11n
90
(K11n
90
)
1
Arc Sequences
6 1 9 8 2 3 11 4 5 1 8
Solving Sequence
2,5
6
1,8
4 9 3 7 11 10
c
5
c
1
c
4
c
8
c
3
c
6
c
11
c
10
c
2
, c
7
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, a, b + u + 1i
I
u
2
= ha
4
2a
2
+ 4, a
2
+ 2u, a
3
+ a
2
+ 2b 2i
I
u
3
= hu
26
+ 2u
25
+ ··· + u + 3, 2612003u
25
+ 2633475u
24
+ ··· + 4443878b 5050430,
3316412u
25
+ 7083841u
24
+ ··· + 13331634a 28261507i
There are 3 irreducible components with 32 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
+ u + 1, a, b + u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
u 1
a
8
=
0
u 1
a
4
=
0
u
a
9
=
0
u 1
a
3
=
0
u
a
7
=
u
u + 1
a
11
=
u
2u 2
a
10
=
0
u 1
a
10
=
0
u 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 = 0.500000 0.866025I
a = 0
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 12.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0
b = 0.500000 0.866025I
1.64493 2.02988I 12.00000 + 3.46410I
3
II. I
u
2
= ha
4
2a
2
+ 4, a
2
+ 2u, a
3
+ a
2
+ 2b 2i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
2
a
2
a
6
=
1
2
a
2
1
2
a
2
a
1
=
1
2
a
2
1
2
a
2
1
a
8
=
a
1
2
a
3
1
2
a
2
+ 1
a
4
=
a
2
2
1
2
a
2
+ a 2
a
9
=
a
1
2
a
2
a 1
a
3
=
0
1
2
a
2
a
7
=
1
2
a
2
1
2
a
2
1
a
11
=
1
2
a
2
a
1
2
a
3
+ a
2
2
a
10
=
a
1
2
a
3
+
1
2
a
2
1
a
10
=
a
1
2
a
3
+
1
2
a
2
1
(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 = 0.500000 + 0.866025I
a = 1.22474 0.70711I
b = 0.50000 2.28024I
3.28987 + 2.02988I 6.00000 3.46410I
u = 0.500000 0.866025I
a = 1.22474 + 0.70711I
b = 0.50000 + 2.28024I
3.28987 2.02988I 6.00000 + 3.46410I
u = 0.500000 0.866025I
a = 1.22474 0.70711I
b = 0.500000 0.548188I
3.28987 2.02988I 6.00000 + 3.46410I
u = 0.500000 + 0.866025I
a = 1.22474 + 0.70711I
b = 0.500000 + 0.548188I
3.28987 + 2.02988I 6.00000 3.46410I
5
III.
I
u
3
= hu
26
+ 2u
25
+ · · · + u + 3, 2.61 × 10
6
u
25
+ 2.63 × 10
6
u
24
+ · · · + 4.44 ×
10
6
b 5.05 ×10
6
, 3.32× 10
6
u
25
+7.08× 10
6
u
24
+· · ·+ 1.33 ×10
7
a 2.83 ×10
7
i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
8
=
0.248763u
25
0.531356u
24
+ ··· 1.84971u + 2.11988
0.587776u
25
0.592607u
24
+ ··· 1.81642u + 1.13649
a
4
=
0.399078u
25
+ 1.03920u
24
+ ··· 0.102018u + 3.27267
0.653747u
25
+ 1.49959u
24
+ ··· + 3.62523u + 0.617162
a
9
=
0.350580u
25
+ 0.793132u
24
+ ··· + 1.83334u + 0.523587
0.588524u
25
+ 0.711798u
24
+ ··· 0.487769u 0.654627
a
3
=
u
4
+ u
2
+ 1
u
4
a
7
=
u
7
2u
5
2u
3
u
7
u
5
+ u
a
11
=
0.254668u
25
+ 0.460389u
24
+ ··· + 3.72725u 0.655503
0.576545u
25
+ 0.400932u
24
+ ··· + 0.277158u 1.33948
a
10
=
0.350580u
25
+ 0.793132u
24
+ ··· + 1.83334u + 0.523587
0.00919400u
25
0.493313u
24
+ ··· 1.63148u 0.930543
a
10
=
0.350580u
25
+ 0.793132u
24
+ ··· + 1.83334u + 0.523587
0.00919400u
25
0.493313u
24
+ ··· 1.63148u 0.930543
(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 = 0.987320 0.168214I
a = 0.72012 + 1.22782I
b = 0.577336 + 0.714767I
5.22414 5.39338I 6.45106 + 2.82273I
u = 0.987320 + 0.168214I
a = 0.72012 1.22782I
b = 0.577336 0.714767I
5.22414 + 5.39338I 6.45106 2.82273I
u = 0.683039 0.071498I
a = 0.45632 1.63078I
b = 0.358747 0.126452I
2.38839 2.21658I 3.00360 + 3.59199I
u = 0.683039 + 0.071498I
a = 0.45632 + 1.63078I
b = 0.358747 + 0.126452I
2.38839 + 2.21658I 3.00360 3.59199I
u = 0.576371 1.269309I
a = 0.786703 + 0.753829I
b = 0.80884 + 2.42539I
8.60064 + 11.02739I 8.81919 5.78425I
u = 0.576371 + 1.269309I
a = 0.786703 0.753829I
b = 0.80884 2.42539I
8.60064 11.02739I 8.81919 + 5.78425I
u = 0.432711 1.187741I
a = 0.759442 0.618449I
b = 1.27695 1.88674I
0.86443 + 6.41567I 7.45843 6.37638I
u = 0.432711 + 1.187741I
a = 0.759442 + 0.618449I
b = 1.27695 + 1.88674I
0.86443 6.41567I 7.45843 + 6.37638I
u = 0.409972 1.042736I
a = 0.653204 + 0.189915I
b = 0.987125 + 0.181713I
0.71901 + 1.35928I 6.71358 0.21049I
u = 0.409972 + 1.042736I
a = 0.653204 0.189915I
b = 0.987125 0.181713I
0.71901 1.35928I 6.71358 + 0.21049I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.376370 1.343313I
a = 0.992733 + 0.198233I
b = 0.835460 + 0.841765I
10.10765 0.68348I 10.33009 + 0.33748I
u = 0.376370 + 1.343313I
a = 0.992733 0.198233I
b = 0.835460 0.841765I
10.10765 + 0.68348I 10.33009 0.33748I
u = 0.280901 0.919746I
a = 0.423297 + 0.414948I
b = 0.356935 + 0.681008I
0.60039 + 1.42912I 6.05587 3.68708I
u = 0.280901 + 0.919746I
a = 0.423297 0.414948I
b = 0.356935 0.681008I
0.60039 1.42912I 6.05587 + 3.68708I
u = 0.086149 0.939073I
a = 1.54009 0.26805I
b = 0.41562 1.68165I
1.87196 0.46648I 9.69334 0.39377I
u = 0.086149 + 0.939073I
a = 1.54009 + 0.26805I
b = 0.41562 + 1.68165I
1.87196 + 0.46648I 9.69334 + 0.39377I
u = 0.232752 1.110797I
a = 0.314705 0.567127I
b = 0.09167 1.71081I
3.76323 2.26383I 13.05428 + 2.02208I
u = 0.232752 + 1.110797I
a = 0.314705 + 0.567127I
b = 0.09167 + 1.71081I
3.76323 + 2.26383I 13.05428 2.02208I
u = 0.339124
a = 1.23653
b = 0.422539
0.865956 11.4730
u = 0.49655 1.32834I
a = 0.178904 + 0.751212I
b = 0.16316 + 1.94567I
13.5263 5.3591I 12.04516 + 3.16064I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.49655 + 1.32834I
a = 0.178904 0.751212I
b = 0.16316 1.94567I
13.5263 + 5.3591I 12.04516 3.16064I
u = 0.541900 0.798242I
a = 1.16213 + 0.83131I
b = 0.06745 + 1.46140I
4.95516 2.19764I 0.54342 + 3.86213I
u = 0.541900 + 0.798242I
a = 1.16213 0.83131I
b = 0.06745 1.46140I
4.95516 + 2.19764I 0.54342 3.86213I
u = 0.714585 0.848170I
a = 0.598743 0.649572I
b = 0.51239 1.52112I
0.16795 2.70526I 6.45261 + 3.54399I
u = 0.714585 + 0.848170I
a = 0.598743 + 0.649572I
b = 0.51239 + 1.52112I
0.16795 + 2.70526I 6.45261 3.54399I
u = 1.01037
a = 1.00132
b = 0.327695
9.37437 9.45944
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u + 1)
2
(u
2
+ u + 1)(u
26
+ 2u
25
+ ··· + u + 3)
c
2
(u
2
+ u + 1)
3
(u
26
+ 16u
25
+ ··· 43u + 9)
c
3
, c
4
, c
8
u
2
(u
2
+ 2)
2
(u
26
+ u
25
+ ··· 8u 4)
c
5
(u
2
+ u + 1)
3
(u
26
+ 2u
25
+ ··· + u + 3)
c
6
(u
2
u + 1)
2
(u
2
+ u + 1)(u
26
+ 2u
25
+ ··· 13u + 3)
c
7
(u 1)
2
(u + 1)
4
(u
26
+ 3u
25
+ ··· + 22u 3)
c
9
u
2
(u
2
+ 2)
2
(u
26
+ u
25
+ ··· + 32u 4)
c
10
(u 1)
4
(u + 1)
2
(u
26
+ 33u
25
+ ··· + 64u + 9)
c
11
(u 1)
4
(u + 1)
2
(u
26
+ 3u
25
+ ··· + 22u 3)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
3
(y
26
+ 16y
25
+ ··· 43y + 9)
c
2
(y
2
+ y + 1)
3
(y
26
8y
25
+ ··· 7123y + 81)
c
3
, c
4
, c
8
y
2
(y + 2)
4
(y
26
+ 21y
25
+ ··· + 64y + 16)
c
6
(y
2
+ y + 1)
3
(y
26
32y
25
+ ··· 187y + 9)
c
7
, c
11
(y 1)
6
(y
26
33y
25
+ ··· 64y + 9)
c
9
y
2
(y + 2)
4
(y
26
39y
25
+ ··· 128y + 16)
c
10
(y 1)
6
(y
26
73y
25
+ ··· + 35108y + 81)
11