
a
5
=
1
0
a
9
=
b
2
b 1
b
2
b 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5b
2
4b 16
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.56984
b = 0.569840
2.75839 15.3442
u = 1.00000
a = 1.21508 1.30714I
b = 0.215080 1.307141I
1.37919 2.82812I 6.82789 + 2.41717I
u = 1.00000
a = 1.21508 + 1.30714I
b = 0.215080 + 1.307141I
1.37919 + 2.82812I 6.82789 2.41717I
3
II.
I
u
2
= hu
18
4u
17
+· · ·+3u1, u
17
3u
16
+· · ·+4a+5, u
17
3u
16
+· · ·+4b+1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
1
4
u
17
+
3
4
u
16
+ ···
3
2
u
5
4
1
4
u
17
+
3
4
u
16
+ ··· +
1
2
u
1
4
a
10
=
11
4
u
17
+
35
4
u
16
+ ··· + 6u
9
4
2u
17
+
13
2
u
16
+ ··· +
9
2
u
5
2
a
7
=
5u
17
+ 18u
16
+ ··· + 11u 10
5
4
u
17
+
21
4
u
16
+ ··· + 4u
15
4
a
8
=
15
4
u
17
+
51
4
u
16
+ ··· + 7u
25
4
5
4
u
17
+
21
4
u
16
+ ··· + 4u
15
4
a
3
=
u
3
u
5
+ u
3
u
a
5
=
u
u
3
+ u
a
9
=
17
4
u
17
+
57
4
u
16
+ ··· + 8u
31
4
11
4
u
17
+
35
4
u
16
+ ··· + 5u
17
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
17
+
23
2
u
16
15u
15
33
2
u
14
+
127
2
u
13
91
2
u
12
68u
11
+
110u
10
11
2
u
9
175
2
u
8
2u
7
+ 53u
6
+ 27u
5
75u
4
14u
3
+ 41u
2
+
21
2
u
29
2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.189206 0.282581I
a = 1.43576 + 0.78808I
b = 0.051790 + 1.344455I
2.07423 1.22055I 3.51872 0.07112I
u = 1.189206 + 0.282581I
a = 1.43576 0.78808I
b = 0.051790 1.344455I
2.07423 + 1.22055I 3.51872 + 0.07112I
u = 1.10588
a = 1.27324
b = 0.334161
2.12974 1.01837
u = 0.509257 0.343539I
a = 0.118556 + 0.636675I
b = 0.380850 0.301405I
0.575696 1.116816I 6.38496 + 6.15764I
u = 0.509257 + 0.343539I
a = 0.118556 0.636675I
b = 0.380850 + 0.301405I
0.575696 + 1.116816I 6.38496 6.15764I
u = 0.405572 0.756937I
a = 0.511486 + 1.207991I
b = 0.12430 1.42145I
4.97233 2.95811I 1.13170 + 3.60082I
u = 0.405572 + 0.756937I
a = 0.511486 1.207991I
b = 0.12430 + 1.42145I
4.97233 + 2.95811I 1.13170 3.60082I
u = 0.441998
a = 2.38885
b = 0.662052
1.60276 5.18594
u = 0.550076 0.259421I
a = 2.10081 + 1.63815I
b = 0.262040 + 1.270410I
2.36168 + 3.34376I 0.22641 4.65236I
u = 0.550076 + 0.259421I
a = 2.10081 1.63815I
b = 0.262040 1.270410I
2.36168 3.34376I 0.22641 + 4.65236I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.841043 1.112378I
a = 0.218185 0.109146I
b = 0.21016 + 1.52072I
12.50877 2.04734I 0.610263 + 0.647242I
u = 0.841043 + 1.112378I
a = 0.218185 + 0.109146I
b = 0.21016 1.52072I
12.50877 + 2.04734I 0.610263 0.647242I
u = 0.889957 0.956699I
a = 0.639432 + 0.360381I
b = 0.658112 + 0.569573I
5.67221 + 1.09047I 3.82592 + 0.42258I
u = 0.889957 + 0.956699I
a = 0.639432 0.360381I
b = 0.658112 0.569573I
5.67221 1.09047I 3.82592 0.42258I
u = 1.023452 0.903197I
a = 1.289146 + 0.364109I
b = 0.740141 0.450201I
5.25155 + 5.76942I 4.89628 5.17142I
u = 1.023452 + 0.903197I
a = 1.289146 0.364109I
b = 0.740141 + 0.450201I
5.25155 5.76942I 4.89628 + 5.17142I
u = 1.13145 0.93287I
a = 1.70614 0.00878I
b = 0.27093 1.49425I
11.5470 + 9.4650I 1.80359 5.12935I
u = 1.13145 + 0.93287I
a = 1.70614 + 0.00878I
b = 0.27093 + 1.49425I
11.5470 9.4650I 1.80359 + 5.12935I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u + 1)
3
(u
18
+ 4u
17
+ ··· 3u 1)
c
2
, c
8
u
3
(u
18
+ u
17
+ ··· + 4u + 8)
c
3
(u 1)
3
(u
18
+ 4u
17
+ ··· + 11u + 1)
c
5
(u
3
+ u
2
1)(u
18
+ 2u
17
+ ··· 5u
2
+ 1)
c
6
, c
9
, c
10
(u
3
u
2
+ 2u 1)(u
18
+ 2u
17
+ ··· 2u 1)
c
7
(u
3
u
2
+ 1)(u
18
+ 2u
17
+ ··· + 18u 17)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
3
(y
18
4y
17
+ ··· 11y + 1)
c
2
, c
8
y
3
(y
18
21y
17
+ ··· 592y + 64)
c
3
(y 1)
3
(y
18
+ 24y
17
+ ··· 11y + 1)
c
5
(y
3
y
2
+ 2y 1)(y
18
+ 22y
17
+ ··· 10y + 1)
c
6
, c
9
, c
10
(y
3
+ 3y
2
+ 2y 1)(y
18
+ 18y
17
+ ··· 10y + 1)
c
7
(y
3
y
2
+ 2y 1)(y
18
+ 10y
17
+ ··· 1106y + 289)
8