11a
366
(K11a
366
)
1
Arc Sequences
8 7 10 9 1 11 2 3 4 5 6
Solving Sequence
1,8 2,5
6 7 3 9 4 11 10
c
1
c
5
c
7
c
2
c
8
c
4
c
11
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
6
\
i=1
I
u
i
I
u
1
= hu
2
+ 1, a + 1, b + ui
I
u
2
= ha
6
+ 2a
5
3a
3
+ 6a
2
+ a + 1, 2a
5
+ 3a
4
2a
3
7a
2
+ 6b + 17a 1,
5a
5
9a
4
+ 5a
3
+ 22a
2
32a + 12u 5i
I
u
3
= hu
6
+ u
5
+ 4u
4
+ 3u
3
+ 4u
2
+ 2u 1, b u, u
5
+ u
4
+ 3u
3
+ 2u
2
+ a + 2ui
I
u
4
= hu
10
u
9
+ 5u
8
5u
7
+ 9u
6
9u
5
+ 6u
4
6u
3
+ u
2
+ 1, u
6
3u
4
2u
2
+ a + 1,
u
9
u
8
+ 5u
7
4u
6
+ 9u
5
6u
4
+ 6u
3
4u
2
+ b + u 1i
I
u
5
= hu
10
u
9
+ 5u
8
5u
7
+ 9u
6
9u
5
+ 6u
4
6u
3
+ u
2
+ 1, b u, u
9
3u
7
u
5
+ 4u
3
+ a + 2u 1i
I
u
6
= hu
10
+ 2u
9
+ 5u
8
+ 6u
7
+ 7u
6
+ 6u
5
+ 4u
4
+ 4u
3
+ 3u
2
+ 3u + 2, u
9
u
7
+ 2u
6
+ u
5
+ 2u
4
+ 2a u + 1,
u
9
+ u
8
+ 3u
7
+ 2u
6
+ 3u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ b + 2u + 1i
There are 6 irreducible components with 44 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
2
+ 1, a + 1, b + ui
(i) Arc colorings
a
1
=
1
0
a
8
=
1
u
a
2
=
u + 1
1
a
5
=
0
u
a
6
=
u
u
a
7
=
u
0
a
3
=
u
1
a
9
=
1
u
a
4
=
u
u 1
a
11
=
0
1
a
10
=
0
1
a
10
=
0
1
(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 = 1.00000I
a = 1.00000
b = 1.00000I
4.93480 4.00000
u = 1.00000I
a = 1.00000
b = 1.00000I
4.93480 4.00000
3
II. I
u
2
= ha
6
+ 2a
5
3a
3
+ 6a
2
+ a + 1, 2a
5
+ 3a
4
2a
3
7a
2
+ 6b + 17a
1, 5a
5
9a
4
+ 5a
3
+ 22a
2
32a + 12u 5i
(i) Arc colorings
a
1
=
1
0
a
8
=
a
1
3
a
5
1
2
a
4
+ ···
17
6
a +
1
6
a
2
=
1
6
a
5
+
1
3
a
4
+ ··· +
1
2
a +
4
3
5
12
a
5
11
12
a
4
+ ···
11
6
a
5
4
a
5
=
0
5
12
a
5
+
3
4
a
4
+ ··· +
8
3
a +
5
12
a
6
=
5
12
a
5
3
4
a
4
+ ···
8
3
a
5
12
5
12
a
5
+
3
4
a
4
+ ··· +
8
3
a +
5
12
a
7
=
5
12
a
5
3
4
a
4
+ ···
8
3
a +
7
12
1
a
3
=
1
4
a
5
7
12
a
4
+ ···
7
3
a +
1
12
1
12
a
5
5
12
a
4
+ ··· + a
17
12
a
9
=
1
4
a
5
7
12
a
4
+ ···
4
3
a +
1
12
5
12
a
5
11
12
a
4
+ ···
11
6
a
5
4
a
4
=
a
1
6
a
5
+
1
6
a
4
+ ··· +
4
3
a
1
2
a
11
=
1
3
a
5
+
1
2
a
4
+ ··· +
11
6
a
1
6
1
3
a
5
1
2
a
4
+ ···
11
6
a +
7
6
a
10
=
1
3
a
5
+
1
2
a
4
+ ··· +
11
6
a
1
6
1
12
a
5
+
1
4
a
4
+ ··· +
5
6
a +
19
12
a
10
=
1
3
a
5
+
1
2
a
4
+ ··· +
11
6
a
1
6
1
12
a
5
+
1
4
a
4
+ ··· +
5
6
a +
19
12
(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 = 0.682328
a = 1.80674 1.16154I
b = 0.341164 + 1.161541I
1.64493 10.0000
u = 0.682328
a = 1.80674 + 1.16154I
b = 0.341164 1.161541I
1.64493 10.0000
u = 0.341164 1.161541I
a = 0.108378 0.368989I
b = 0.341164 + 1.161541I
1.64493 10.0000
u = 0.341164 + 1.161541I
a = 0.108378 + 0.368989I
b = 0.341164 1.161541I
1.64493 10.0000
u = 0.341164 + 1.161541I
a = 0.915113 0.792552I
b = 0.682328
1.64493 10.0000
u = 0.341164 1.161541I
a = 0.915113 + 0.792552I
b = 0.682328
1.64493 10.0000
5
III.
I
u
3
= hu
6
+ u
5
+ 4u
4
+ 3u
3
+ 4u
2
+ 2u 1, b u, u
5
+ u
4
+ 3u
3
+ 2u
2
+ a + 2ui
(i) Arc colorings
a
1
=
1
0
a
8
=
u
5
u
4
3u
3
2u
2
2u
u
a
2
=
u
4
+ u
3
+ 2u
2
+ 2u
u
2
a
5
=
0
u
a
6
=
u
u
a
7
=
u
3
2u
u
3
+ u
a
3
=
u
3
+ 2u
u
4
+ 2u
2
a
9
=
1
u
4
u
3
2u
2
2u + 1
a
4
=
u
u
5
+ u
4
+ 2u
3
+ 2u
2
a
11
=
u
2
+ 1
u
2
a
10
=
u
2
+ 1
u
4
2u
2
a
10
=
u
2
+ 1
u
4
2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.800464
a = 1.77620
b = 0.800464
5.50851 17.3138
u = 0.37587 1.37813I
a = 1.76299 0.36235I
b = 0.37587 1.37813I
7.7894 12.7681I 4.48012 + 7.54465I
u = 0.37587 + 1.37813I
a = 1.76299 + 0.36235I
b = 0.37587 + 1.37813I
7.7894 + 12.7681I 4.48012 7.54465I
u = 0.13297 1.45639I
a = 0.743382 0.969198I
b = 0.13297 1.45639I
14.9383 + 4.7754I 0.31743 3.39879I
u = 0.13297 + 1.45639I
a = 0.743382 + 0.969198I
b = 0.13297 + 1.45639I
14.9383 4.7754I 0.31743 + 3.39879I
u = 0.286259
a = 0.815416
b = 0.286259
0.468566 21.0911
7
IV.
I
u
4
= hu
10
u
9
+ · · · + u
2
+ 1, u
6
3u
4
2u
2
+ a + 1, u
9
u
8
+ · · · + b 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
u
6
+ 3u
4
+ 2u
2
1
u
9
+ u
8
5u
7
+ 4u
6
9u
5
+ 6u
4
6u
3
+ 4u
2
u + 1
a
2
=
u
9
5u
7
8u
5
2u
3
+ u
2
+ 4u + 2
u
7
+ 3u
5
+ 2u
3
u
2
u 2
a
5
=
0
u
a
6
=
u
u
a
7
=
u
3
2u
u
3
+ u
a
3
=
u
3
+ 2u
u
9
3u
7
2u
5
+ u
3
1
a
9
=
1
u
9
+ 2u
8
5u
7
+ 8u
6
9u
5
+ 10u
4
6u
3
+ 3u
2
u + 1
a
4
=
u
u
9
3u
7
u
5
+ 3u
3
1
a
11
=
u
2
+ 1
u
2
a
10
=
u
2
+ 1
u
4
2u
2
a
10
=
u
2
+ 1
u
4
2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.309318 0.396943I
a = 1.282228 + 0.387959I
b = 0.060791 + 1.179490I
2.84181 1.23169I 8.90177 + 5.44908I
u = 0.309318 + 0.396943I
a = 1.282228 0.387959I
b = 0.060791 1.179490I
2.84181 + 1.23169I 8.90177 5.44908I
u = 0.280829 1.292557I
a = 0.320301 0.345722I
b = 0.480814 + 1.084508I
5.70347 3.47839I 4.80497 + 2.79515I
u = 0.280829 + 1.292557I
a = 0.320301 + 0.345722I
b = 0.480814 1.084508I
5.70347 + 3.47839I 4.80497 2.79515I
u = 0.057928 1.351674I
a = 0.674534 + 0.158285I
b = 0.642886 + 0.580182I
8.22706 2.31006I 3.13631 + 3.52133I
u = 0.057928 + 1.351674I
a = 0.674534 0.158285I
b = 0.642886 0.580182I
8.22706 + 2.31006I 3.13631 3.52133I
u = 0.347624 1.331993I
a = 1.055417 + 0.534294I
b = 0.871979 + 0.168588I
2.90872 + 8.28632I 8.17560 6.14881I
u = 0.347624 + 1.331993I
a = 1.055417 0.534294I
b = 0.871979 0.168588I
2.90872 8.28632I 8.17560 + 6.14881I
u = 0.800451 0.099834I
a = 1.58104 1.11119I
b = 0.350885 + 1.264619I
1.58679 + 4.14585I 12.98134 3.97600I
u = 0.800451 + 0.099834I
a = 1.58104 + 1.11119I
b = 0.350885 1.264619I
1.58679 4.14585I 12.98134 + 3.97600I
9
V. I
u
5
= hu
10
u
9
+ · · · + u
2
+ 1, b u, u
9
3u
7
u
5
+ 4u
3
+ a + 2u 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
u
9
+ 3u
7
+ u
5
4u
3
2u + 1
u
a
2
=
u
9
2u
8
+ 5u
7
8u
6
+ 9u
5
10u
4
+ 6u
3
3u
2
+ u
u
2
a
5
=
0
u
a
6
=
u
u
a
7
=
u
3
2u
u
3
+ u
a
3
=
u
9
2u
8
+ 5u
7
8u
6
+ 9u
5
11u
4
+ 6u
3
5u
2
+ u
u
4
+ 2u
2
a
9
=
u
9
+ 4u
7
+ 5u
5
3u + 1
u
7
3u
5
2u
3
+ u
a
4
=
2u
9
2u
8
+ 10u
7
9u
6
+ 18u
5
15u
4
+ 12u
3
10u
2
+ u 1
u
9
5u
7
+ u
6
9u
5
+ 4u
4
5u
3
+ 5u
2
+ 2u + 1
a
11
=
u
2
+ 1
u
2
a
10
=
u
2
+ 1
u
4
2u
2
a
10
=
u
2
+ 1
u
4
2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.309318 0.396943I
a = 1.13379 + 1.02738I
b = 0.309318 0.396943I
2.84181 1.23169I 8.90177 + 5.44908I
u = 0.309318 + 0.396943I
a = 1.13379 1.02738I
b = 0.309318 + 0.396943I
2.84181 + 1.23169I 8.90177 5.44908I
u = 0.280829 1.292557I
a = 2.04111 0.91614I
b = 0.280829 1.292557I
5.70347 3.47839I 4.80497 + 2.79515I
u = 0.280829 + 1.292557I
a = 2.04111 + 0.91614I
b = 0.280829 + 1.292557I
5.70347 + 3.47839I 4.80497 2.79515I
u = 0.057928 1.351674I
a = 0.52441 1.84172I
b = 0.057928 1.351674I
8.22706 2.31006I 3.13631 + 3.52133I
u = 0.057928 + 1.351674I
a = 0.52441 + 1.84172I
b = 0.057928 + 1.351674I
8.22706 + 2.31006I 3.13631 3.52133I
u = 0.347624 1.331993I
a = 1.91462 0.52432I
b = 0.347624 1.331993I
2.90872 + 8.28632I 8.17560 6.14881I
u = 0.347624 + 1.331993I
a = 1.91462 + 0.52432I
b = 0.347624 + 1.331993I
2.90872 8.28632I 8.17560 + 6.14881I
u = 0.800451 0.099834I
a = 1.78469 + 0.12581I
b = 0.800451 0.099834I
1.58679 + 4.14585I 12.98134 3.97600I
u = 0.800451 + 0.099834I
a = 1.78469 0.12581I
b = 0.800451 + 0.099834I
1.58679 4.14585I 12.98134 + 3.97600I
11
VI. I
u
6
=
hu
10
+2u
9
+· · ·+3u+2, u
9
u
7
+2u
6
+u
5
+2u
4
+2au+1, u
9
+u
8
+· · ·+b+1i
(i) Arc colorings
a
1
=
1
0
a
8
=
1
2
u
9
+
1
2
u
7
+ ··· +
1
2
u
1
2
u
9
u
8
3u
7
2u
6
3u
5
2u
4
2u
3
2u
2
2u 1
a
2
=
3
2
u
9
+ 2u
8
+ ··· +
3
2
u +
7
2
u
9
2u
8
4u
7
4u
6
4u
5
2u
4
2u
3
2u
2
2u 3
a
5
=
0
u
a
6
=
u
u
a
7
=
u
3
2u
u
3
+ u
a
3
=
1
2
u
9
+
1
2
u
7
+ ···
1
2
u
1
2
u
9
2u
8
3u
7
4u
6
2u
5
3u
4
u
3
2u
2
2u 1
a
9
=
1
2
u
9
3
2
u
7
+ ··· +
1
2
u +
1
2
u
9
+ 2u
8
+ 4u
7
+ 4u
6
+ 4u
5
+ 2u
4
+ 2u
3
+ u
2
+ 2u + 1
a
4
=
3
2
u
9
+ 2u
8
+ ··· +
5
2
u +
5
2
u
9
2u
8
5u
7
5u
6
7u
5
4u
4
4u
3
3u
2
u 3
a
11
=
u
2
+ 1
u
2
a
10
=
u
2
+ 1
u
4
2u
2
a
10
=
u
2
+ 1
u
4
2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 0.871979 0.168588I
a = 1.45312 1.11826I
b = 0.347624 + 1.331993I
2.90872 8.28632I 8.17560 + 6.14881I
u = 0.871979 + 0.168588I
a = 1.45312 + 1.11826I
b = 0.347624 1.331993I
2.90872 + 8.28632I 8.17560 6.14881I
u = 0.480814 1.084508I
a = 0.060819 0.521949I
b = 0.280829 + 1.292557I
5.70347 + 3.47839I 4.80497 2.79515I
u = 0.480814 + 1.084508I
a = 0.060819 + 0.521949I
b = 0.280829 1.292557I
5.70347 3.47839I 4.80497 + 2.79515I
u = 0.350885 1.264619I
a = 1.004172 + 0.634393I
b = 0.800451 + 0.099834I
1.58679 4.14585I 12.98134 + 3.97600I
u = 0.350885 + 1.264619I
a = 1.004172 0.634393I
b = 0.800451 0.099834I
1.58679 + 4.14585I 12.98134 3.97600I
u = 0.060791 1.179490I
a = 0.352899 + 0.448635I
b = 0.309318 + 0.396943I
2.84181 + 1.23169I 8.90177 5.44908I
u = 0.060791 + 1.179490I
a = 0.352899 0.448635I
b = 0.309318 0.396943I
2.84181 1.23169I 8.90177 + 5.44908I
u = 0.642886 0.580182I
a = 0.915208 0.578010I
b = 0.057928 + 1.351674I
8.22706 + 2.31006I 3.13631 3.52133I
u = 0.642886 + 0.580182I
a = 0.915208 + 0.578010I
b = 0.057928 1.351674I
8.22706 2.31006I 3.13631 + 3.52133I
13
VII. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
9
, c
11
(u
2
+ 1)(u
3
+ u + 1)
2
(u
6
+ u
5
+ 4u
4
+ 3u
3
+ 4u
2
+ 2u 1)
(u
10
u
9
+ 5u
8
5u
7
+ 9u
6
9u
5
+ 6u
4
6u
3
+ u
2
+ 1)
2
(u
10
+ 2u
9
+ 5u
8
+ 6u
7
+ 7u
6
+ 6u
5
+ 4u
4
+ 4u
3
+ 3u
2
+ 3u + 2)
c
8
, c
10
u
2
(u 1)
6
(u
6
+ u
5
u
4
+ 3u
3
+ 4u
2
12u 4)
(u
10
+ 2u
9
u
8
5u
7
3u
6
+ 4u
5
+ 12u
4
+ 13u
3
+ 5u
2
+ u + 2)
3
14
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
9
, c
11
(y + 1)
2
(1 + y + 2y
2
+ y
3
)
2
(y
6
+ 7y
5
+ ··· 12y + 1)
(y
10
+ 6y
9
+ 15y
8
+ 18y
7
+ 7y
6
6y
5
6y
4
+ y
2
+ 3y + 4)
(1 + 2y + 13y
2
6y
3
44y
4
21y
5
+ 41y
6
+ 59y
7
+ 33y
8
+ 9y
9
+ y
10
)
2
c
8
, c
10
y
2
(y 1)
6
(y
6
3y
5
+ 3y
4
y
3
+ 96y
2
176y + 16)
(4 + 19y + 47y
2
69y
3
+ 16y
4
+ 32y
5
17y
6
11y
7
+ 15y
8
6y
9
+ y
10
)
3
15