11n
1
(K11n
1
)
1
Arc Sequences
5 1 9 2 3 9 11 4 1 7 10
Solving Sequence
1,5
2 3
6,9
10 4 8 11 7
c
1
c
2
c
5
c
9
c
4
c
8
c
11
c
7
c
3
, c
6
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
6
+ a
5
a
4
+ 3a
2
+ 2a + 1, a
5
a
4
+ a
3
+ 3a
2
+ 5b + 2a 2, 2a
5
+ 3a
4
3a
3
+ a
2
+ 9a + 5u + 6i
I
u
2
= hu
19
4u
18
+ ··· 12u
2
+ 1, 2u
18
+ 7u
17
+ ··· + 4b + 5, 5u
18
18u
17
+ ··· + 4a 5i
There are 2 irreducible components with 25 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
+ 3a
2
+ 2a + 1, a
5
a
4
+ a
3
+ 3a
2
+ 5b + 2a
2, 2a
5
+ 3a
4
3a
3
+ a
2
+ 9a + 5u + 6i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
2
5
a
5
3
5
a
4
+ ···
9
5
a
6
5
a
2
=
1
2
5
a
5
3
5
a
4
+ ···
9
5
a
1
5
a
3
=
2
5
a
5
+
3
5
a
4
+ ··· +
9
5
a +
6
5
2
5
a
5
3
5
a
4
+ ···
9
5
a
1
5
a
6
=
1
0
a
9
=
a
1
5
a
5
+
1
5
a
4
+ ···
2
5
a +
2
5
a
10
=
1
5
a
5
+
1
5
a
4
+ ··· +
3
5
a +
2
5
1
5
a
5
+
1
5
a
4
+ ···
2
5
a +
2
5
a
4
=
2
5
a
5
+
3
5
a
4
+ ··· +
9
5
a +
6
5
2
5
a
5
3
5
a
4
+ ···
9
5
a
1
5
a
8
=
a
1
5
a
5
+
1
5
a
4
+ ···
2
5
a +
2
5
a
11
=
a
2
+ 1
2
5
a
5
+
2
5
a
4
+ ···
4
5
a
1
5
a
7
=
2
5
a
5
2
5
a
4
+ ··· +
4
5
a
4
5
2
5
a
5
+
2
5
a
4
+ ···
4
5
a
1
5
a
7
=
2
5
a
5
2
5
a
4
+ ··· +
4
5
a
4
5
2
5
a
5
+
2
5
a
4
+ ···
4
5
a
1
5
(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 = 1.239557 0.467306I
b = 0.215080 + 1.307141I
3.02413 + 4.85801I 4.03424 5.28153I
u = 0.500000 + 0.866025I
a = 1.239557 + 0.467306I
b = 0.215080 1.307141I
3.02413 4.85801I 4.03424 + 5.28153I
u = 0.500000 + 0.866025I
a = 0.284920 0.493496I
b = 0.569840
1.11345 2.02988I 12.72167 + 1.07831I
u = 0.500000 0.866025I
a = 0.284920 + 0.493496I
b = 0.569840
1.11345 + 2.02988I 12.72167 1.07831I
u = 0.500000 + 0.866025I
a = 1.024478 0.839835I
b = 0.215080 + 1.307141I
3.02413 + 0.79824I 2.74410 0.29766I
u = 0.500000 0.866025I
a = 1.024478 + 0.839835I
b = 0.215080 1.307141I
3.02413 0.79824I 2.74410 + 0.29766I
3
II. I
u
2
=
hu
19
4u
18
+· · ·12u
2
+1, 2u
18
+7u
17
+· · ·+4b+5, 5u
18
18u
17
+· · ·+4a5i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
5
2u
3
u
u
5
+ u
3
+ u
a
9
=
5
4
u
18
+
9
2
u
17
+ ··· +
11
2
u +
5
4
1
2
u
18
7
4
u
17
+ ···
5
4
u
5
4
a
10
=
0.750000u
18
+ 2.75000u
17
+ ··· 19.2500u
2
+ 4.25000u
1
2
u
18
7
4
u
17
+ ···
5
4
u
5
4
a
4
=
u
u
3
+ u
a
8
=
3
4
u
18
+
5
2
u
17
+ ··· +
9
2
u +
3
4
1
2
u
18
9
4
u
17
+ ··· +
1
4
u
3
4
a
11
=
1
4
u
17
+
3
4
u
16
+ ··· +
3
4
u +
7
4
1
4
u
18
u
17
+ ··· u
1
4
a
7
=
1
4
u
18
3
4
u
17
+ ···
7
4
u 1
1
4
u
18
+ u
17
+ ··· + 2u +
1
4
a
7
=
1
4
u
18
3
4
u
17
+ ···
7
4
u 1
1
4
u
18
+ u
17
+ ··· + 2u +
1
4
(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 = 0.578849 0.831148I
a = 0.236375 + 0.103692I
b = 0.223009 0.136441I
0.57171 + 2.29308I 0.43534 5.97155I
u = 0.578849 + 0.831148I
a = 0.236375 0.103692I
b = 0.223009 + 0.136441I
0.57171 2.29308I 0.43534 + 5.97155I
u = 0.379573 1.066794I
a = 0.384017 0.248046I
b = 0.118852 0.503818I
1.34954 + 2.72131I 3.10172 4.42849I
u = 0.379573 + 1.066794I
a = 0.384017 + 0.248046I
b = 0.118852 + 0.503818I
1.34954 2.72131I 3.10172 + 4.42849I
u = 0.202383
a = 1.82553
b = 0.369456
0.846922 12.0185
u = 0.066477 0.849480I
a = 0.390573 + 0.852872I
b = 0.698534 + 0.388480I
0.275217 0.309939I 6.84413 + 1.19842I
u = 0.066477 + 0.849480I
a = 0.390573 0.852872I
b = 0.698534 0.388480I
0.275217 + 0.309939I 6.84413 1.19842I
u = 0.105421 1.309259I
a = 0.964760 0.228937I
b = 0.401445 1.238985I
6.74492 + 0.62050I 0.784704 1.156660I
u = 0.105421 + 1.309259I
a = 0.964760 + 0.228937I
b = 0.401445 + 1.238985I
6.74492 0.62050I 0.784704 + 1.156660I
u = 0.239802 1.225754I
a = 1.218926 + 0.275212I
b = 0.62964 + 1.42811I
5.41009 5.31951I 2.17850 + 4.32462I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.239802 + 1.225754I
a = 1.218926 0.275212I
b = 0.62964 1.42811I
5.41009 + 5.31951I 2.17850 4.32462I
u = 0.493476 0.148245I
a = 0.12816 + 2.14991I
b = 0.255468 1.079929I
2.12600 + 2.49879I 4.77209 3.99040I
u = 0.493476 + 0.148245I
a = 0.12816 2.14991I
b = 0.255468 + 1.079929I
2.12600 2.49879I 4.77209 + 3.99040I
u = 0.54417 1.43391I
a = 1.179384 + 0.320996I
b = 0.18150 1.86580I
17.9983 2.5634I 2.10495 + 0.56524I
u = 0.54417 + 1.43391I
a = 1.179384 0.320996I
b = 0.18150 + 1.86580I
17.9983 + 2.5634I 2.10495 0.56524I
u = 0.58478 1.39635I
a = 1.225594 0.379250I
b = 0.18714 + 1.93313I
17.6117 9.7005I 2.62109 + 4.88323I
u = 0.58478 + 1.39635I
a = 1.225594 + 0.379250I
b = 0.18714 1.93313I
17.6117 + 9.7005I 2.62109 4.88323I
u = 1.158441 0.036055I
a = 0.02963 + 1.59757I
b = 0.02328 1.85176I
13.35978 + 3.50957I 4.64823 2.14006I
u = 1.158441 + 0.036055I
a = 0.02963 1.59757I
b = 0.02328 + 1.85176I
13.35978 3.50957I 4.64823 + 2.14006I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
3
(u
19
+ 4u
18
+ ··· + 12u
2
1)
c
2
(u
2
+ u + 1)
3
(u
19
+ 14u
18
+ ··· + 24u 1)
c
3
, c
8
u
6
(u
19
+ u
18
+ ··· + 160u 64)
c
4
(u
2
u + 1)
3
(u
19
+ 4u
18
+ ··· + 12u
2
1)
c
5
(u
2
+ u + 1)
3
(u
19
+ 4u
18
+ ··· + u + 2)
c
6
(u
3
+ u
2
+ 2u + 1)
2
(u
19
+ 3u
18
+ ··· + 2759u 937)
c
7
(u
3
u
2
+ 1)
2
(u
19
+ 3u
18
+ ··· + u + 1)
c
9
(u
3
+ u
2
+ 2u + 1)
2
(u
19
+ 3u
18
+ ··· + 11u + 1)
c
10
(u
3
+ u
2
1)
2
(u
19
+ 3u
18
+ ··· + u + 1)
c
11
(u
3
u
2
+ 2u 1)
2
(u
19
+ 3u
18
+ ··· + 11u + 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
3
(y
19
+ 14y
18
+ ··· + 24y 1)
c
2
(y
2
+ y + 1)
3
(y
19
14y
18
+ ··· + 612y 1)
c
3
, c
8
y
6
(y
19
+ 35y
18
+ ··· 15360y 4096)
c
5
(y
2
+ y + 1)
3
(y
19
42y
18
+ ··· + 13y 4)
c
6
(y
3
+ 3y
2
+ 2y 1)
2
(y
19
+ 89y
18
+ ··· + 1.53948 × 10
7
y 877969)
c
7
, c
10
(y
3
y
2
+ 2y 1)
2
(y
19
3y
18
+ ··· + 11y 1)
c
9
, c
11
(y
3
+ 3y
2
+ 2y 1)
2
(y
19
+ 29y
18
+ ··· + 11y 1)
8