10
135
(K10n
5
)
1
Arc Sequences
3 9 1 7 4 9 5 2 7 8
Solving Sequence
2,8
9 3
1,5
7 4 6 10
c
8
c
2
c
1
c
7
c
4
c
5
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
3
u
2
+ 2u 1, u
2
+ a u + 2, u
2
+ b + u 1i
I
u
2
= hu
21
2u
20
+ ··· + 7u 85,
4.14212 × 10
35
u
20
+ 8.57345 × 10
35
u
19
+ ··· + 4.58432 × 10
37
b 1.63308 × 10
37
,
8.04820 × 10
37
u
20
1.53060 × 10
38
u
19
+ ··· + 3.89667 × 10
39
a 1.77848 × 10
39
i
There are 2 irreducible components with 24 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
+ 2u 1, u
2
+ a u + 2, u
2
+ b + u 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
u
2
+ u 2
u
2
u + 1
a
9
=
1
u
2
u + 1
a
3
=
u
2
u + 2
u
a
1
=
0
u
2
a
5
=
0
u
a
7
=
u
2
+ u 2
u
2
+ 1
a
4
=
u
2
u + 2
u
2
+ u 1
a
6
=
u
2
+ u 2
u
2
+ 1
a
10
=
1
u
2
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7u
2
5u + 5
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.215080 1.307141I
a = 0.122561 0.744862I
b = 0.877439 + 0.744862I
4.66906 2.82812I 7.71191 + 2.59975I
u = 0.215080 + 1.307141I
a = 0.122561 + 0.744862I
b = 0.877439 0.744862I
4.66906 + 2.82812I 7.71191 2.59975I
u = 0.569840
a = 1.75488
b = 0.754878
0.531480 4.42382
3
II. I
u
2
=
hu
21
2u
20
+· · ·+7u85, 4.14×10
35
u
20
+8.57×10
35
u
19
+· · ·+4.58×10
37
b
1.63 × 10
37
, 8.05 × 10
37
u
20
1.53 × 10
38
u
19
+ · · · + 3.90 × 10
39
a 1.78 × 10
39
i
(i) Arc colorings
a
2
=
1
0
a
8
=
0.0206540u
20
+ 0.0392797u
19
+ ··· + 3.40406u + 0.456411
0.00903541u
20
0.0187017u
19
+ ··· 2.62821u + 0.356231
a
9
=
0.0116186u
20
+ 0.0205780u
19
+ ··· + 0.775853u + 0.812642
0.00903541u
20
0.0187017u
19
+ ··· 2.62821u + 0.356231
a
3
=
0.00300327u
20
0.00620089u
19
+ ··· 0.566824u + 1.17775
0.000909868u
20
0.00553251u
19
+ ··· + 0.585631u + 1.41382
a
1
=
0.0161310u
20
0.0448934u
19
+ ··· 1.18001u + 2.92372
0.000649318u
20
0.00275159u
19
+ ··· + 0.919236u 0.0686905
a
5
=
0
u
a
7
=
0.0206540u
20
+ 0.0392797u
19
+ ··· + 3.40406u + 0.456411
0.000846197u
20
0.00536711u
19
+ ··· 0.886816u + 0.528644
a
4
=
0.00665576u
20
0.0234770u
19
+ ··· + 0.928091u + 2.55427
0.00524729u
20
+ 0.00608328u
19
+ ··· + 1.62657u + 0.531450
a
6
=
0.0303553u
20
0.0576998u
19
+ ··· 7.33398u 0.641791
0.00804957u
20
0.0206389u
19
+ ··· 0.110740u + 1.14078
a
10
=
0.0124242u
20
+ 0.0129921u
19
+ ··· + 3.70821u + 2.66745
0.00524729u
20
0.00608328u
19
+ ··· 1.62657u 0.531450
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0.00376892u
20
0.0149564u
19
+ ··· 1.12863u + 1.16821
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.02434 2.16479I
a = 0.534577 0.190790I
b = 1.169826 + 0.051846I
8.83595 3.51416I 4.91512 + 2.66916I
u = 1.02434 + 2.16479I
a = 0.534577 + 0.190790I
b = 1.169826 0.051846I
8.83595 + 3.51416I 4.91512 2.66916I
u = 0.703419 0.890089I
a = 0.313344 0.202642I
b = 0.853051 + 0.160221I
1.46918 + 0.34630I 5.96536 0.53554I
u = 0.703419 + 0.890089I
a = 0.313344 + 0.202642I
b = 0.853051 0.160221I
1.46918 0.34630I 5.96536 + 0.53554I
u = 0.569601 0.261224I
a = 2.05326 + 0.46735I
b = 0.567882 + 0.851579I
2.42497 4.94435I 1.24866 + 2.70559I
u = 0.569601 + 0.261224I
a = 2.05326 0.46735I
b = 0.567882 0.851579I
2.42497 + 4.94435I 1.24866 2.70559I
u = 0.072252 0.712453I
a = 0.506395 + 1.227569I
b = 0.707761 + 0.560391I
1.83472 0.21101I 3.18710 + 0.57244I
u = 0.072252 + 0.712453I
a = 0.506395 1.227569I
b = 0.707761 0.560391I
1.83472 + 0.21101I 3.18710 0.57244I
u = 0.117256 0.594050I
a = 1.68359 0.96506I
b = 0.427156 + 0.796867I
3.29052 + 1.36266I 0.18856 2.27516I
u = 0.117256 + 0.594050I
a = 1.68359 + 0.96506I
b = 0.427156 0.796867I
3.29052 1.36266I 0.18856 + 2.27516I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.13316 1.77819I
a = 0.012577 + 0.714869I
b = 0.848992 0.598239I
3.02655 2.36605I 0.59037 + 2.67274I
u = 0.13316 + 1.77819I
a = 0.012577 0.714869I
b = 0.848992 + 0.598239I
3.02655 + 2.36605I 0.59037 2.67274I
u = 0.43502 2.72022I
a = 0.364037 0.352429I
b = 1.075838 + 0.689537I
3.96319 + 10.68717I 0.56681 6.96141I
u = 0.43502 + 2.72022I
a = 0.364037 + 0.352429I
b = 1.075838 0.689537I
3.96319 10.68717I 0.56681 + 6.96141I
u = 0.439691 1.065143I
a = 0.164036 0.516848I
b = 0.882737 + 0.780973I
3.85955 2.93752I 2.97600 + 3.43881I
u = 0.439691 + 1.065143I
a = 0.164036 + 0.516848I
b = 0.882737 0.780973I
3.85955 + 2.93752I 2.97600 3.43881I
u = 0.555881
a = 1.51829
b = 0.289436
1.20998 9.37190
u = 0.93625 1.78325I
a = 0.434228 + 0.145921I
b = 0.951460 0.595395I
1.06863 + 4.45806I 0.43689 6.14529I
u = 0.93625 + 1.78325I
a = 0.434228 0.145921I
b = 0.951460 + 0.595395I
1.06863 4.45806I 0.43689 + 6.14529I
u = 1.03030 1.04737I
a = 0.839884 + 0.137633I
b = 1.083584 0.616829I
5.21503 3.89686I 2.41425 + 2.65107I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.03030 + 1.04737I
a = 0.839884 0.137633I
b = 1.083584 + 0.616829I
5.21503 + 3.89686I 2.41425 2.65107I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
3
+ u
2
+ 2u + 1)(u
21
+ 8u
20
+ ··· + 17u + 1)
c
2
(u
3
+ u
2
1)(u
21
+ 2u
20
+ ··· + u 1)
c
3
(u
3
u
2
+ 2u 1)(u
21
+ 8u
20
+ ··· + 17u + 1)
c
4
(u 1)
3
(u
21
+ 4u
20
+ ··· 2u 1)
c
5
(u + 1)
3
(u
21
+ 6u
20
+ ··· 2u + 1)
c
6
, c
9
u
3
(u
21
+ u
20
+ ··· + 4u 8)
c
7
(u + 1)
3
(u
21
+ 4u
20
+ ··· 2u 1)
c
8
(u
3
u
2
+ 1)(u
21
+ 2u
20
+ ··· + u 1)
c
10
(u
3
u
2
+ 2u 1)(u
21
+ 2u
20
+ ··· + 3u + 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
(y
3
+ 3y
2
+ 2y 1)(y
21
+ 12y
20
+ ··· + 137y 1)
c
2
, c
8
(y
3
y
2
+ 2y 1)(y
21
8y
20
+ ··· + 17y 1)
c
4
, c
7
(y 1)
3
(y
21
6y
20
+ ··· 2y 1)
c
5
(y 1)
3
(y
21
+ 22y
20
+ ··· + 66y 1)
c
6
, c
9
y
3
(y
21
+ 21y
20
+ ··· 176y 64)
c
10
1.00000000
(1y
3
+ 3.00000000y
2
+ 2.00000000y 1.000000000)
(1y
21
24.0000000y
20
+ ··· + 17.0000000y 1.000000000)
8