11a
223
(K11a
223
)
1
Arc Sequences
6 1 10 8 9 2 11 5 3 4 7
Solving Sequence
1,6
2 3
7,9
10 5 8 4 11
c
1
c
2
c
6
c
9
c
5
c
8
c
4
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
4
2u
2
+ 2, u
3
+ 2a 2u, u
3
u
2
+ b + u + 1i
I
u
2
= ha
24
a
23
+ ··· 5a + 5, 5.99446 × 10
18
b 1.10005 × 10
18
a
23
+ ··· + 1.65091 × 10
18
a + 5.25563 × 10
18
,
5.99446 × 10
18
u 2.35199 × 10
18
a
23
+ ··· 3.65357 × 10
18
a + 2.59004 × 10
18
i
I
u
3
= hu
20
+ 3u
19
+ ··· + 6u + 2, u
19
+ u
18
+ ··· + 2a 2, 3u
19
+ 7u
18
+ ··· + b + 7i
I
v
1
= hv + 1, b + 1, ai
There are 4 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
4
2u
2
+ 2, u
3
+ 2a 2u, u
3
u
2
+ b + u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
7
=
u
u
3
+ u
a
9
=
1
2
u
3
+ u
u
3
+ u
2
u 1
a
10
=
1
2
u
3
+ u
2
+ u 1
u
3
u 1
a
5
=
1
2
u
3
+ u
u
3
+ u
2
1
a
8
=
0
u
a
4
=
1
2
u
3
+ u
u
3
+ u
2
u 1
a
11
=
u
2
1
u
2
a
11
=
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 = 1.098684 0.455090I
a = 0.776887 + 0.321797I
b = 0.455090 0.098684I
5.75727 3.66386I 16.0000 + 4.0000I
u = 1.098684 + 0.455090I
a = 0.776887 0.321797I
b = 0.455090 + 0.098684I
5.75727 + 3.66386I 16.0000 4.0000I
u = 1.098684 0.455090I
a = 0.776887 + 0.321797I
b = 0.45509 2.09868I
5.75727 + 3.66386I 16.0000 4.0000I
u = 1.098684 + 0.455090I
a = 0.776887 0.321797I
b = 0.45509 + 2.09868I
5.75727 3.66386I 16.0000 + 4.0000I
3
II.
I
u
2
= ha
24
a
23
+· · ·5a+5, 5.99×10
18
b1.10×10
18
a
23
+· · ·+1.65×10
18
a+
5.26 × 10
18
, 5.99 × 10
18
u 2.35 × 10
18
a
23
+ · · · 3.65 × 10
18
a + 2.59 × 10
18
i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
0.392360a
23
+ 0.0861072a
22
+ ··· + 0.609490a 0.432072
a
2
=
1
0.162360a
23
0.0596316a
22
+ ··· 0.113539a 0.382886
a
3
=
0.162360a
23
+ 0.0596316a
22
+ ··· + 0.113539a + 1 .38289
0.162360a
23
0.0596316a
22
+ ··· 0.113539a 0.382886
a
7
=
0.392360a
23
0.0861072a
22
+ ··· 0.609490a + 0.432072
0.140760a
23
+ 0.0121953a
22
+ ··· 0.484192a + 1 .25186
a
9
=
a
0.183512a
23
0.107451a
22
+ ··· 0.275406a 0.876747
a
10
=
0.299186a
23
0.0885839a
22
+ ··· + 1.20274a 1.52291
0.437815a
23
0.129870a
22
+ ··· 2.90602a + 1 .09449
a
5
=
0.408869a
23
0.161236a
22
+ ··· + 0.430536a 2.39234
0.0205051a
23
+ 0.0743279a
22
+ ··· 0.271704a + 0 .195624
a
8
=
0.112379a
23
+ 0.0486234a
22
+ ··· + 1 .60100a 0.0873573
0.191444a
23
0.274158a
22
+ ··· 0.213052a 1.76300
a
4
=
0.553239a
23
0.209762a
22
+ ··· + 1.45843a 2.81914
0.233643a
23
0.156066a
22
+ ··· 2.12190a 0.668049
a
11
=
0.251385a
23
0.235442a
22
+ ··· + 0.525906a 1.67456
0.136894a
23
0.0790884a
22
+ ··· 1.36055a 0.307545
a
11
=
0.251385a
23
0.235442a
22
+ ··· + 0.525906a 1.67456
0.136894a
23
0.0790884a
22
+ ··· 1.36055a 0.307545
(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 = 0.961384 + 0.208970I
a = 1.266854 0.118461I
b = 1.67659 + 1.11377I
5.02961 + 0.71593I 15.9565 0.6487I
u = 0.961384 0.208970I
a = 1.266854 + 0.118461I
b = 1.67659 1.11377I
5.02961 0.71593I 15.9565 + 0.6487I
u = 0.463636 0.458719I
a = 1.17997 1.28744I
b = 0.632108 + 0.554683I
1.85256 0.35310I 5.33308 + 0.62981I
u = 0.463636 + 0.458719I
a = 1.17997 + 1.28744I
b = 0.632108 0.554683I
1.85256 + 0.35310I 5.33308 0.62981I
u = 1.228552 0.484706I
a = 1.130542 0.326132I
b = 1.96382 + 1.57386I
10.84802 + 7.80134I 14.3661 5.6398I
u = 1.228552 + 0.484706I
a = 1.130542 + 0.326132I
b = 1.96382 1.57386I
10.84802 7.80134I 14.3661 + 5.6398I
u = 0.958024 + 0.460561I
a = 0.488261 0.325826I
b = 0.38736 1.59157I
3.21312 4.24921I 9.82351 + 6.98310I
u = 0.958024 0.460561I
a = 0.488261 + 0.325826I
b = 0.38736 + 1.59157I
3.21312 + 4.24921I 9.82351 6.98310I
u = 1.238641 + 0.435356I
a = 0.271181 0.846630I
b = 0.738747 0.020204I
11.20505 + 1.48234I 15.1526 0.6754I
u = 1.238641 0.435356I
a = 0.271181 + 0.846630I
b = 0.738747 + 0.020204I
11.20505 1.48234I 15.1526 + 0.6754I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.049813 + 0.844037I
a = 0.01718 1.79571I
b = 0.446051 + 0.235907I
7.33005 + 3.01307I 11.36825 2.63251I
u = 0.049813 0.844037I
a = 0.01718 + 1.79571I
b = 0.446051 0.235907I
7.33005 3.01307I 11.36825 + 2.63251I
u = 0.463636 + 0.458719I
a = 0.110838 0.568585I
b = 1.176271 + 0.491504I
1.85256 + 0.35310I 5.33308 0.62981I
u = 0.463636 0.458719I
a = 0.110838 + 0.568585I
b = 1.176271 0.491504I
1.85256 0.35310I 5.33308 + 0.62981I
u = 1.228552 0.484706I
a = 0.682674 0.525224I
b = 1.62850 1.12901I
10.84802 + 7.80134I 14.3661 5.6398I
u = 1.228552 + 0.484706I
a = 0.682674 + 0.525224I
b = 1.62850 + 1.12901I
10.84802 7.80134I 14.3661 + 5.6398I
u = 0.961384 0.208970I
a = 0.822893 0.596955I
b = 0.601903 0.871254I
5.02961 0.71593I 15.9565 + 0.6487I
u = 0.961384 + 0.208970I
a = 0.822893 + 0.596955I
b = 0.601903 + 0.871254I
5.02961 + 0.71593I 15.9565 0.6487I
u = 0.049813 0.844037I
a = 1.028290 0.794481I
b = 1.337899 + 0.284281I
7.33005 3.01307I 11.36825 + 2.63251I
u = 0.049813 + 0.844037I
a = 1.028290 + 0.794481I
b = 1.337899 0.284281I
7.33005 + 3.01307I 11.36825 2.63251I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.238641 0.435356I
a = 1.102583 0.432921I
b = 1.62832 + 0.63402I
11.20505 1.48234I 15.1526 + 0.6754I
u = 1.238641 + 0.435356I
a = 1.102583 + 0.432921I
b = 1.62832 0.63402I
11.20505 + 1.48234I 15.1526 0.6754I
u = 0.958024 + 0.460561I
a = 1.106712 0.376059I
b = 0.78641 + 1.88338I
3.21312 4.24921I 9.82351 + 6.98310I
u = 0.958024 0.460561I
a = 1.106712 + 0.376059I
b = 0.78641 1.88338I
3.21312 + 4.24921I 9.82351 6.98310I
7
III.
I
u
3
= hu
20
+3u
19
+· · ·+6u+2, u
19
+u
18
+· · ·+2a2, 3u
19
+7u
18
+· · ·+b+7i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
7
=
u
u
3
+ u
a
9
=
1
2
u
19
1
2
u
18
+ ··· 2u
2
+ 1
3u
19
7u
18
+ ··· 15u 7
a
10
=
1
2
u
19
1
2
u
18
+ ··· 4u
3
u
2
2u
19
5u
18
+ ··· 11u 5
a
5
=
1
2
u
19
+
3
2
u
18
+ ··· + 2 u + 1
u
19
2u
18
+ ··· 2u 1
a
8
=
u
7
+ 2u
5
2u
3
u
9
+ 3u
7
3u
5
+ u
a
4
=
3
2
u
19
+
7
2
u
18
+ ··· + 6 u + 3
u
19
2u
18
+ ··· 3u 1
a
11
=
u
4
u
2
+ 1
u
6
2u
4
+ u
2
a
11
=
u
4
u
2
+ 1
u
6
2u
4
+ u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.236099 0.531142I
a = 1.200236 0.238508I
b = 1.66403 + 2.29208I
18.0142 12.6981I 15.8020 + 6.4148I
u = 1.236099 + 0.531142I
a = 1.200236 + 0.238508I
b = 1.66403 2.29208I
18.0142 + 12.6981I 15.8020 6.4148I
u = 1.160541 0.458172I
a = 0.516427 0.208738I
b = 1.27395 0.65297I
3.93117 5.51600I 10.44810 + 8.22749I
u = 1.160541 + 0.458172I
a = 0.516427 + 0.208738I
b = 1.27395 + 0.65297I
3.93117 + 5.51600I 10.44810 8.22749I
u = 0.977557 0.624357I
a = 1.160204 + 0.595294I
b = 0.14127 2.41691I
9.37386 6.54808I 13.7315 + 5.5285I
u = 0.977557 + 0.624357I
a = 1.160204 0.595294I
b = 0.14127 + 2.41691I
9.37386 + 6.54808I 13.7315 5.5285I
u = 0.773452 0.404695I
a = 0.388315 + 0.473780I
b = 0.870710 + 0.755791I
0.87704 1.78379I 2.58390 + 5.68445I
u = 0.773452 + 0.404695I
a = 0.388315 0.473780I
b = 0.870710 0.755791I
0.87704 + 1.78379I 2.58390 5.68445I
u = 0.544915 0.735723I
a = 0.91679 1.36955I
b = 0.98707 + 1.43802I
8.11263 + 1.41331I 12.03617 0.10296I
u = 0.544915 + 0.735723I
a = 0.91679 + 1.36955I
b = 0.98707 1.43802I
8.11263 1.41331I 12.03617 + 0.10296I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.128827 0.901492I
a = 0.08022 + 1.79613I
b = 1.066020 0.754502I
14.6659 + 7.5175I 13.03534 3.27786I
u = 0.128827 + 0.901492I
a = 0.08022 1.79613I
b = 1.066020 + 0.754502I
14.6659 7.5175I 13.03534 + 3.27786I
u = 0.058790 0.660109I
a = 0.530045 0.680805I
b = 0.492607 0.068781I
0.84744 + 1.30386I 6.93259 5.24353I
u = 0.058790 + 0.660109I
a = 0.530045 + 0.680805I
b = 0.492607 + 0 .068781I
0.84744 1.30386I 6.93259 + 5.24353I
u = 0.695075
a = 0.478434
b = 0.196184
0.859562 12.8976
u = 1.145210 0.438306I
a = 0.317417 + 0.392871I
b = 0.211300 0.188051I
4.07199 + 2.60865I 11.03085 + 0.93775I
u = 1.145210 + 0.438306I
a = 0.317417 0.392871I
b = 0.211300 + 0.188051I
4.07199 2.60865I 11.03085 0.93775I
u = 1.21457
a = 1.33810
b = 0.733877
14.1194 17.9241
u = 1.280146 0.384189I
a = 1.126134 0.558334I
b = 0.686001 + 0 .239307I
19.0845 3.0881I 16.9887 + 0.4542I
u = 1.280146 + 0.384189I
a = 1.126134 + 0.558334I
b = 0.686001 0.239307I
19.0845 + 3.0881I 16.9887 0.4542I
10
IV. I
v
1
= hv + 1, b + 1, ai
(i) Arc colorings
a
1
=
1
0
a
6
=
1
0
a
2
=
1
0
a
3
=
1
0
a
7
=
1
0
a
9
=
0
1
a
10
=
1
1
a
5
=
1
1
a
8
=
1
0
a
4
=
0
1
a
11
=
1
0
a
11
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
11
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
3.28987 12.0000
12
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
u(u
4
2u
2
+ 2)
(1 2u + u
2
+ 3u
3
4u
4
+ 2u
5
+ 2u
6
6u
7
+ 3u
8
+ 4u
9
3u
10
u
11
+ u
12
)
2
(u
20
+ 3u
19
+ ··· + 6 u + 2)
c
2
u(u
2
+ 2u + 2)
2
(1 + 2u + 5u
2
+ 5u
3
10u
4
32u
5
28u
6
+ 8u
7
+ 41u
8
+ 42u
9
+ 23u
10
+ 7u
11
+ u
12
)
2
(u
20
+ 11u
19
+ ··· + 4u + 4)
c
3
, c
8
(u 1)
4
(u + 1)(u
20
+ u
19
+ ··· 2u 1)(u
24
+ u
23
+ ··· 10u + 5)
c
4
, c
5
, c
9
c
10
(u 1)(u + 1)
4
(u
20
+ u
19
+ ··· 2u 1)(u
24
+ u
23
+ ··· 10u + 5)
c
7
, c
11
u(u
4
+ 2u
2
+ 2)
(1 2u + 5u
2
17u
3
+ 38u
4
56u
5
+ 64u
6
56u
7
+ 41u
8
22u
9
+ 11u
10
3u
11
+ u
12
)
2
(u
20
+ 9u
19
+ ··· 110u 22)
13
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
y(y
2
2y + 2)
2
(1 2y + 5y
2
5y
3
10y
4
+ 32y
5
28y
6
8y
7
+ 41y
8
42y
9
+ 23y
10
7y
11
+ y
12
)
2
(y
20
11y
19
+ ··· 4y + 4)
c
2
y(y
2
+ 4)
2
(1 + 6y 15y
2
53y
3
+ 190y
4
256y
5
+ 260y
6
192y
7
+ 149y
8
46y
9
+ 23y
10
3y
11
+ y
12
)
2
(y
20
3y
19
+ ··· 208y + 16)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y 1)
5
(y
20
27y
19
+ ··· 10y + 1)(y
24
21y
23
+ ··· 220y + 25)
c
7
, c
11
y(y
2
+ 2y + 2)
2
(1 + 6y + 33y
2
5y
3
+ 38y
4
+ 168y
5
+ 292y
6
+ 392y
7
+ 365y
8
+ 210y
9
+ 71y
10
+ 13y
11
+ y
12
)
2
(y
20
+ 17y
19
+ ··· 4004y + 484)
14