11n
14
(K11n
14
)
1
Arc Sequences
5 1 9 2 3 10 11 3 6 7 9
Solving Sequence
3,9 4,6
10 7 11 1 2 5 8
c
3
c
9
c
6
c
10
c
11
c
2
c
5
c
8
c
1
, c
4
, c
7
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
4
+ a
3
+ 2a
2
a + 1, b, a
3
+ 2a
2
+ 2a + 2u 1i
I
u
2
= hu
26
3u
25
+ ··· 11u + 2,
4.48569 × 10
23
u
25
+ 1.49428 × 10
24
u
24
+ ··· + 3.41287 × 10
25
b 4.73072 × 10
24
,
1.03148 × 10
25
u
25
3.10386 × 10
25
u
24
+ ··· + 3.41287 × 10
25
a 9.41886 × 10
25
i
There are 2 irreducible components with 30 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
4
+ a
3
+ 2a
2
a + 1, b, a
3
+ 2a
2
+ 2a + 2u 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
a
0
a
4
=
1
0
a
6
=
0
1
2
a
3
a
2
a +
1
2
a
10
=
a
1
2
a
3
+ a
1
2
a
7
=
1
2
a
3
+ a
2
+
1
2
1
2
a
3
a +
1
2
a
11
=
1
2
a
3
+ a
2
+
1
2
1
2
a
3
+ a
2
+ a
1
2
a
1
=
0
1
2
a
3
+ a
2
+ a
1
2
a
2
=
1
1
2
a
3
+ a
2
+ a +
1
2
a
5
=
1
2
a
3
a
2
a +
1
2
1
2
a
3
a
2
a +
1
2
a
8
=
a
0
a
8
=
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 = 0.500000 0.866025I
a = 0.80902 1.40126I
b = 0
8.88264 + 2.02988I 13.5000 5.4006I
u = 0.500000 + 0.866025I
a = 0.80902 + 1.40126I
b = 0
8.88264 2.02988I 13.5000 + 5.4006I
u = 0.500000 + 0.866025I
a = 0.309017 0.535233I
b = 0
0.98696 2.02988I 13.50000 + 1.52761I
u = 0.500000 0.866025I
a = 0.309017 + 0.535233I
b = 0
0.98696 + 2.02988I 13.50000 1.52761I
3
II. I
u
2
=
hu
26
3u
25
+· · ·11u+2, 4.49×10
23
u
25
+1.49×10
24
u
24
+· · ·+3.41×10
25
b
4.73 × 10
24
, 1.03 × 10
25
u
25
3.10 × 10
25
u
24
+ · · · + 3.41 × 10
25
a 9.42 × 10
25
i
(i) Arc colorings
a
3
=
1
0
a
9
=
0.302232u
25
+ 0.909459u
24
+ ··· 12.0193u + 2.75981
0.0131435u
25
0.0437836u
24
+ ··· + 2.52319u + 0.138614
a
4
=
0.0726343u
25
0.258082u
24
+ ··· + 7.84100u 1.19405
0.0365577u
25
+ 0.109811u
24
+ ··· 2.42263u + 0.217909
a
6
=
0
u
a
10
=
0.302232u
25
+ 0.909459u
24
+ ··· 12.0193u + 2.75981
0.0187109u
25
0.0558617u
24
+ ··· + 1.88834u + 0.144140
a
7
=
0.256474u
25
+ 0.768857u
24
+ ··· 9.19278u + 1.38268
0.0187109u
25
0.0558617u
24
+ ··· + 1.88834u + 0.144140
a
11
=
0.187698u
25
+ 0.587988u
24
+ ··· 7.75839u + 2.46154
u
a
1
=
0.108955u
25
0.290306u
24
+ ··· + 2.82947u + 1.22413
0.0403172u
25
+ 0.126519u
24
+ ··· 0.210851u 0.218384
a
2
=
0.131680u
25
0.345666u
24
+ ··· + 2.78713u + 1.28611
0.0175918u
25
+ 0.0711603u
24
+ ··· 0.253196u 0.156404
a
5
=
u
u
a
8
=
0.315376u
25
+ 0.953243u
24
+ ··· 14.5425u + 2.62119
0.0131435u
25
0.0437836u
24
+ ··· + 2.52319u + 0.138614
a
8
=
0.315376u
25
+ 0.953243u
24
+ ··· 14.5425u + 2.62119
0.0131435u
25
0.0437836u
24
+ ··· + 2.52319u + 0.138614
(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 = 1.59480 0.23303I
a = 0.255679 0.457783I
b = 0.15314 1.46790I
4.15442 2.25820I 7.09524 + 3.00458I
u = 1.59480 + 0.23303I
a = 0.255679 + 0.457783I
b = 0.15314 + 1.46790I
4.15442 + 2.25820I 7.09524 3.00458I
u = 1.54625 0.70491I
a = 0.764118 + 0.487826I
b = 0.46334 + 1.44028I
2.54423 5.47373I 10.67253 + 2.88121I
u = 1.54625 + 0.70491I
a = 0.764118 0.487826I
b = 0.46334 1.44028I
2.54423 + 5.47373I 10.67253 2.88121I
u = 1.191587 0.262632I
a = 1.125708 0.330748I
b = 1.242204 + 0.129924I
3.94867 3.99401I 9.09163 + 3.57778I
u = 1.191587 + 0.262632I
a = 1.125708 + 0.330748I
b = 1.242204 0.129924I
3.94867 + 3.99401I 9.09163 3.57778I
u = 0.536127 0.217517I
a = 0.070226 + 0.884348I
b = 0.861165 + 0.293695I
1.40190 2.19157I 3.35211 + 5.42014I
u = 0.536127 + 0.217517I
a = 0.070226 0.884348I
b = 0.861165 0.293695I
1.40190 + 2.19157I 3.35211 5.42014I
u = 0.460465 1.052825I
a = 0.029394 0.305810I
b = 0.007894 0.508582I
0.44924 2.24817I 0.24032 + 5.78182I
u = 0.460465 + 1.052825I
a = 0.029394 + 0.305810I
b = 0.007894 + 0.508582I
0.44924 + 2.24817I 0.24032 5.78182I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.36258 1.62911I
a = 0.171404 + 0.865575I
b = 0.077948 + 1.101212I
6.47513 2.78553I 10.00226 + 3.18308I
u = 0.36258 + 1.62911I
a = 0.171404 0.865575I
b = 0.077948 1.101212I
6.47513 + 2.78553I 10.00226 3.18308I
u = 0.020892 0.242346I
a = 0.82583 + 3.08557I
b = 0.568442 0.590625I
0.141642 + 0.491245I 7.19488 1.21216I
u = 0.020892 + 0.242346I
a = 0.82583 3.08557I
b = 0.568442 + 0.590625I
0.141642 0.491245I 7.19488 + 1.21216I
u = 0.195879
a = 2.28864
b = 0.442855
0.706372 14.0845
u = 0.266246 0.474227I
a = 1.42314 2.65603I
b = 0.352840 + 0.587473I
8.45818 + 1.37920I 7.13203 + 2.69707I
u = 0.266246 + 0.474227I
a = 1.42314 + 2.65603I
b = 0.352840 0.587473I
8.45818 1.37920I 7.13203 2.69707I
u = 0.956110
a = 1.55931
b = 1.04159
7.14521 13.5408
u = 1.239851 0.375242I
a = 1.012839 + 0.524585I
b = 0.24847 + 1.60710I
2.20853 + 1.36342I 6.29553 0.38377I
u = 1.239851 + 0.375242I
a = 1.012839 0.524585I
b = 0.24847 1.60710I
2.20853 1.36342I 6.29553 + 0.38377I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.56800 0.06124I
a = 0.257064 0.495904I
b = 0.10895 1.65464I
8.25588 2.37235I 3.41364 + 0.56644I
u = 1.56800 + 0.06124I
a = 0.257064 + 0.495904I
b = 0.10895 + 1.65464I
8.25588 + 2.37235I 3.41364 0.56644I
u = 1.73126 0.24397I
a = 0.287489 0.453974I
b = 0.38599 1.61721I
7.73283 + 7.28919I 4.65018 5.96812I
u = 1.73126 + 0.24397I
a = 0.287489 + 0.453974I
b = 0.38599 + 1.61721I
7.73283 7.28919I 4.65018 + 5.96812I
u = 1.78957 0.55743I
a = 0.760995 + 0.371694I
b = 0.62734 + 1.51647I
0.54439 + 10.83966I 8.04696 6.04188I
u = 1.78957 + 0.55743I
a = 0.760995 0.371694I
b = 0.62734 1.51647I
0.54439 10.83966I 8.04696 + 6.04188I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
2
(u
26
+ 3u
25
+ ··· 3u + 1)
c
2
(u
2
+ u + 1)
2
(u
26
+ 15u
25
+ ··· 23u + 1)
c
3
, c
8
u
4
(u
26
+ u
25
+ ··· 16u 16)
c
4
(u
2
u + 1)
2
(u
26
+ 3u
25
+ ··· 3u + 1)
c
5
(u
2
+ u + 1)
2
(u
26
+ 3u
25
+ ··· + 11u + 2)
c
6
, c
7
(u
2
u 1)
2
(u
26
+ 3u
25
+ ··· + u 1)
c
9
, c
10
(u
2
+ u 1)
2
(u
26
+ 3u
25
+ ··· + u 1)
c
11
(u
2
+ u 1)
2
(u
26
+ 3u
25
+ ··· + 3u 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
2
(y
26
+ 15y
25
+ ··· 23y + 1)
c
2
(y
2
+ y + 1)
2
(y
26
5y
25
+ ··· 795y + 1)
c
3
, c
8
y
4
(y
26
+ 25y
25
+ ··· + 1664y + 256)
c
5
(y
2
+ y + 1)
2
(y
26
25y
25
+ ··· + 7y + 4)
c
6
, c
7
, c
9
c
10
(y
2
3y + 1)
2
(y
26
29y
25
+ ··· 19y + 1)
c
11
(y
2
3y + 1)
2
(y
26
+ 31y
25
+ ··· 19y + 1)
9