11n
20
(K11n
20
)
1
Arc Sequences
5 1 9 2 3 10 11 1 3 8 7
Solving Sequence
1,5
2
3,7
11 8 9 4 10 6
c
1
c
2
c
11
c
7
c
8
c
3
c
10
c
6
c
4
, c
5
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
6
+ 7a
5
+ 19a
4
+ 26a
3
+ 22a
2
+ 15a + 7, a
5
+ 4a
4
+ 7a
3
+ 10a
2
+ 5b + 12a + 4,
2a
5
+ 13a
4
+ 29a
3
+ 30a
2
+ 24a + 5u + 18i
I
u
2
= hu
17
+ 4u
16
+ ··· + 3u + 1,
u
16
3u
15
5u
14
4u
13
7u
12
10u
11
10u
10
7u
8
8u
7
3u
6
+ 3u
5
5u
4
+ 10u
2
+ 4a 9u + 4,
u
16
+ 4u
15
+ ··· + 4b + 1i
There are 2 irreducible components with 23 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
= ha
6
+ 7a
5
+ 19a
4
+ 26a
3
+ 22a
2
+ 15a + 7, a
5
+ 4a
4
+ 7a
3
+ 10a
2
+
5b + 12a + 4, 2a
5
+ 13a
4
+ 29a
3
+ 30a
2
+ 24a + 5u + 18i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
2
5
a
5
13
5
a
4
+ ···
24
5
a
18
5
a
2
=
1
2
5
a
5
13
5
a
4
+ ···
24
5
a
13
5
a
3
=
2
5
a
5
+
13
5
a
4
+ ··· +
24
5
a +
18
5
2
5
a
5
13
5
a
4
+ ···
24
5
a
13
5
a
7
=
a
1
5
a
5
4
5
a
4
+ ···
12
5
a
4
5
a
11
=
3
5
a
5
+
12
5
a
4
+ ··· +
11
5
a +
12
5
2
5
a
5
8
5
a
4
+ ···
9
5
a
8
5
a
8
=
a
5
+ 5a
4
+ 8a
3
+ 5a
2
+ 3a + 2
1
5
a
5
4
5
a
4
+ ··· +
3
5
a +
1
5
a
9
=
4
5
a
5
+
21
5
a
4
+ ··· +
18
5
a +
11
5
1
5
a
5
4
5
a
4
+ ··· +
3
5
a +
1
5
a
4
=
2
5
a
5
+
13
5
a
4
+ ··· +
24
5
a +
18
5
2
5
a
5
13
5
a
4
+ ···
24
5
a
13
5
a
10
=
4
5
a
5
+
21
5
a
4
+ ··· +
18
5
a +
11
5
1
5
a
5
4
5
a
4
+ ··· +
3
5
a +
1
5
a
6
=
1
0
a
6
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 2.23956 0.46731I
b = 0.215080 + 1.307141I
3.02413 + 4.85801I 1.45566 6.64456I
u = 0.500000 + 0.866025I
a = 2.23956 + 0.46731I
b = 0.215080 1.307141I
3.02413 4.85801I 1.45566 + 6.64456I
u = 0.500000 + 0.866025I
a = 1.284920 0.493496I
b = 0.569840
1.11345 2.02988I 5.85715 + 4.49037I
u = 0.500000 0.866025I
a = 1.284920 + 0.493496I
b = 0.569840
1.11345 + 2.02988I 5.85715 4.49037I
u = 0.500000 + 0.866025I
a = 0.024478 0.839835I
b = 0.215080 + 1.307141I
3.02413 + 0.79824I 2.09851 + 0.12339I
u = 0.500000 0.866025I
a = 0.024478 + 0.839835I
b = 0.215080 1.307141I
3.02413 0.79824I 2.09851 0.12339I
3
II.
I
u
2
= hu
17
+ 4 u
16
+· · ·+3u+1, u
16
3u
15
+· · ·+4a+4, u
16
+4u
15
+· · ·+4b+1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
7
=
1
4
u
16
+
3
4
u
15
+ ··· +
9
4
u 1
1
4
u
16
u
15
+ ···
3
4
u
1
4
a
11
=
1
4
u
16
+
7
4
u
15
+ ··· +
11
4
u +
5
2
3
4
u
16
+
9
4
u
15
+ ··· + 2u
2
+
5
4
u
a
8
=
11
4
u
16
39
4
u
15
+ ···
27
4
u 4
u
16
+
15
4
u
15
+ ··· + 3u +
7
4
a
9
=
7
4
u
16
6u
15
+ ···
15
4
u
9
4
u
16
+
15
4
u
15
+ ··· + 3u +
7
4
a
4
=
u
u
3
u
a
10
=
13
4
u
16
10u
15
+ ···
25
4
u
15
4
2u
16
+
25
4
u
15
+ ··· + 4u +
9
4
a
6
=
u
5
2u
3
u
u
5
+ u
3
+ u
a
6
=
u
5
2u
3
u
u
5
+ u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.09163 0.95076I
a = 0.96960 1.67272I
b = 0.55805 + 1.32231I
11.91545 1.84478I 3.96952 + 0.75367I
u = 1.09163 + 0.95076I
a = 0.96960 + 1.67272I
b = 0.55805 1.32231I
11.91545 + 1.84478I 3.96952 0.75367I
u = 1.05476 1.04416I
a = 2.07888 0.43065I
b = 1.052716 0.047013I
15.8539 + 3.8626I 6.39661 2.12816I
u = 1.05476 + 1.04416I
a = 2.07888 + 0.43065I
b = 1.052716 + 0.047013I
15.8539 3.8626I 6.39661 + 2.12816I
u = 0.97902 1.09233I
a = 2.11700 + 1.07061I
b = 0.50759 1.37481I
11.4057 + 9.4106I 3.33658 4.76975I
u = 0.97902 + 1.09233I
a = 2.11700 1.07061I
b = 0.50759 + 1.37481I
11.4057 9.4106I 3.33658 + 4.76975I
u = 0.509232 0.371578I
a = 2.67025 1.66568I
b = 0.281522 + 1.323870I
2.51924 + 3.59257I 1.69678 1.62034I
u = 0.509232 + 0.371578I
a = 2.67025 + 1.66568I
b = 0.281522 1.323870I
2.51924 3.59257I 1.69678 + 1.62034I
u = 0.494210
a = 2.62173
b = 0.710942
1.69761 6.54343
u = 0.045204 0.831305I
a = 0.842729 0.566333I
b = 0.095288 1.269798I
4.44298 1.97657I 3.41444 + 3.62302I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.045204 + 0.831305I
a = 0.842729 + 0.566333I
b = 0.095288 + 1.269798I
4.44298 + 1.97657I 3.41444 3.62302I
u = 0.559459 0.682406I
a = 0.584057 + 0.456170I
b = 0.122694 0.403191I
0.106087 1.407485I 0.91901 + 2.91397I
u = 0.559459 + 0.682406I
a = 0.584057 0.456170I
b = 0.122694 + 0.403191I
0.106087 + 1.407485I 0.91901 2.91397I
u = 0.591154 1.059120I
a = 1.71916 0.51588I
b = 0.219268 + 0.999289I
1.31705 3.54605I 2.16487 + 2.95335I
u = 0.591154 + 1.059120I
a = 1.71916 + 0.51588I
b = 0.219268 0.999289I
1.31705 + 3.54605I 2.16487 2.95335I
u = 0.685929 0.418499I
a = 0.330311 + 0.852864I
b = 0.228042 0.683004I
0.22550 1.43526I 4.15937 + 3.64291I
u = 0.685929 + 0.418499I
a = 0.330311 0.852864I
b = 0.228042 + 0.683004I
0.22550 + 1.43526I 4.15937 3.64291I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
3
(u
17
+ 4u
16
+ ··· + 3u + 1)
c
2
(u
2
+ u + 1)
3
(u
17
+ 2u
16
+ ··· + 3u 1)
c
3
, c
9
u
6
(u
17
+ u
16
+ ··· 96u 64)
c
4
(u
2
u + 1)
3
(u
17
+ 4u
16
+ ··· + 3u + 1)
c
5
(u
2
+ u + 1)
3
(u
17
+ 4u
16
+ ··· + 557u 137)
c
6
, c
8
(u
3
u
2
+ 1)
2
(u
17
+ 3u
16
+ ··· + 2u + 1)
c
7
, c
10
, c
11
(u
3
u
2
+ 2u 1)
2
(u
17
+ 3u
16
+ ··· 2u 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
3
(y
17
+ 2y
16
+ ··· + 3y 1)
c
2
(y
2
+ y + 1)
3
(y
17
+ 30y
16
+ ··· + 3y 1)
c
3
, c
9
y
6
(y
17
+ 35y
16
+ ··· + 9216y 4096)
c
5
(y
2
+ y + 1)
3
(y
17
+ 58y
16
+ ··· 518053y 18769)
c
6
, c
8
(y
3
y
2
+ 2y 1)
2
(y
17
31y
16
+ ··· 14y 1)
c
7
, c
10
, c
11
(y
3
+ 3y
2
+ 2y 1)
2
(y
17
+ 13y
16
+ ··· 14y 1)
8