11n
53
(K11n
53
)
1
Arc Sequences
4 1 8 2 9 10 4 1 11 6 7
Solving Sequence
7,10
6 11
1,4
2 3 9 5 8
c
6
c
10
c
11
c
1
c
2
c
9
c
5
c
8
c
3
, c
4
, c
7
Representation Ideals
I = I
u
1
\
I
v
1
I
u
1
= hu
23
+ u
22
+ ··· + 32u + 32, 3.31708 × 10
25
u
22
+ 8.27738 × 10
25
u
21
+ ··· + 1.29432 × 10
27
a 2.01433 × 10
27
,
8.91772 × 10
25
u
22
+ 1.37401 × 10
26
u
21
+ ··· + 2.58863 × 10
27
b 4.38457 × 10
27
i
I
v
1
= h−v
3
+ b + 2v, v
5
+ v
4
2v
3
v
2
+ v 1, ai
There are 2 irreducible components with 28 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
23
+u
22
+· · ·+32u+32, 3.32×10
25
u
22
+8.28×10
25
u
21
+· · ·+1.29×10
27
a
2.01 × 10
27
, 8.92 × 10
25
u
22
+ 1.37 × 10
26
u
21
+ · · · + 2.59 × 10
27
b 4.38 × 10
27
i
(i) Arc colorings
a
7
=
0
u
a
10
=
0.0256281u
22
0.0639518u
21
+ ··· + 6.80037u + 1.55629
0.0344495u
22
0.0530787u
21
+ ··· + 4.19086u + 1.69378
a
6
=
0.0225902u
22
0.0188226u
21
+ ··· + 3.37798u + 2.21916
0.0102620u
22
0.0193633u
21
+ ··· + 0.476411u + 0.0791350
a
11
=
0.00970061u
22
0.0254070u
21
+ ··· 0.0897565u 1.17800
0.00742192u
22
0.00272547u
21
+ ··· + 0.485063u + 0.285039
a
1
=
0.00970061u
22
0.0254070u
21
+ ··· 0.0897565u 1.17800
0.00456961u
22
0.0157254u
21
+ ··· 0.327962u 0.217567
a
4
=
1
0
a
2
=
0.00513100u
22
0.00968167u
21
+ ··· + 0.238205u 0.960433
0.00456961u
22
0.0157254u
21
+ ··· 0.327962u 0.217567
a
3
=
u
2
1
u
2
a
9
=
0.0138226u
22
+ 0.0208365u
21
+ ··· + 0.896559u 0.671863
0.00455068u
22
+ 0.00587223u
21
+ ··· + 0.796241u 0.164192
a
5
=
0.00513100u
22
+ 0.00968167u
21
+ ··· 0.238205u + 0.960433
0.00324806u
22
0.0176451u
21
+ ··· 0.637775u 0.363188
a
8
=
u
u
a
8
=
u
u
(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 = 1.105496 0.395247I
a = 0.084041 + 0.972018I
b = 0.15192 + 3.13181I
0.82985 + 3 .19017I 0.92409 4.72756I
u = 1.105496 + 0 .395247I
a = 0.084041 0.972018I
b = 0.15192 3.13181I
0.82985 3.19017I 0.92409 + 4.72756I
u = 0.57374 1.89861I
a = 0.485150 0.448663I
b = 2.81610 2.77328I
8.35091 + 10.39485I 0.03516 5.66347I
u = 0.57374 + 1 .89861I
a = 0.485150 + 0.448663I
b = 2.81610 + 2.77328I
8.35091 10.39485I 0.03516 + 5 .66347I
u = 0.562756
a = 1.27903
b = 0.379024
1.81820 6.35978
u = 0.349805 0.349927I
a = 0.69346 1.46706I
b = 0.009371 0.594952I
0.390924 + 1.217985I 4.26810 5.45735I
u = 0.349805 + 0 .349927I
a = 0.69346 + 1.46706I
b = 0.009371 + 0 .594952I
0.390924 1.217985I 4.26810 + 5.45735I
u = 0.34057 1.76704I
a = 0.704218 0.016799I
b = 3.13054 + 1 .00295I
6.23110 + 1 .56382I 1.72734 0.04821I
u = 0.34057 + 1 .76704I
a = 0.704218 + 0.016799I
b = 3.13054 1.00295I
6.23110 1.56382I 1.72734 + 0.04821I
u = 0.143374 1.329067I
a = 0.862822 0.293705I
b = 1.37956 1.12053I
0.55803 6.09572I 0.93537 + 6.19372I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.143374 + 1.329067I
a = 0.862822 + 0.293705I
b = 1.37956 + 1.12053I
0.55803 + 6 .09572I 0.93537 6.19372I
u = 0.09416 2.13003I
a = 0.357605 + 0.344657I
b = 0.95917 + 1.21374I
14.1192 2.6463I 3.65714 + 2 .84707I
u = 0.09416 + 2.13003I
a = 0.357605 0.344657I
b = 0.95917 1.21374I
14.1192 + 2.6463I 3.65714 2.84707I
u = 0.215406 0.997383I
a = 1.103343 0.129187I
b = 0.963592 0.044916I
1.62847 + 1 .16584I 3.47504 0.42481I
u = 0.215406 + 0.997383I
a = 1.103343 + 0.129187I
b = 0.963592 + 0.044916I
1.62847 1.16584I 3.47504 + 0.42481I
u = 0.247796 1.071449I
a = 0.129135 0.600598I
b = 0.268593 0.514264I
1.75612 + 1 .64275I 3.33585 2.40342I
u = 0.247796 + 1.071449I
a = 0.129135 + 0.600598I
b = 0.268593 + 0.514264I
1.75612 1.64275I 3.33585 + 2 .40342I
u = 0.45428 1.93201I
a = 0.229112 0.389086I
b = 0.630634 + 0.372852I
11.07970 5.13429I 2.96602 + 2 .08249I
u = 0.45428 + 1.93201I
a = 0.229112 + 0.389086I
b = 0.630634 0.372852I
11.07970 + 5.13429I 2.96602 2.08249I
u = 0.608111 0.025252I
a = 0.66639 + 1.97201I
b = 0.13770 + 1.49354I
5.15559 4.27437I 9.78577 + 3.28837I
4
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.608111 + 0.025252I
a = 0.66639 1.97201I
b = 0.13770 1.49354I
5.15559 + 4 .27437I 9.78577 3.28837I
u = 0.674614 0.656598I
a = 0.729660 0.035075I
b = 0.817939 + 0.104980I
2.46377 + 0 .78545I 4.30142 0.09221I
u = 0.674614 + 0.656598I
a = 0.729660 + 0.035075I
b = 0.817939 0.104980I
2.46377 0.78545I 4.30142 + 0 .09221I
5
II. I
v
1
= h−v
3
+ b + 2v, v
5
+ v
4
2v
3
v
2
+ v 1, ai
(i) Arc colorings
a
7
=
v
0
a
10
=
0
v
3
2v
a
6
=
v
v
2
+ 1
a
11
=
v
4
v
2
+ v 1
1
a
1
=
v
4
2v
2
+ v 1
1
a
4
=
1
0
a
2
=
v
4
2v
2
+ v
1
a
3
=
1
0
a
9
=
v
4
+ 2v
2
v + 1
v
a
5
=
v
4
+ 2v
2
v + 1
1
a
8
=
v
0
a
8
=
v
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.41878 0.21917I
a = 0
b = 0.186078 0.874646I
4.22763 + 4.40083I 0.31681 3.97407I
v = 1.41878 + 0.21917I
a = 0
b = 0.186078 + 0.874646I
4.22763 4.40083I 0.31681 + 3.97407I
v = 0.309916 0.549911I
a = 0
b = 0.871221 + 1 .107662I
1.31583 1.53058I 0.02124 + 2.62456I
v = 0.309916 + 0.549911I
a = 0
b = 0.871221 1.107662I
1.31583 + 1.53058I 0.02124 2.62456I
v = 1.21774
a = 0
b = 0.629714
0.756147 2.67610
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
23
+ 6u
22
+ ··· + 4u + 1)
c
2
(u + 1)
5
(u
23
+ 32u
22
+ ··· 16u + 1)
c
3
, c
7
u
5
(u
23
+ u
22
+ ··· + 32u + 32)
c
4
(u + 1)
5
(u
23
+ 6u
22
+ ··· + 4u + 1)
c
5
(u
5
u
4
+ ··· + u + 1)(u
23
+ 2u
22
+ ··· + 30u + 9)
c
6
(u
5
+ u
4
+ ··· + u + 1)(u
23
+ 2u
22
+ ··· 2u 1)
c
8
(u
5
u
4
+ ··· + u + 1)(u
23
+ 24u
21
+ ··· + 2u 1)
c
9
(u
5
+ 3u
4
+ ··· u 1)(u
23
+ 12u
22
+ ··· 2u 1)
c
10
(u
5
u
4
+ ··· + u 1)(u
23
+ 2u
22
+ ··· 2u 1)
c
11
(u
5
+ u
4
+ ··· + u 1)(u
23
+ 2u
22
+ ··· + 30u + 9)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
23
32y
22
+ ··· 16y 1)
c
2
(y 1)
5
(y
23
76y
22
+ ··· + 772y 1)
c
3
, c
7
y
5
(y
23
+ 33y
22
+ ··· + 3584y 1024)
c
5
, c
11
(y
5
5y
4
+ ··· y 1)(y
23
12y
22
+ ··· 162y 81)
c
6
, c
10
(y
5
+ 3y
4
+ ··· y 1)(y
23
+ 12y
22
+ ··· 2y 1)
c
8
(y
5
5y
4
+ ··· y 1)(y
23
+ 48y
22
+ ··· 2y 1)
c
9
(y
5
y
4
+ ··· + 3y 1)(y
23
+ 24y
21
+ ··· 6y 1)
9