11n
54
(K11n
54
)
1
Arc Sequences
4 1 7 2 9 10 3 1 11 7 6
Solving Sequence
7,10 4,11
3 6 1 2 5 9 8
c
10
c
3
c
6
c
11
c
2
c
4
c
9
c
8
c
1
, c
5
, c
7
Representation Ideals
I = I
u
1
\
I
v
1
I
u
1
= hu
27
+ u
26
+ ··· + 128u + 64, 2.25883 × 10
34
u
26
+ 6.71521 × 10
34
u
25
+ ··· + 4.81127 × 10
36
b 1.68358 × 10
36
,
3.23949 × 10
35
u
26
1.41213 × 10
35
u
25
+ ··· + 9.62254 × 10
36
a + 1.69205 × 10
37
i
I
v
1
= hv
4
v
2
+ b + 1, v
6
+ v
5
v
4
2v
3
+ v + 1, ai
There are 2 irreducible components with 33 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
27
+u
26
+· · ·+128u+64, 2.26×10
34
u
26
+6.72×10
34
u
25
+· · ·+4.81×10
36
b
1.68× 10
36
, 3.24 × 10
35
u
26
1.41 ×10
35
u
25
+· · · +9.62 ×10
36
a + 1.69× 10
37
i
(i) Arc colorings
a
7
=
0
u
a
10
=
0.0336657u
26
+ 0.0146752u
25
+ ··· + 0.158474u 1.75842
0.00469488u
26
0.0139573u
25
+ ··· + 4.87475u + 0.349924
a
4
=
1
0
a
11
=
0.0336657u
26
+ 0.0146752u
25
+ ··· + 0.158474u 1.75842
0.0142207u
26
0.0154764u
25
+ ··· + 5.15092u + 1.56531
a
3
=
1
u
2
a
6
=
0.0371213u
26
0.0414337u
25
+ ··· 10.4975u 2.60349
0.0413740u
26
0.0246607u
25
+ ··· + 2.44720u + 2.28218
a
1
=
0.0118157u
26
+ 0.0162794u
25
+ ··· + 4.26934u + 2.02994
0.00887124u
26
+ 0.00394903u
25
+ ··· + 5.49294u + 2.17103
a
2
=
0.0206870u
26
+ 0.0123303u
25
+ ··· 1.22360u 0.141092
0.00887124u
26
+ 0.00394903u
25
+ ··· + 5.49294u + 2.17103
a
5
=
0.00426849u
26
+ 0.0131278u
25
+ ··· + 8.43473u + 3.91529
0.0160842u
26
+ 0.00315151u
25
+ ··· 4.16539u 1.88536
a
9
=
0.00661984u
26
0.00830158u
25
+ ··· 2.05650u 1.94409
0.00835665u
26
0.0259380u
25
+ ··· 2.78903u 1.32397
a
8
=
u
u
3
+ u
a
8
=
u
u
3
+ 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 = 0.916936 0.740217I
a = 0.978245 + 0.140900I
b = 1.94712 + 0.16829I
1.53114 4.74698I 2.05877 + 5.37624I
u = 0.916936 + 0.740217I
a = 0.978245 0.140900I
b = 1.94712 0.16829I
1.53114 + 4.74698I 2.05877 5.37624I
u = 0.756621 0.053719I
a = 0.947761 + 0.339256I
b = 1.280090 0.221628I
1.077413 0.093546I 6.21440 0.06252I
u = 0.756621 + 0.053719I
a = 0.947761 0.339256I
b = 1.280090 + 0.221628I
1.077413 + 0.093546I 6.21440 + 0.06252I
u = 0.591296
a = 0.672828
b = 0.867515
1.09734 8.57437
u = 0.52916 1.90711I
a = 0.462320 0.451922I
b = 2.81680 2.09025I
9.8976 11.4401I 3.60368 + 6.65783I
u = 0.52916 + 1.90711I
a = 0.462320 + 0.451922I
b = 2.81680 + 2.09025I
9.8976 + 11.4401I 3.60368 6.65783I
u = 0.31974 1.83076I
a = 0.087922 + 0.626844I
b = 0.92863 + 2.70225I
5.98407 3.79755I 6.46791 + 2.18250I
u = 0.31974 + 1.83076I
a = 0.087922 0.626844I
b = 0.92863 2.70225I
5.98407 + 3.79755I 6.46791 2.18250I
u = 0.14157 2.04029I
a = 0.286400 0.546539I
b = 1.41202 2.19032I
10.52734 + 3.36992I 2.71394 2.50695I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.14157 + 2.04029I
a = 0.286400 + 0.546539I
b = 1.41202 + 2.19032I
10.52734 3.36992I 2.71394 + 2.50695I
u = 0.099608 0.648052I
a = 0.67781 1.53140I
b = 0.014762 1.397672I
2.30226 0.55935I 5.41913 0.15707I
u = 0.099608 + 0.648052I
a = 0.67781 + 1.53140I
b = 0.014762 + 1.397672I
2.30226 + 0.55935I 5.41913 + 0.15707I
u = 0.065099 0.869686I
a = 0.521552 + 1.307380I
b = 0.62639 + 1.84157I
0.58075 + 6.65503I 3.43691 7.46005I
u = 0.065099 + 0.869686I
a = 0.521552 1.307380I
b = 0.62639 1.84157I
0.58075 6.65503I 3.43691 + 7.46005I
u = 0.210916 0.867503I
a = 0.837971 + 0.172931I
b = 0.285706 0.077541I
1.66847 1.93992I 0.12713 + 2.72762I
u = 0.210916 + 0.867503I
a = 0.837971 0.172931I
b = 0.285706 + 0.077541I
1.66847 + 1.93992I 0.12713 2.72762I
u = 0.24190 2.02768I
a = 0.434808 0.207968I
b = 0.506970 0.092441I
12.17430 + 2.19817I 0.63423 2.08830I
u = 0.24190 + 2.02768I
a = 0.434808 + 0.207968I
b = 0.506970 + 0.092441I
12.17430 2.19817I 0.63423 + 2.08830I
u = 0.247086 0.712064I
a = 0.909883 + 0.972277I
b = 1.87921 0.63112I
2.84916 + 1.60658I 4.88146 5.04321I
4
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.247086 + 0.712064I
a = 0.909883 0.972277I
b = 1.87921 + 0.63112I
2.84916 1.60658I 4.88146 + 5.04321I
u = 0.45786 1.94824I
a = 0.356806 0.323757I
b = 0.086723 + 0.461433I
11.82207 + 5.91141I 1.03607 2.33228I
u = 0.45786 + 1.94824I
a = 0.356806 + 0.323757I
b = 0.086723 0.461433I
11.82207 5.91141I 1.03607 + 2.33228I
u = 0.682025 0.833060I
a = 0.129773 + 0.803497I
b = 0.052629 + 0.131638I
2.76848 0.13713I 0.773547 + 0.780119I
u = 0.682025 + 0.833060I
a = 0.129773 0.803497I
b = 0.052629 0.131638I
2.76848 + 0.13713I 0.773547 0.780119I
u = 0.784593 0.468587I
a = 0.633364 1.013310I
b = 1.47514 + 0.10710I
2.13464 3.70052I 7.14700 + 4.32876I
u = 0.784593 + 0.468587I
a = 0.633364 + 1.013310I
b = 1.47514 0.10710I
2.13464 + 3.70052I 7.14700 4.32876I
5
II. I
v
1
= hv
4
v
2
+ b + 1, v
6
+ v
5
v
4
2v
3
+ v + 1, ai
(i) Arc colorings
a
7
=
v
0
a
10
=
0
v
4
+ v
2
1
a
4
=
1
0
a
11
=
v
5
+ 2v
3
+ v
2
v 1
v
4
+ v
2
1
a
3
=
1
0
a
6
=
v
v
2
+ 1
a
1
=
v
5
+ v
3
+ v
2
v 1
1
a
2
=
v
5
+ v
3
+ v
2
v
1
a
5
=
v
5
v
3
v
2
+ v + 1
1
a
9
=
v
5
v
3
v
2
+ v + 1
v
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 = 1.073950 0.558752I
a = 0
b = 0.573950 0.818891I
1.64493 + 5.69302I 6.76721 4.86918I
v = 1.073950 + 0.558752I
a = 0
b = 0.573950 + 0.818891I
1.64493 5.69302I 6.76721 + 4.86918I
v = 0.428243 0.664531I
a = 0
b = 1.000936 + 0.863088I
3.53554 0.92430I 12.63596 0.09369I
v = 0.428243 + 0.664531I
a = 0
b = 1.000936 0.863088I
3.53554 + 0.92430I 12.63596 + 0.09369I
v = 1.002193 0.295542I
a = 0
b = 0.573013 + 0.494098I
0.245672 0.924305I 2.59683 + 0.69886I
v = 1.002193 + 0.295542I
a = 0
b = 0.573013 0.494098I
0.245672 + 0.924305I 2.59683 0.69886I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
6
(u
27
+ 7u
26
+ ··· 5u 1)
c
2
(u + 1)
6
(u
27
+ 3u
26
+ ··· + 5u + 1)
c
3
, c
7
u
6
(u
27
+ u
26
+ ··· + 128u + 64)
c
4
(u + 1)
6
(u
27
+ 7u
26
+ ··· 5u 1)
c
5
(u
6
u
5
+ ··· u + 1)(u
27
+ 2u
26
+ ··· + 2u 1)
c
6
(u
6
+ u
5
+ ··· + u + 1)(u
27
+ 2u
26
+ ··· + 4u + 1)
c
8
(u
6
u
5
+ ··· u + 1)(u
27
+ 8u
26
+ ··· + 16990u + 565)
c
9
(u
6
3u
5
+ ··· u + 1)(u
27
+ 12u
26
+ ··· + 12u + 1)
c
10
(u
6
u
5
+ ··· u + 1)(u
27
+ 2u
26
+ ··· + 4u + 1)
c
11
(u
6
3u
5
+ ··· u + 1)(u
27
+ 6u
26
+ ··· + 48u + 5)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
6
(y
27
3y
26
+ ··· + 5y 1)
c
2
(y 1)
6
(y
27
+ 49y
26
+ ··· 23y 1)
c
3
, c
7
y
6
(y
27
+ 39y
26
+ ··· 28672y 4096)
c
5
(y
6
3y
5
+ ··· y + 1)(y
27
36y
26
+ ··· + 12y 1)
c
6
, c
10
(y
6
3y
5
+ ··· y + 1)(y
27
12y
26
+ ··· + 12y 1)
c
8
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
27
72y
26
+ ··· + 171458760y 319225)
c
9
(y
6
+ y
5
+ ··· + 3y + 1)(y
27
+ 8y
26
+ ··· + 32y 1)
c
11
(y
6
+ y
5
+ ··· + 3y + 1)(y
27
4y
26
+ ··· + 504y 25)
9