11n
69
(K11n
69
)
1
Arc Sequences
4 1 8 2 10 9 11 4 6 1 7
Solving Sequence
4,8
9
3,11
7 1 2 5 6 10
c
8
c
3
c
7
c
11
c
2
c
4
c
6
c
10
c
1
, c
5
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
\
I
v
1
I
u
1
= ha
6
+ 3a
4
a
3
+ a
2
+ a 1, a
5
+ a
4
+ 4a
3
+ a
2
+ 2a + 2u 1, a
5
+ a
4
2a
3
+ 3a
2
+ 2b + 2a 1i
I
u
2
= hu
20
+ 2u
19
+ ··· + 112u + 64,
17520363573484u
19
30313572228603u
18
+ ··· + 2543787008583104b 442949904703936,
93935953407515u
19
+ 170357340159806u
18
+ ··· + 5087574017166208a 4178460022729760i
I
v
1
= hv
4
v
3
+ 5v
2
+ v + 2, v
3
+ 2v
2
+ 4b 5v + 4, ai
There are 3 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
6
+ 3a
4
a
3
+ a
2
+ a 1, a
5
+ a
4
+ 4a
3
+ a
2
+ 2a + 2u
1, a
5
+ a
4
2a
3
+ 3a
2
+ 2b + 2a 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
1
2
a
5
1
2
a
4
+ ··· a +
1
2
a
9
=
1
2
a
5
+
1
2
a
4
+ ··· + a
1
2
1
2
a
5
1
2
a
4
+ ··· a +
1
2
a
3
=
1
1
2
a
5
+
1
2
a
4
+ ··· + a
3
2
a
11
=
a
1
2
a
5
1
2
a
4
+ ··· a +
1
2
a
7
=
1
2
a
5
1
2
a
4
+ a
3
3
2
a
2
+
1
2
1
2
a
5
+
1
2
a
4
+ ··· a
1
2
a
1
=
1
2
a
5
+
1
2
a
4
+ ··· + a
1
2
1
2
a
5
1
2
a
4
+ ··· a +
1
2
a
2
=
a
5
+ a
4
+ 4a
3
+ a
2
+ 2a 1
1
2
a
5
1
2
a
4
+ ··· a +
1
2
a
5
=
a
5
+ a
4
+ 4a
3
+ a
2
+ 2a 2
1
2
a
5
1
2
a
4
+ ··· a +
3
2
a
6
=
a
5
+ 2a
3
a
2
2a + 1
a
5
2a
3
+ a
2
+ a 1
a
10
=
a
5
a
4
+ 2a
3
4a
2
+ a + 1
1
2
a
5
+
1
2
a
4
+ ··· a
1
2
a
10
=
a
5
a
4
+ 2a
3
4a
2
+ a + 1
1
2
a
5
+
1
2
a
4
+ ··· a
1
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 = 1.61803
a = 0.704116
b = 0.0980714
8.88264 10.0000
u = 0.618034
a = 0.27666 1.68750I
b = 1.43598 + 2.26458I
0.986960 10.0000
u = 0.618034
a = 0.27666 + 1.68750I
b = 1.43598 2.26458I
0.986960 10.0000
u = 1.61803
a = 0.352058 0.868217I
b = 0.66707 + 1.85736I
8.88264 10.0000
u = 1.61803
a = 0.352058 + 0.868217I
b = 0.66707 1.85736I
8.88264 10.0000
u = 0.618034
a = 0.553325
b = 0.364102
0.986960 10.0000
3
II. I
u
2
=
hu
20
+2u
19
+· · ·+112u+64, 1.75×10
13
u
19
3.03×10
13
u
18
+· · ·+2.54×10
15
b
4.43 × 10
14
, 9.39 × 10
13
u
19
+ 1.70 × 10
14
u
18
+ · · · + 5.09 × 10
15
a 4.18 × 10
15
i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
9
=
u
u
a
3
=
1
u
2
a
11
=
0.0184638u
19
0.0334850u
18
+ ··· 5.59696u + 0.821307
0.00688751u
19
+ 0.0119167u
18
+ ··· + 1.66358u + 0.174130
a
7
=
0.00412161u
19
0.0208251u
18
+ ··· 4.99864u 3.54213
0.00147297u
19
+ 0.00519601u
18
+ ··· + 1.30383u + 0.495070
a
1
=
0.00944101u
19
+ 0.0161658u
18
+ ··· + 1.03095u + 0.796495
0.00317617u
19
+ 0.00588576u
18
+ ··· + 0.410496u + 0.216538
a
2
=
0.00626484u
19
+ 0.0102800u
18
+ ··· + 0.620452u + 0.579958
0.00317617u
19
+ 0.00588576u
18
+ ··· + 0.410496u + 0.216538
a
5
=
0.00774647u
19
+ 0.0156318u
18
+ ··· + 1.14144u + 1.18687
0.00169454u
19
+ 0.000533951u
18
+ ··· 0.110492u 0.390378
a
6
=
0.00270212u
19
0.0151195u
18
+ ··· 3.67196u 2.88089
0.0000534837u
19
0.000509593u
18
+ ··· 0.0228522u 0.166170
a
10
=
0.00155754u
19
+ 0.0115742u
18
+ ··· + 0.551041u + 1.57152
0.0111325u
19
+ 0.0216207u
18
+ ··· + 2.69742u + 0.238205
a
10
=
0.00155754u
19
+ 0.0115742u
18
+ ··· + 0.551041u + 1.57152
0.0111325u
19
+ 0.0216207u
18
+ ··· + 2.69742u + 0.238205
(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 = 1.93768 0.60339I
a = 0.057267 + 0.617600I
b = 0.37765 2.56572I
2.03730 5.64317I 8.76092 + 6.80873I
u = 1.93768 + 0.60339I
a = 0.057267 0.617600I
b = 0.37765 + 2.56572I
2.03730 + 5.64317I 8.76092 6.80873I
u = 1.80160 0.49892I
a = 0.079382 0.800981I
b = 0.28710 + 2.35034I
13.3225 5.9895I 12.03384 + 3.05262I
u = 1.80160 + 0.49892I
a = 0.079382 + 0.800981I
b = 0.28710 2.35034I
13.3225 + 5.9895I 12.03384 3.05262I
u = 1.47115 0.51322I
a = 0.676403 + 0.266051I
b = 0.0545436 0.1210817I
4.62345 6.11364I 7.31633 + 3.70196I
u = 1.47115 + 0.51322I
a = 0.676403 0.266051I
b = 0.0545436 + 0.1210817I
4.62345 + 6.11364I 7.31633 3.70196I
u = 0.424240 0.658500I
a = 0.479222 0.024714I
b = 1.227044 + 0.657457I
0.977750 + 0.984957I 3.11347 + 0.07087I
u = 0.424240 + 0.658500I
a = 0.479222 + 0.024714I
b = 1.227044 0.657457I
0.977750 0.984957I 3.11347 0.07087I
u = 0.125280 0.430080I
a = 1.51027 + 1.09286I
b = 0.083291 0.380059I
0.440396 1.279685I 4.66436 + 4.97948I
u = 0.125280 + 0.430080I
a = 1.51027 1.09286I
b = 0.083291 + 0.380059I
0.440396 + 1.279685I 4.66436 4.97948I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.048547 0.635889I
a = 1.51003 1.09425I
b = 0.419207 + 0.119153I
5.68828 + 2.93127I 2.45037 0.45578I
u = 0.048547 + 0.635889I
a = 1.51003 + 1.09425I
b = 0.419207 0.119153I
5.68828 2.93127I 2.45037 + 0.45578I
u = 0.76178 1.87180I
a = 0.583738 + 0.210762I
b = 1.19849 0.95547I
5.41434 1.81549I 13.14101 + 3.54833I
u = 0.76178 + 1.87180I
a = 0.583738 0.210762I
b = 1.19849 + 0.95547I
5.41434 + 1.81549I 13.14101 3.54833I
u = 0.993380 0.445842I
a = 0.452214 + 0.233679I
b = 0.197673 + 0.384531I
1.73140 + 2.57417I 2.05095 4.12677I
u = 0.993380 + 0.445842I
a = 0.452214 0.233679I
b = 0.197673 0.384531I
1.73140 2.57417I 2.05095 + 4.12677I
u = 1.38118 0.37600I
a = 0.331529 + 0.771900I
b = 0.23714 2.17041I
4.30761 1.95796I 13.39097 + 1.55059I
u = 1.38118 + 0.37600I
a = 0.331529 0.771900I
b = 0.23714 + 2.17041I
4.30761 + 1.95796I 13.39097 1.55059I
u = 1.67216 0.94363I
a = 0.121716 + 0.768504I
b = 0.23112 2.52832I
9.1950 + 11.9560I 9.10352 6.09824I
u = 1.67216 + 0.94363I
a = 0.121716 0.768504I
b = 0.23112 + 2.52832I
9.1950 11.9560I 9.10352 + 6.09824I
6
III. I
v
1
= hv
4
v
3
+ 5v
2
+ v + 2, v
3
+ 2v
2
+ 4b 5v + 4, ai
(i) Arc colorings
a
4
=
1
0
a
8
=
v
0
a
9
=
v
0
a
3
=
1
0
a
11
=
0
1
4
v
3
1
2
v
2
+
5
4
v 1
a
7
=
v
1
4
v
3
+
3
4
v + 1
a
1
=
1
4
v
3
1
4
v +
1
2
1
a
2
=
1
4
v
3
1
4
v +
3
2
1
a
5
=
1
4
v
3
+
1
4
v
1
2
1
a
6
=
1
4
v
3
1
2
v
2
+
1
4
v
1
2
1
4
v
3
+
3
4
v + 1
a
10
=
1
4
v
3
v
2
+
1
4
v
1
2
1
2
v
1
2
a
10
=
1
4
v
3
v
2
+
1
4
v
1
2
1
2
v
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 0.152300 0.614030I
a = 0
b = 0.971274 0.813859I
1.85594 1.41510I 12.38954 + 3.92814I
v = 0.152300 + 0.614030I
a = 0
b = 0.971274 + 0.813859I
1.85594 + 1.41510I 12.38954 3.92814I
v = 0.65230 2.13814I
a = 0
b = 0.278726 + 0.483420I
5.14581 + 3.16396I 10.48546 5.24252I
v = 0.65230 + 2.13814I
a = 0
b = 0.278726 0.483420I
5.14581 3.16396I 10.48546 + 5.24252I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
2
+ u 1)
3
(u
20
+ 2u
19
+ ··· + 35u + 4)
c
2
(u + 1)
4
(u
2
+ 3u + 1)
3
(u
20
+ 22u
19
+ ··· + 353u + 16)
c
3
, c
8
u
4
(u
2
+ u 1)
3
(u
20
2u
19
+ ··· 112u + 64)
c
4
(u + 1)
4
(u
2
+ u 1)
3
(u
20
+ 2u
19
+ ··· + 35u + 4)
c
5
, c
6
(u
4
u
3
+ 3u
2
2u + 1)(u
6
+ 2u
4
3u
3
+ u
2
3u + 1)
(u
20
+ 2u
19
+ ··· + 2u + 1)
c
7
(u
4
u
3
+ u
2
+ 1)(u
6
+ 2u
4
+ ··· 3u + 1)(u
20
+ 2u
19
+ ··· + 2u + 1)
c
9
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
6
+ 2u
4
3u
3
+ u
2
3u + 1)
(u
20
+ 2u
19
+ ··· + 2u + 1)
c
10
(u
4
u
3
+ 3u
2
2u + 1)(u
6
+ 4u
5
+ 6u
4
3u
3
13u
2
7u + 1)
(u
20
+ 14u
19
+ ··· + 4u + 1)
c
11
(u
4
+ u
3
+ u
2
+ 1)(u
6
+ 2u
4
+ ··· 3u + 1)(u
20
+ 2u
19
+ ··· + 2u + 1)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
4
(y
2
3y + 1)
3
(y
20
22y
19
+ ··· 353y + 16)
c
2
(y 1)
4
(y
2
7y + 1)
3
(y
20
46y
19
+ ··· + 223775y + 256)
c
3
, c
8
y
4
(y
2
3y + 1)
3
(y
20
18y
19
+ ··· + 45824y + 4096)
c
5
, c
6
, c
9
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
6
+ 4y
5
+ 6y
4
3y
3
13y
2
7y + 1)
(y
20
+ 14y
19
+ ··· + 4y + 1)
c
7
, c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
6
+ 4y
5
+ 6y
4
3y
3
13y
2
7y + 1)
(y
20
+ 14y
19
+ ··· + 4y + 1)
c
10
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
6
4y
5
+ ··· 75y + 1)
(y
20
14y
19
+ ··· + 56y + 1)
10