11n
44
(K11n
44
)
1
Arc Sequences
4 1 7 2 10 11 4 6 1 8 9
Solving Sequence
1,4
2
5,10
6 9 8 11 7 3
c
1
c
4
c
5
c
9
c
8
c
11
c
6
c
3
c
2
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
3
+ u
2
1, b 1, 2u
2
+ a 4u 3i
I
u
2
= hb
5
+ b
4
2b
3
b
2
+ b 1, u 1, b + a 1i
I
u
3
= hu
41
+ 7u
40
+ ··· + 2u + 1, 4.55071 × 10
32
u
40
3.04971 × 10
33
u
39
+ ··· + 3.54634 × 10
33
b + 1.97956 × 10
32
,
1.27956 × 10
33
u
40
7.27594 × 10
33
u
39
+ ··· + 3.54634 × 10
33
a 1.00211 × 10
34
i
There are 3 irreducible components with 49 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
3
+ u
2
1, b 1, 2u
2
+ a 4u 3i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
2
+ u 1
a
10
=
2u
2
+ 4u + 3
1
a
6
=
7u
2
+ 11u + 9
2u
2
+ 3u
a
9
=
2u
2
+ 4u + 2
1
a
8
=
u
2u
2
+ u 2
a
11
=
2u
2
+ 4u + 3
1
a
7
=
u
u
2
+ u 1
a
3
=
u
2
+ 1
u
2
a
3
=
u
2
+ 1
u
2
(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 = 0.877439 0.744862I
a = 0.079596 0.365165I
b = 1.00000
4.66906 2.82812I 4.03193 6.06881I
u = 0.877439 + 0.744862I
a = 0.079596 + 0.365165I
b = 1.00000
4.66906 + 2.82812I 4.03193 + 6.06881I
u = 0.754878
a = 7.15919
b = 1.00000
0.531480 84.9361
3
II. I
u
2
= hb
5
+ b
4
2b
3
b
2
+ b 1, u 1, b + a 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
1
0
a
10
=
b + 1
b
a
6
=
b
2
+ b 1
b
2
a
9
=
1
b
a
8
=
0
b
4
b
3
b
2
+ 2b 1
a
11
=
b + 1
b
2
a
7
=
0
b
4
b
3
b
2
+ 2b 1
a
3
=
0
1
a
3
=
0
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 = 1.00000
a = 0.418784 0.219165I
b = 1.41878 0.21917I
4.22763 + 4.40083I 8.55516 1.78781I
u = 1.00000
a = 0.418784 + 0.219165I
b = 1.41878 + 0.21917I
4.22763 4.40083I 8.55516 + 1.78781I
u = 1.00000
a = 1.30992 0.54991I
b = 0.309916 0.549911I
1.31583 1.53058I 8.42731 + 4.45807I
u = 1.00000
a = 1.30992 + 0.54991I
b = 0.309916 + 0.549911I
1.31583 + 1.53058I 8.42731 4.45807I
u = 1.00000
a = 2.21774
b = 1.21774
0.756147 3.96494
5
III. I
u
3
=
hu
41
+7u
40
+· · ·+2u+1, 4.55×10
32
u
40
3.05×10
33
u
39
+· · ·+3.55×10
33
b+
1.98× 10
32
, 1.28× 10
33
u
40
7.28 ×10
33
u
39
+· · · +3.55 ×10
33
a 1.00× 10
34
i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
10
=
0.360812u
40
+ 2.05167u
39
+ ··· 8.17453u + 2.82576
0.128321u
40
+ 0.859959u
39
+ ··· 3.73417u 0.0558198
a
6
=
2.57419u
40
+ 13.6219u
39
+ ··· 13.2764u + 5.50487
0.824828u
40
4.46073u
39
+ ··· 3.42158u 0.703781
a
9
=
0.232491u
40
+ 1.19171u
39
+ ··· 4.44036u + 2.88158
0.128321u
40
+ 0.859959u
39
+ ··· 3.73417u 0.0558198
a
8
=
1.89468u
40
+ 10.8589u
39
+ ··· 3.67648u + 4.04534
0.427689u
40
2.12225u
39
+ ··· 3.33185u + 0.204351
a
11
=
0.101116u
40
0.258346u
39
+ ··· + 7.23983u 3.55682
0.153838u
40
+ 0.879491u
39
+ ··· + 4.81960u + 0.645350
a
7
=
1.89468u
40
+ 10.8589u
39
+ ··· 3.67648u + 4.04534
4.09416u
40
22.1611u
39
+ ··· 6.24482u 2.19947
a
3
=
u
2
+ 1
u
2
a
3
=
u
2
+ 1
u
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 = 1.68426 0.19522I
a = 0.197462 + 0.763635I
b = 0.402639 + 0.808887I
11.04374 1.47634I 7.66064 1.60453I
u = 1.68426 + 0.19522I
a = 0.197462 0.763635I
b = 0.402639 0.808887I
11.04374 + 1.47634I 7.66064 + 1.60453I
u = 1.61056 0.23381I
a = 0.423955 + 1.249646I
b = 0.32746 + 1.40800I
11.27278 7.05517I 7.07836 + 4.93484I
u = 1.61056 + 0.23381I
a = 0.423955 1.249646I
b = 0.32746 1.40800I
11.27278 + 7.05517I 7.07836 4.93484I
u = 1.58766 0.38818I
a = 0.05254 1.45672I
b = 1.36138 0.75276I
7.9395 14.4828I 4.83661 + 7.30920I
u = 1.58766 + 0.38818I
a = 0.05254 + 1.45672I
b = 1.36138 + 0.75276I
7.9395 + 14.4828I 4.83661 7.30920I
u = 1.56483 0.05519I
a = 0.05994 + 1.52905I
b = 1.049241 + 0.368090I
6.84034 1.09870I 9.88199 5.22097I
u = 1.56483 + 0.05519I
a = 0.05994 1.52905I
b = 1.049241 0.368090I
6.84034 + 1.09870I 9.88199 + 5.22097I
u = 1.53570 0.38602I
a = 0.149350 1.179903I
b = 1.157159 0.624027I
8.77885 6.88775I 5.65103 + 4.66706I
u = 1.53570 + 0.38602I
a = 0.149350 + 1.179903I
b = 1.157159 + 0.624027I
8.77885 + 6.88775I 5.65103 4.66706I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.51445 0.11394I
a = 0.47763 + 1.44493I
b = 1.22246 + 1.14875I
4.95010 4.48342I 3.30708 + 5.30527I
u = 1.51445 + 0.11394I
a = 0.47763 1.44493I
b = 1.22246 1.14875I
4.95010 + 4.48342I 3.30708 5.30527I
u = 1.48400
a = 0.697477
b = 1.91952
2.98279 1.32719
u = 0.817513 0.853964I
a = 0.255534 0.434819I
b = 0.895543 0.128230I
4.46595 3.11596I 9.1421 + 11.7493I
u = 0.817513 + 0.853964I
a = 0.255534 + 0.434819I
b = 0.895543 + 0.128230I
4.46595 + 3.11596I 9.1421 11.7493I
u = 0.762796 0.059538I
a = 0.685914 0.388453I
b = 1.354297 0.223700I
4.65237 4.48889I 10.07507 + 5.98728I
u = 0.762796 + 0.059538I
a = 0.685914 + 0.388453I
b = 1.354297 + 0.223700I
4.65237 + 4.48889I 10.07507 5.98728I
u = 0.213366 0.037411I
a = 1.99525 2.11282I
b = 0.146268 0.552391I
0.05575 + 1.50352I 0.38191 4.17550I
u = 0.213366 + 0.037411I
a = 1.99525 + 2.11282I
b = 0.146268 + 0.552391I
0.05575 1.50352I 0.38191 + 4.17550I
u = 0.059485 0.186213I
a = 2.78956 + 5.11758I
b = 1.278259 + 0.126441I
2.56340 0.10081I 4.27921 2.25595I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.059485 + 0.186213I
a = 2.78956 5.11758I
b = 1.278259 0.126441I
2.56340 + 0.10081I 4.27921 + 2.25595I
u = 0.380775 0.454242I
a = 0.670485 + 0.032746I
b = 1.007361 0.614689I
1.43126 + 2.56358I 1.01752 7.87421I
u = 0.380775 + 0.454242I
a = 0.670485 0.032746I
b = 1.007361 + 0.614689I
1.43126 2.56358I 1.01752 + 7.87421I
u = 0.387590 0.911908I
a = 0.971331 + 0.325235I
b = 0.764541 + 0.597354I
2.61027 + 2.03740I 5.72892 3.65159I
u = 0.387590 + 0.911908I
a = 0.971331 0.325235I
b = 0.764541 0.597354I
2.61027 2.03740I 5.72892 + 3.65159I
u = 0.508644 1.042803I
a = 0.681098 + 0.649137I
b = 1.178244 + 0.659530I
1.17487 + 9.23550I 3.04368 7.03311I
u = 0.508644 + 1.042803I
a = 0.681098 0.649137I
b = 1.178244 0.659530I
1.17487 9.23550I 3.04368 + 7.03311I
u = 0.612280 0.220916I
a = 0.42857 4.33330I
b = 0.908176 + 0.005657I
0.484163 + 0.158339I 13.38590 1.00156I
u = 0.612280 + 0.220916I
a = 0.42857 + 4.33330I
b = 0.908176 0.005657I
0.484163 0.158339I 13.38590 + 1.00156I
u = 0.695393 0.752192I
a = 0.0733729 0.1016162I
b = 0.382217 0.951284I
3.58550 + 3.36599I 6.85826 4.39505I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.695393 + 0.752192I
a = 0.0733729 + 0.1016162I
b = 0.382217 + 0.951284I
3.58550 3.36599I 6.85826 + 4.39505I
u = 0.737003
a = 0.918962
b = 0.00861004
1.10369 8.82467
u = 0.858145 0.924978I
a = 0.0136488 + 0.1120601I
b = 0.908643 0.588751I
2.16828 2.66511I 5.14222 + 3.41368I
u = 0.858145 + 0.924978I
a = 0.0136488 0.1120601I
b = 0.908643 + 0.588751I
2.16828 + 2.66511I 5.14222 3.41368I
u = 0.883212
a = 5.41324
b = 1.04711
0.458131 57.1146
u = 1.149305 0.071261I
a = 2.67006 0.97064I
b = 0.783716 0.351647I
0.578838 + 1.255807I 2.38019 + 0.06658I
u = 1.149305 + 0.071261I
a = 2.67006 + 0.97064I
b = 0.783716 + 0.351647I
0.578838 1.255807I 2.38019 0.06658I
u = 1.46523 0.11671I
a = 0.138455 + 1.392323I
b = 0.596164 + 0.809978I
5.33009 0.54259I 7.21463 + 0.52853I
u = 1.46523 + 0.11671I
a = 0.138455 1.392323I
b = 0.596164 0.809978I
5.33009 + 0.54259I 7.21463 0.52853I
u = 1.60618 0.13682I
a = 0.489280 + 1.112378I
b = 1.046201 + 0.671367I
3.96071 + 6.10430I 4.84103 5.05221I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.60618 + 0.13682I
a = 0.489280 1.112378I
b = 1.046201 0.671367I
3.96071 6.10430I 4.84103 + 5.05221I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
3
+ u
2
1)(u
41
+ 7u
40
+ ··· + 2u + 1)
c
2
(u + 1)
5
(u
3
+ u
2
+ 2u + 1)(u
41
+ 43u
40
+ ··· + 12u + 1)
c
3
u
5
(u
3
u
2
+ 2u 1)(u
41
+ 2u
40
+ ··· + 96u + 32)
c
4
(u + 1)
5
(u
3
u
2
+ 1)(u
41
+ 7u
40
+ ··· + 2u + 1)
c
5
(u
3
+ 2u
2
3u + 1)(u
5
u
4
2u
3
+ u
2
+ u + 1)
(u
41
+ 16u
39
+ ··· 1085u 79)
c
6
(u
3
+ 2u
2
3u + 1)(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
(u
41
+ 4u
40
+ ··· 237u + 191)
c
7
u
5
(u
3
+ u
2
+ 2u + 1)(u
41
+ 2u
40
+ ··· + 96u + 32)
c
8
(u
3
3u
2
+ 2u + 1)(u
5
3u
4
+ 4u
3
u
2
u + 1)
(u
41
+ 3u
40
+ ··· 2u 1)
c
9
(u + 1)
3
(u
5
u
4
+ ··· + u + 1)(u
41
+ 5u
40
+ ··· + 119u + 1)
c
10
u
3
(u
5
u
4
+ ··· + u 1)(u
41
+ 6u
40
+ ··· + 156u + 8)
c
11
(u 1)
3
(u
5
+ u
4
+ ··· + u 1)(u
41
+ 5u
40
+ ··· + 119u + 1)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
3
y
2
+ 2y 1)(y
41
43y
40
+ ··· + 12y 1)
c
2
(y 1)
5
(y
3
+ 3y
2
+ 2y 1)(y
41
83y
40
+ ··· + 1144y 1)
c
3
, c
7
y
5
(y
3
+ 3y
2
+ 2y 1)(y
41
30y
40
+ ··· + 3584y 1024)
c
5
(y
3
10y
2
+ 5y 1)(y
5
5y
4
+ 8y
3
3y
2
y 1)
(y
41
+ 32y
40
+ ··· + 183721y 6241)
c
6
(y
3
10y
2
+ 5y 1)(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
(y
41
+ 8y
40
+ ··· + 999709y 36481)
c
8
(y
3
5y
2
+ 10y 1)(y
5
y
4
+ 8y
3
3y
2
+ 3y 1)
(y
41
11y
40
+ ··· + 26y 1)
c
9
, c
11
(y 1)
3
(y
5
5y
4
+ ··· y 1)(y
41
21y
40
+ ··· + 13495y 1)
c
10
y
3
(y
5
+ 3y
4
+ ··· y 1)(y
41
18y
40
+ ··· + 7824y 64)
12