11n
51
(K11n
51
)
1
Arc Sequences
4 1 7 2 9 10 3 1 11 6 7
Solving Sequence
4,7 3,10
6 11 1 2 5 9 8
c
3
c
6
c
10
c
11
c
2
c
4
c
9
c
8
c
1
, c
5
, c
7
Representation Ideals
I = I
u
1
\
I
v
1
I
u
1
= hu
19
+ u
18
+ ··· + 32u + 32, 3.89161 × 10
18
u
18
1.25291 × 10
19
u
17
+ ··· + 1.19579 × 10
21
a 2.43773 × 10
21
,
4.27694 × 10
19
u
18
+ 2.93614 × 10
19
u
17
+ ··· + 2.39158 × 10
21
b + 7.99100 × 10
20
i
I
v
1
= h−v
3
+ b + 2v, v
5
+ v
4
2v
3
v
2
+ v 1, ai
There are 2 irreducible components with 24 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
19
+u
18
+· · ·+32u+32, 3.89×10
18
u
18
1.25×10
19
u
17
+· · ·+1.20×10
21
a
2.44 × 10
21
, 4.28 × 10
19
u
18
+ 2.94 × 10
19
u
17
+ · · · + 2.39 × 10
21
b + 7.99 × 10
20
i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
10
=
0.00325443u
18
+ 0.0104777u
17
+ ··· + 1.47789u + 2.03859
0.0178833u
18
0.0122770u
17
+ ··· + 0.528252u 0.334131
a
6
=
0.0334283u
18
+ 0.0315554u
17
+ ··· + 2.33117u 0.293509
0.00547265u
18
+ 0.00812761u
17
+ ··· + 0.354335u + 0.00511905
a
11
=
0.0126757u
18
0.0158711u
17
+ ··· 0.965637u + 0.712666
0.0119895u
18
0.00312216u
17
+ ··· 0.280593u 0.182519
a
1
=
0.0126757u
18
0.0158711u
17
+ ··· 0.965637u + 0.712666
0.0154120u
18
0.0118073u
17
+ ··· 0.788470u 0.284774
a
2
=
0.00273632u
18
0.00406381u
17
+ ··· 0.177167u + 0.997440
0.0154120u
18
0.0118073u
17
+ ··· 0.788470u 0.284774
a
5
=
0.0246651u
18
0.0189933u
17
+ ··· 1.24623u + 0.530148
0.0119895u
18
+ 0.00312216u
17
+ ··· + 0.280593u + 0.182519
a
9
=
0.00225903u
18
+ 0.00481394u
17
+ ··· + 0.299894u + 0.940474
0.00680013u
18
0.00393377u
17
+ ··· + 0.909878u 0.0875624
a
8
=
u
u
3
+ u
a
8
=
u
u
3
+ u
(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 = 0.66687 1.26708I
a = 0.868526 0.181292I
b = 2.01097 0.37234I
6.61331 2.23643I 3.68670 + 1.85634I
u = 0.66687 + 1 .26708I
a = 0.868526 + 0.181292I
b = 2.01097 + 0.37234I
6.61331 + 2.23643I 3.68670 1.85634I
u = 0.559824
a = 0.520700
b = 0.696990
1.12234 9.25558
u = 0.55184 2.12917I
a = 0.624151 0.075817I
b = 3.70077 0.64779I
17.2204 9.5042I 1.84766 + 4.86373I
u = 0.55184 + 2 .12917I
a = 0.624151 + 0.075817I
b = 3.70077 + 0 .64779I
17.2204 + 9.5042I 1.84766 4.86373I
u = 0.377279 0.570351I
a = 0.928450 + 0.852722I
b = 1.60892 0.88722I
0.904771 + 0 .899537I 0.47063 + 1.75855I
u = 0.377279 + 0 .570351I
a = 0.928450 0.852722I
b = 1.60892 + 0 .88722I
0.904771 0.899537I 0.47063 1.75855I
u = 0.33784 2.24339I
a = 0.540132 0.287132I
b = 3.20814 0.98987I
17.5665 + 0.7457I 2.31305 0.82283I
u = 0.33784 + 2 .24339I
a = 0.540132 + 0.287132I
b = 3.20814 + 0.98987I
17.5665 0.7457I 2.31305 + 0.82283I
u = 0.118930 1.200608I
a = 0.902998 + 0.509171I
b = 1.28119 + 1 .32428I
5.36309 + 6.16703I 2.22619 6.06641I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.118930 + 1.200608I
a = 0.902998 0.509171I
b = 1.28119 1.32428I
5.36309 6.16703I 2.22619 + 6 .06641I
u = 0.113652 0.545215I
a = 1.25949 1.01771I
b = 0.107580 0.561696I
0.05095 1.76235I 0.18768 + 4 .49049I
u = 0.113652 + 0.545215I
a = 1.25949 + 1.01771I
b = 0.107580 + 0.561696I
0.05095 + 1 .76235I 0.18768 4.49049I
u = 0.349333 1.064248I
a = 0.342118 + 0.593985I
b = 0.144940 + 0.182376I
2.51975 1.53406I 0.31883 + 1 .85733I
u = 0.349333 + 1.064248I
a = 0.342118 0.593985I
b = 0.144940 0.182376I
2.51975 + 1 .53406I 0.31883 1.85733I
u = 0.42263 2.12489I
a = 0.069295 0.437160I
b = 0.299792 + 0.326685I
13.37814 + 4.28212I 1.00628 2.00074I
u = 0.42263 + 2.12489I
a = 0.069295 + 0.437160I
b = 0.299792 0.326685I
13.37814 4.28212I 1.00628 + 2 .00074I
u = 0.947060 0.059794I
a = 0.339607 + 1.065378I
b = 0.667953 + 0.197417I
1.26128 + 2 .36565I 1.22099 4.76618I
u = 0.947060 + 0.059794I
a = 0.339607 1.065378I
b = 0.667953 0.197417I
1.26128 2.36565I 1.22099 + 4 .76618I
4
II. I
v
1
= h−v
3
+ b + 2v, v
5
+ v
4
2v
3
v
2
+ v 1, ai
(i) Arc colorings
a
4
=
1
0
a
7
=
v
0
a
3
=
1
0
a
10
=
0
v
3
2v
a
6
=
v
v
2
+ 1
a
11
=
v
4
v
2
+ v 1
1
a
1
=
v
4
2v
2
+ v 1
1
a
2
=
v
4
2v
2
+ v
1
a
5
=
v
4
+ 2v
2
v + 1
1
a
9
=
v
4
+ 2v
2
v + 1
v
a
8
=
v
0
a
8
=
v
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.41878 0.21917I
a = 0
b = 0.186078 0.874646I
4.22763 + 4.40083I 1.17182 3.02310I
v = 1.41878 + 0.21917I
a = 0
b = 0.186078 + 0.874646I
4.22763 4.40083I 1.17182 + 3.02310I
v = 0.309916 0.549911I
a = 0
b = 0.871221 + 1 .107662I
1.31583 1.53058I 6.99101 + 6.23673I
v = 0.309916 + 0.549911I
a = 0
b = 0.871221 1.107662I
1.31583 + 1.53058I 6.99101 6.23673I
v = 1.21774
a = 0
b = 0.629714
0.756147 2.36161
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
19
+ 6u
18
+ ··· 6u 1)
c
2
(u + 1)
5
(u
19
+ 22u
17
+ ··· + 10u + 1)
c
3
, c
7
u
5
(u
19
+ u
18
+ ··· + 32u + 32)
c
4
(u + 1)
5
(u
19
+ 6u
18
+ ··· 6u 1)
c
5
(u
5
u
4
+ ··· + u + 1)(u
19
+ 2u
18
+ ··· + 2u 1)
c
6
(u
5
+ u
4
+ ··· + u + 1)(u
19
+ 2u
18
+ ··· + 4u + 1)
c
8
(u
5
u
4
+ ··· + u + 1)(u
19
+ 8u
18
+ ··· + 3614u 53)
c
9
(u
5
+ 3u
4
+ ··· u 1)(u
19
+ 12u
18
+ ··· + 8u 1)
c
10
(u
5
u
4
+ ··· + u 1)(u
19
+ 2u
18
+ ··· + 4u + 1)
c
11
(u
5
+ u
4
+ ··· + u 1)(u
19
+ 2u
18
+ ··· + 2u 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
19
+ 22y
17
+ ··· + 10y 1)
c
2
(y 1)
5
(y
19
+ 44y
18
+ ··· 82y 1)
c
3
, c
7
y
5
(y
19
+ 33y
18
+ ··· 6656y 1024)
c
5
, c
11
(y
5
5y
4
+ ··· y 1)(y
19
28y
18
+ ··· + 8y 1)
c
6
, c
10
(y
5
+ 3y
4
+ ··· y 1)(y
19
+ 12y
18
+ ··· + 8y 1)
c
8
(y
5
5y
4
+ ··· y 1)(y
19
88y
18
+ ··· + 1.13576 × 10
7
y 2809)
c
9
(y
5
y
4
+ ··· + 3y 1)(y
19
8y
18
+ ··· + 148y 1)
8