10
162
(K10n
40
)
1
Arc Sequences
5 7 9 8 10 2 1 4 2 7
Solving Sequence
3,9 4,7
2 10 1 6 5 8
c
3
c
2
c
9
c
10
c
6
c
5
c
8
c
1
, c
4
, c
7
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
5
+ u
3
u
2
u + 1, a + 1, u
4
+ u
3
+ 2u
2
+ b 1i
I
u
2
= hu
10
+ 7u
8
+ u
7
+ 20u
6
+ 6u
5
+ 25u
4
+ 8u
3
+ 10u
2
+ 2u + 1, a + 1,
13u
9
+ 5u
8
86u
7
+ 18u
6
224u
5
+ 4u
4
222u
3
2u
2
+ 9b 33u + 4i
I
u
3
= hu
12
u
11
+ 4u
10
5u
9
+ 2u
8
9u
7
5u
6
+ 7u
5
4u
4
+ 42u
3
+ 2u
2
+ 26u 1,
3342u
11
+ 1642u
10
+ ··· + 100385b + 49662,
902014u
11
+ 814306u
10
+ ··· + 702695a 22886944i
There are 3 irreducible components with 27 representations.
1
The knot diagram image is adapter from “C. Livingston and A. H. Mo ore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= hu
5
+ u
3
u
2
u + 1, a + 1, u
4
+ u
3
+ 2u
2
+ b 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
1
u
4
u
3
2u
2
+ 1
a
4
=
u
4
+ u
3
+ 2u
2
u
4
+ u
2
u 2
a
7
=
0
u
a
2
=
1
u
2
a
10
=
u
4
+ u
3
+ u
2
2
u
4
u
3
u
2
a
1
=
u
4
+ u
3
+ u
2
2
u
4
u
3
u
2
+ 1
a
6
=
u
u
3
+ u
a
5
=
u
4
+ u
2
2u 1
u
4
u
3
u
2
+ 1
a
8
=
2u
4
u
3
3u
2
+ u + 2
u
4
+ u
3
+ 2u
2
+ u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
3
+ 3u
2
+ u + 9
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.862442
a = 1.00000
b = 0.399372
3.66375 11.0105
u = 0.286139 1.377343I
a = 1.00000
b = 0.351694 0.989493I
2.68365 + 1.36579I 1.66321 1.28728I
u = 0.286139 + 1.377343I
a = 1.00000
b = 0.351694 + 0.989493I
2.68365 1.36579I 1.66321 + 1.28728I
u = 0.717360 0.267040I
a = 1.00000
b = 0.15201 + 1.49915I
9.07644 + 2.10101I 10.83155 1.02320I
u = 0.717360 + 0.267040I
a = 1.00000
b = 0.15201 1.49915I
9.07644 2.10101I 10.83155 + 1.02320I
3
II. I
u
2
= hu
10
+ 7u
8
+ · · · + 2u + 1, a + 1, 13u
9
+ 5u
8
+ · · · + 9b + 4i
(i) Arc colorings
a
3
=
1
0
a
9
=
1
13
9
u
9
5
9
u
8
+ ··· +
11
3
u
4
9
a
4
=
13
9
u
9
+
5
9
u
8
+ ···
11
3
u +
13
9
4
3
u
9
+
1
3
u
8
+ ··· + 7u +
5
3
a
7
=
0
u
a
2
=
1
u
2
a
10
=
13
9
u
9
+
5
9
u
8
+ ···
11
3
u
5
9
2u
9
u
8
+ 13u
7
4u
6
+ 33u
5
4u
4
+ 31u
3
2u
2
+ 4u 1
a
1
=
13
9
u
9
+
5
9
u
8
+ ···
11
3
u
5
9
13
9
u
9
5
9
u
8
+ ··· +
11
3
u
4
9
a
6
=
u
u
3
+ u
a
5
=
5
9
u
9
+
5
9
u
8
+ ··· +
4
3
u +
13
9
u
9
u
8
6u
7
7u
6
16u
5
19u
4
19u
3
16u
2
4u 2
a
8
=
1
9
u
9
8
9
u
8
+ ···
10
3
u
28
9
2
3
u
9
+
1
3
u
8
+ ··· + 2u +
5
3
(ii) Obstruction class = 1
(iii) Cusp Shapes =
19
3
u
9
5
3
u
8
+
131
3
u
7
6u
6
+
362
3
u
5
+
8
3
u
4
+ 134u
3
+
17
3
u
2
+ 31u +
5
3
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.28935 1.60768I
a = 1.00000
b = 1.012223 0.338453I
3.41629 + 5.60135I 2.31471 5.03009I
u = 0.28935 + 1.60768I
a = 1.00000
b = 1.012223 + 0.338453I
3.41629 5.60135I 2.31471 + 5.03009I
u = 0.154622 1.375598I
a = 1.00000
b = 0.728811 1.048010I
1.336136 + 0.440636I 5.86082 + 0.80149I
u = 0.154622 + 1.375598I
a = 1.00000
b = 0.728811 + 1.048010I
1.336136 0.440636I 5.86082 0.80149I
u = 0.148293 0.384893I
a = 1.00000
b = 0.246304 0.538008I
0.201388 + 1.011141I 3.39938 6.83831I
u = 0.148293 + 0.384893I
a = 1.00000
b = 0.246304 + 0.538008I
0.201388 1.011141I 3.39938 + 6.83831I
u = 0.061135 0.618058I
a = 1.00000
b = 0.10841 + 1.59746I
7.86026 2.34852I 3.25800 + 2.98056I
u = 0.061135 + 0.618058I
a = 1.00000
b = 0.10841 1.59746I
7.86026 + 2.34852I 3.25800 2.98056I
u = 0.65340 1.59789I
a = 1.00000
b = 0.40425 + 1.49580I
2.44804 10.69336I 5.16708 + 5.74333I
u = 0.65340 + 1.59789I
a = 1.00000
b = 0.40425 1.49580I
2.44804 + 10.69336I 5.16708 5.74333I
5
III. I
u
3
= hu
12
u
11
+ · · · + 26u 1, 3342u
11
+ 1642u
10
+ · · · + 100385b +
49662, 9.02 × 10
5
u
11
+ 8.14 × 10
5
u
10
+ · · · + 7.03 × 10
5
a 2.29 × 10
7
i
(i) Arc colorings
a
3
=
1
0
a
9
=
1.28365u
11
1.15883u
10
+ ··· + 7.27260u + 32.5702
0.0332918u
11
0.0163570u
10
+ ··· 1.94056u 0.494715
a
4
=
0.872563u
11
+ 0.864391u
10
+ ··· 2.95672u 17.5988
0.122568u
11
0.0696419u
10
+ ··· + 1.09534u + 0.277372
a
7
=
0
u
a
2
=
1
u
2
a
10
=
1.23460u
11
1.10414u
10
+ ··· + 7.25158u + 33.1898
0.0533375u
11
0.127816u
10
+ ··· 2.13642u 0.489069
a
1
=
1.23460u
11
1.10414u
10
+ ··· + 7.25158u + 33.1898
0.0490455u
11
0.0546923u
10
+ ··· + 0.0210233u 0.619532
a
6
=
u
u
3
+ u
a
5
=
1.61953u
11
+ 1.66858u
10
+ ··· 2.77131u 42.0868
0.0892763u
11
0.0859989u
10
+ ··· 0.845216u + 0.782657
a
8
=
1.70881u
11
1.75458u
10
+ ··· + 1.92609u + 42.8695
0.0892763u
11
+ 0.0859989u
10
+ ··· + 0.845216u 0.782657
(ii) Obstruction class = 1
(iii) Cusp Shapes =
35848
100385
u
11
+
34532
100385
u
10
+ ··· +
19964
5905
u
113498
100385
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.25968
a = 0.0303719
b = 0.569840
2.83439 1.01951
u = 0.690381 1.179909I
a = 1.090317 + 0.512447I
b = 0.215080 + 1.307141I
0.92371 + 2.82812I 5.50976 2.97945I
u = 0.690381 + 1.179909I
a = 1.090317 0.512447I
b = 0.215080 1.307141I
0.92371 2.82812I 5.50976 + 2.97945I
u = 0.080877 0.888165I
a = 0.20145 1.70790I
b = 0.215080 + 1.307141I
6.97197 + 2.82812I 5.50976 2.97945I
u = 0.080877 + 0.888165I
a = 0.20145 + 1.70790I
b = 0.215080 1.307141I
6.97197 2.82812I 5.50976 + 2.97945I
u = 0.0382588
a = 32.9252
b = 0.569840
2.83439 1.01951
u = 0.14809 1.64026I
a = 0.751221 + 0.353072I
b = 0.215080 1.307141I
0.92371 2.82812I 5.50976 + 2.97945I
u = 0.14809 + 1.64026I
a = 0.751221 0.353072I
b = 0.215080 + 1.307141I
0.92371 + 2.82812I 5.50976 2.97945I
u = 0.23327 1.46064I
a = 0.950258 0.311465I
b = 0.569840
5.06130 1.01951
u = 0.23327 + 1.46064I
a = 0.950258 + 0.311465I
b = 0.569840
5.06130 1.01951
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.50060 0.31705I
a = 0.068114 + 0.577481I
b = 0.215080 + 1.307141I
6.97197 + 2.82812I 5.50976 2.97945I
u = 1.50060 + 0.31705I
a = 0.068114 0.577481I
b = 0.215080 1.307141I
6.97197 2.82812I 5.50976 + 2.97945I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
(u
5
u
4
u
3
+ u
2
+ 1)
(u
10
+ u
9
3u
8
3u
7
+ 7u
6
+ 5u
5
6u
4
4u
3
+ 3u
2
+ 3u + 1)
(u
12
+ u
11
+ ··· 10u 11)
c
2
, c
5
(u
5
+ u
3
u
2
u + 1)
(u
10
+ 7u
8
u
7
+ 20u
6
6u
5
+ 25u
4
8u
3
+ 10u
2
2u + 1)
(u
12
+ u
11
+ ··· 26u 1)
c
3
, c
4
(1 + 2u u
2
+ u
3
)
4
(u
5
+ 3u
3
+ 2u + 1)(u
10
+ 5u
9
+ ··· + 18u + 4)
c
6
(u
5
+ u
3
+ u
2
u 1)
(u
10
+ 7u
8
u
7
+ 20u
6
6u
5
+ 25u
4
8u
3
+ 10u
2
2u + 1)
(u
12
+ u
11
+ ··· 26u 1)
c
8
(1 + 2u u
2
+ u
3
)
4
(u
5
+ 3u
3
+ 2u 1)(u
10
+ 5u
9
+ ··· + 18u + 4)
c
9
(u
2
u 1)
6
(u
5
2u
4
+ ··· + 2u + 1)(u
10
+ 9u
9
+ ··· + 20u + 8)
c
10
(u
5
+ u
4
u
3
u
2
1)
(u
10
+ u
9
3u
8
3u
7
+ 7u
6
+ 5u
5
6u
4
4u
3
+ 3u
2
+ 3u + 1)
(u
12
+ u
11
+ ··· 10u 11)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
, c
10
(y
5
3y
4
+ ··· 2y 1)(y
10
7y
9
+ ··· 3y + 1)
(y
12
5y
11
+ ··· 452y + 121)
c
2
, c
5
, c
6
(y
5
+ 2y
4
+ ··· + 3y 1)(y
10
+ 14y
9
+ ··· + 16y + 1)
(y
12
+ 7y
11
+ ··· 680y + 1)
c
3
, c
4
, c
8
(y
3
+ 3y
2
+ 2y 1)
4
(y
5
+ 6y
4
+ 13y
3
+ 12y
2
+ 4y 1)
(y
10
+ 9y
9
+ ··· + 68y + 16)
c
9
(y
2
3y + 1)
6
(y
5
2y
4
3y
3
+ 4y
2
+ 8y 1)
(y
10
5y
9
+ ··· + 496y + 64)
10