11n
61
(K11n
61
)
1
Arc Sequences
4 1 8 2 9 10 1 4 11 7 6
Solving Sequence
6,10
7 11 1
4,8
3 2 5 9
c
6
c
10
c
11
c
7
c
3
c
2
c
4
c
9
c
1
, c
5
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
6
+ a
5
a
4
2a
3
+ a + 1, u 1, a
5
+ a
3
+ a
2
+ bi
I
u
2
= hu
14
7u
13
+ 10u
12
+ 29u
11
75u
10
21u
9
+ 127u
8
7u
7
19u
6
48u
5
36u
4
14u
3
7u
2
+ 4u 1,
9u
13
+ 56u
12
+ ··· + 32b 15, 29u
13
+ 198u
12
+ ··· + 32a 105i
There are 2 irreducible components with 20 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
+ a
5
a
4
2a
3
+ a + 1, u 1, a
5
+ a
3
+ a
2
+ bi
(i) Arc colorings
a
6
=
0
1
a
10
=
a
a
5
a
3
a
2
a
7
=
a
2
a
5
+ a
3
a
a
11
=
a
3
+ a
a
3
a 1
a
1
=
a
3
+ a
1
a
4
=
1
0
a
8
=
a
5
a
3
+ a
0
a
3
=
1
0
a
2
=
a
3
+ a + 1
1
a
5
=
a
3
a
1
a
9
=
a
5
a
3
+ a
0
a
9
=
a
5
a
3
+ a
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 = 1.00000
a = 1.073950 0.558752I
b = 1.30674 1.20014I
1.64493 + 5.69302I 6.76721 6.15196I
u = 1.00000
a = 1.073950 + 0.558752I
b = 1.30674 + 1.20014I
1.64493 5.69302I 6.76721 + 6.15196I
u = 1.00000
a = 0.428243 0.664531I
b = 0.315740 0.200172I
3.53554 0.92430I 10.03026 + 0.88960I
u = 1.00000
a = 0.428243 + 0.664531I
b = 0.315740 + 0.200172I
3.53554 + 0.92430I 10.03026 0.88960I
u = 1.00000
a = 1.002193 0.295542I
b = 1.49099 + 0.22339I
0.245672 0.924305I 5.20252 + 1.68215I
u = 1.00000
a = 1.002193 + 0.295542I
b = 1.49099 0.22339I
0.245672 + 0.924305I 5.20252 1.68215I
3
II. I
u
2
= hu
14
7u
13
+ · · · + 4u 1, 9u
13
+ 56u
12
+ · · · + 32b
15, 29u
13
+ 198u
12
+ · · · + 32a 105i
(i) Arc colorings
a
6
=
0
u
a
10
=
0.906250u
13
6.18750u
12
+ ··· 8.59375u + 3.28125
9
32
u
13
7
4
u
12
+ ···
35
32
u +
15
32
a
7
=
1.03125u
13
7u
12
+ ··· 10.9688u + 2.59375
0.0937500u
13
+ 0.0625000u
12
+ ··· 0.968750u + 0.406250
a
11
=
0.0312500u
13
+ 0.187500u
12
+ ··· + 3.09375u + 1.96875
0.0312500u
13
0.187500u
12
+ ··· + 0.906250u + 0.0312500
a
1
=
0.0312500u
13
+ 0.187500u
12
+ ··· + 3.09375u + 1.96875
u
a
4
=
1
0
a
8
=
0.906250u
13
5.68750u
12
+ ··· 8.46875u + 1.15625
1
2
u
13
11
4
u
12
+ ···
7
4
u +
1
2
a
3
=
u
3
+ 2u + 2
u
3
+ u
a
2
=
0.0312500u
13
+ 0.187500u
12
+ ··· + 2.09375u + 1.96875
u
a
5
=
0.0312500u
13
+ 0.187500u
12
+ ··· + 2.09375u + 0.968750
u
2
a
9
=
0.406250u
13
2.93750u
12
+ ··· 6.71875u + 0.656250
1
2
u
13
11
4
u
12
+ ···
7
4
u +
1
2
a
9
=
0.406250u
13
2.93750u
12
+ ··· 6.71875u + 0.656250
1
2
u
13
11
4
u
12
+ ···
7
4
u +
1
2
(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 = 1.59669 0.17157I
a = 0.485887 + 0.454240I
b = 0.68045 + 1.42827I
3.47956 2.50408I 8.95669 + 2.99860I
u = 1.59669 + 0.17157I
a = 0.485887 0.454240I
b = 0.68045 1.42827I
3.47956 + 2.50408I 8.95669 2.99860I
u = 0.681509
a = 0.616983
b = 0.766335
1.01289 10.2625
u = 0.355616 0.529402I
a = 1.199836 0.307263I
b = 0.232166 0.732517I
1.309152 0.137583I 8.56031 + 0.56305I
u = 0.355616 + 0.529402I
a = 1.199836 + 0.307263I
b = 0.232166 + 0.732517I
1.309152 + 0.137583I 8.56031 0.56305I
u = 0.036725 0.627532I
a = 0.98809 1.55836I
b = 0.218281 0.941378I
0.01563 4.65799I 4.40917 + 5.70687I
u = 0.036725 + 0.627532I
a = 0.98809 + 1.55836I
b = 0.218281 + 0.941378I
0.01563 + 4.65799I 4.40917 5.70687I
u = 0.191801 0.163474I
a = 2.05110 + 3.46237I
b = 0.322920 + 0.607924I
1.63965 + 1.19495I 1.59955 1.11588I
u = 0.191801 + 0.163474I
a = 2.05110 3.46237I
b = 0.322920 0.607924I
1.63965 1.19495I 1.59955 + 1.11588I
u = 2.12501
a = 0.592627
b = 3.61664
13.7717 5.12958
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 2.21915 0.28216I
a = 0.461684 + 0.368144I
b = 2.88479 + 1.99117I
18.4364 + 8.2751I 7.93412 4.24282I
u = 2.21915 + 0.28216I
a = 0.461684 0.368144I
b = 2.88479 1.99117I
18.4364 8.2751I 7.93412 + 4.24282I
u = 2.28288 0.17435I
a = 0.249180 0.384061I
b = 0.093281 + 0.190265I
19.1238 + 2.3664I 10.04321 0.09569I
u = 2.28288 + 0.17435I
a = 0.249180 + 0.384061I
b = 0.093281 0.190265I
19.1238 2.3664I 10.04321 + 0.09569I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u 1)
6
(u
14
+ 7u
13
+ ··· 4u 1)
c
2
(u + 1)
6
(u
14
+ 29u
13
+ ··· + 2u + 1)
c
3
, c
8
u
6
(u
14
+ u
13
+ ··· + 64u 64)
c
5
(u
6
+ u
5
+ ··· + u + 1)(u
14
+ 2u
13
+ ··· + 3u + 1)
c
6
, c
10
(u
6
u
5
+ ··· u + 1)(u
14
+ 2u
13
+ ··· u + 1)
c
7
(u
6
u
5
+ ··· u + 1)(u
14
+ 2u
13
+ ··· + 3u + 1)
c
9
(u
6
+ 3u
5
+ ··· + u + 1)(u
14
+ 6u
13
+ ··· + 9u + 1)
c
11
(u
6
3u
5
+ ··· u + 1)(u
14
+ 6u
13
+ ··· + u 5)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
6
(y
14
29y
13
+ ··· 2y + 1)
c
2
(y 1)
6
(y
14
129y
13
+ ··· + 462y + 1)
c
3
, c
8
y
6
(y
14
39y
13
+ ··· + 8192y + 4096)
c
5
, c
7
(y
6
3y
5
+ ··· y + 1)(y
14
30y
13
+ ··· 9y + 1)
c
6
, c
10
(y
6
3y
5
+ ··· y + 1)(y
14
6y
13
+ ··· 9y + 1)
c
9
(y
6
+ y
5
+ ··· + 3y + 1)(y
14
+ 6y
13
+ ··· 25y + 1)
c
11
(y
6
+ y
5
+ ··· + 3y + 1)(y
14
6y
13
+ ··· 301y + 25)
8