11a
362
(K11a
362
)
1
Arc Sequences
8 7 9 1 11 10 2 4 3 6 5
Solving Sequence
1,8 2,4
5 9 3 7 11 6 10
c
1
c
4
c
8
c
3
c
7
c
11
c
5
c
10
c
2
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
4
+ 3u
2
+ 1, u
2
+ a + 2, u
3
+ b + 2ui
I
u
2
= ha
12
a
11
4a
10
+ 12a
9
+ 17a
8
21a
7
22a
6
+ 18a
5
+ 12a
4
9a
3
8a
2
+ a + 5,
2211a
11
+ 7979b + ··· + 14136a + 12898, 7396a
11
+ 7979u + ··· + 40693a + 12954i
I
u
3
= hu
11
3u
10
+ 12u
9
25u
8
+ 50u
7
72u
6
+ 88u
5
83u
4
+ 61u
3
33u
2
+ 13u 2,
u
5
u
4
+ 4u
3
3u
2
+ b + 3u 1,
u
10
3u
9
+ 12u
8
25u
7
+ 50u
6
72u
5
+ 86u
4
81u
3
+ 53u
2
+ 2a 27u + 7i
There are 3 irreducible components with 27 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
4
+ 3u
2
+ 1, u
2
+ a + 2, u
3
+ b + 2ui
(i) Arc colorings
a
1
=
1
0
a
8
=
u
2
2
u
3
2u
a
2
=
u
3
+ 3u + 1
1
a
4
=
0
u
a
5
=
u
u
a
9
=
u
2
2
u
3
+ u
2
2u + 1
a
3
=
u
3
+ 3u
1
a
7
=
u
3
2u
0
a
11
=
u
2
+ 1
u
2
a
6
=
u
3
2u
u
3
+ u
a
10
=
0
u
2
+ 1
a
10
=
0
u
2
+ 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 = 0.618034I
a = 1.61803
b = 1.00000I
4.27683 0
u = 0.618034I
a = 1.61803
b = 1.00000I
4.27683 0
u = 1.61803I
a = 0.618034
b = 1.00000I
12.1725 0
u = 1.61803I
a = 0.618034
b = 1.00000I
12.1725 0
3
II. I
u
2
= ha
12
a
11
+ · · · + a + 5, 2211a
11
+ 7979b + · · · + 14136a +
12898, 7396a
11
+ 7979u + · · · + 40693a + 12954i
(i) Arc colorings
a
1
=
1
0
a
8
=
a
0.277102a
11
+ 0.0600326a
10
+ ··· 1.77165a 1.61649
a
2
=
0.337135a
11
0.368843a
10
+ ··· 1.89360a 0.385512
0.695200a
11
+ 0.515603a
10
+ ··· + 4.57501a + 0.717634
a
4
=
0
0.926933a
11
1.02080a
10
+ ··· 5.10001a 1.62351
a
5
=
0.926933a
11
+ 1.02080a
10
+ ··· + 5.10001a + 1.62351
0.926933a
11
1.02080a
10
+ ··· 5.10001a 1.62351
a
9
=
a
0.495927a
11
0.390525a
10
+ ··· 4.75686a 0.872666
a
3
=
1.16756a
11
1.71901a
10
+ ··· 4.54079a + 0.469357
2.37686a
11
+ 2.49818a
10
+ ··· + 10.4190a + 2.99674
a
7
=
0.0975060a
11
0.636045a
10
+ ··· + 0.641183a + 1.85951
0.0274470a
11
+ 1.55308a
10
+ ··· 3.82316a 7.24966
a
11
=
0.148766a
11
+ 0.367590a
10
+ ··· + 0.803233a 2.13398
0.148766a
11
0.367590a
10
+ ··· 0.803233a + 3.13398
a
6
=
0.0700589a
11
+ 0.917032a
10
+ ··· 3.18198a 5.39015
0.996992a
11
+ 0.103772a
10
+ ··· + 8.28199a + 7.01366
a
10
=
0.273468a
11
+ 0.565359a
10
+ ··· 0.137611a 1.15867
0.124702a
11
0.197769a
10
+ ··· + 0.940845a 0.975310
a
10
=
0.273468a
11
+ 0.565359a
10
+ ··· 0.137611a 1.15867
0.124702a
11
0.197769a
10
+ ··· + 0.940845a 0.975310
(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 = 0.321608 0.359079I
a = 1.283470 0.331087I
b = 0.010658 + 1.201246I
3.03178 1.10871I 7.53615 + 6.18117I
u = 0.321608 + 0.359079I
a = 1.283470 + 0.331087I
b = 0.010658 1.201246I
3.03178 + 1.10871I 7.53615 6.18117I
u = 0.142924 1.159516I
a = 1.229042 0.097653I
b = 0.778448 + 0.629355I
7.93269 2.65597I 2.41885 + 3.39809I
u = 0.142924 + 1.159516I
a = 1.229042 + 0.097653I
b = 0.778448 0.629355I
7.93269 + 2.65597I 2.41885 3.39809I
u = 0.03547 + 1.77530I
a = 0.388668 0.582415I
b = 0.11496 1.62096I
18.6443 + 3.4272I 2.04500 2.25224I
u = 0.03547 1.77530I
a = 0.388668 + 0.582415I
b = 0.11496 + 1.62096I
18.6443 3.4272I 2.04500 + 2.25224I
u = 0.142924 1.159516I
a = 0.616165 0.595407I
b = 0.06243 1.43905I
7.93269 2.65597I 2.41885 + 3.39809I
u = 0.142924 + 1.159516I
a = 0.616165 + 0.595407I
b = 0.06243 + 1.43905I
7.93269 + 2.65597I 2.41885 3.39809I
u = 0.03547 1.77530I
a = 0.913988 0.046495I
b = 1.047749 0.669346I
18.6443 3.4272I 2.04500 + 2.25224I
u = 0.03547 + 1.77530I
a = 0.913988 + 0.046495I
b = 1.047749 + 0.669346I
18.6443 + 3.4272I 2.04500 2.25224I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.321608 + 0.359079I
a = 1.87103 1.64610I
b = 0.293888 + 0.567347I
3.03178 + 1.10871I 7.53615 6.18117I
u = 0.321608 0.359079I
a = 1.87103 + 1.64610I
b = 0.293888 0.567347I
3.03178 1.10871I 7.53615 + 6.18117I
6
III. I
u
3
=
hu
11
3u
10
+· · ·+13u2, u
5
u
4
+4u
3
3u
2
+b+3u1, u
10
3u
9
+· · ·+2a+7i
(i) Arc colorings
a
1
=
1
0
a
8
=
1
2
u
10
+
3
2
u
9
+ ··· +
27
2
u
7
2
u
5
+ u
4
4u
3
+ 3u
2
3u + 1
a
2
=
1
2
u
10
+
1
2
u
9
+ ··· +
3
2
u +
1
2
u
10
2u
9
+ 9u
8
14u
7
+ 28u
6
32u
5
+ 35u
4
26u
3
+ 15u
2
6u + 1
a
4
=
0
u
a
5
=
u
u
a
9
=
1
2
u
10
+
3
2
u
9
+ ··· +
27
2
u
7
2
u
6
2u
5
+ 5u
4
7u
3
+ 6u
2
4u + 1
a
3
=
1
2
u
10
1
2
u
9
+ ···
3
2
u +
1
2
u
9
+ 2u
8
8u
7
+ 12u
6
21u
5
+ 22u
4
20u
3
+ 12u
2
5u + 1
a
7
=
u
5
4u
3
3u
u
5
+ 3u
3
+ u
a
11
=
u
2
+ 1
u
2
a
6
=
u
3
2u
u
3
+ u
a
10
=
u
4
+ 3u
2
+ 1
u
4
2u
2
a
10
=
u
4
+ 3u
2
+ 1
u
4
2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.00060 1.67078I
a = 0.370485 0.247020I
b = 0.412939 0.618853I
10.82277 + 1.46957I 5.57474 4.71346I
u = 0.00060 + 1.67078I
a = 0.370485 + 0.247020I
b = 0.412939 + 0.618853I
10.82277 1.46957I 5.57474 + 4.71346I
u = 0.083837 0.832172I
a = 0.510718 + 0.381336I
b = 0.360154 + 0.393035I
1.95559 + 1.25455I 6.26218 5.85654I
u = 0.083837 + 0.832172I
a = 0.510718 0.381336I
b = 0.360154 0.393035I
1.95559 1.25455I 6.26218 + 5.85654I
u = 0.11546 1.78711I
a = 0.945447 + 0.142265I
b = 0.36341 + 1.67319I
13.1804 + 8.7652I 0.57808 3.37097I
u = 0.11546 + 1.78711I
a = 0.945447 0.142265I
b = 0.36341 1.67319I
13.1804 8.7652I 0.57808 + 3.37097I
u = 0.278813
a = 1.12775
b = 0.314433
0.496230 19.9866
u = 0.425370 1.197894I
a = 1.267223 + 0.263212I
b = 0.22374 1.62996I
15.5882 + 6.3668I 0.98879 3.90232I
u = 0.425370 + 1.197894I
a = 1.267223 0.263212I
b = 0.22374 + 1.62996I
15.5882 6.3668I 0.98879 + 3.90232I
u = 0.735323 0.410034I
a = 0.99542 1.61358I
b = 0.07033 + 1.59466I
10.55845 + 2.37127I 3.60289 2.68530I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.735323 + 0.410034I
a = 0.99542 + 1.61358I
b = 0.07033 1.59466I
10.55845 2.37127I 3.60289 + 2.68530I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
3
c
7
, c
8
, c
9
(u
2
+ 1)
2
(u
11
+ 8u
9
+ ··· + 2u 1)
(u
12
+ u
11
+ ··· + 6u + 5)
c
4
, c
5
, c
6
c
10
, c
11
(u
4
+ 3u
2
+ 1)(u
6
u
5
+ 5u
4
4u
3
+ 6u
2
3u + 1)
2
(u
11
+ 3u
10
+ ··· + 13u + 2)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
3
c
7
, c
8
, c
9
(y + 1)
4
(y
11
+ 16y
10
+ ··· + 6y 1)(y
12
+ 11y
11
+ ··· + 124y + 25)
c
4
, c
5
, c
6
c
10
, c
11
(y
2
+ 3y + 1)
2
(y
6
+ 9y
5
+ 29y
4
+ 40y
3
+ 22y
2
+ 3y + 1)
2
(y
11
+ 15y
10
+ ··· + 37y 4)
10