11n
50
(K11n
50
)
1
Arc Sequences
6 1 7 10 2 9 11 4 6 1 8
Solving Sequence
1,6
2 3
5,8
11 7 10 4 9
c
1
c
2
c
5
c
11
c
7
c
10
c
4
c
9
c
3
, c
6
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
20
+ 3u
19
+ ··· + 6u + 1,
37910139708041u
19
201150163381549u
18
+ ··· + 6011452100077376b 7292662285169845,
4.50097 × 10
15
u
19
1.28986 × 10
16
u
18
+ ··· + 3.00573 × 10
15
a 7.74836 × 10
15
i
I
u
2
= hb
4
b
2
+ 1, b
2
+ u, b
3
+ a 1i
I
u
3
= hb
4
b
2
+ 1, b
2
+ u + 1, b
3
b
2
+ a + 1i
There are 3 irreducible components with 28 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
20
+ 3u
19
+ · · · +6u + 1, 3.79 × 10
13
u
19
2.01 × 10
14
u
18
+ · · · + 6.01 × 10
15
b
7.29× 10
15
, 4.50×10
15
u
19
1.29 × 10
16
u
18
+· · · +3.01 × 10
15
a 7.75 ×10
15
i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
8
=
1.49747u
19
+ 4.29134u
18
+ ··· + 24.6486u + 2.57787
0.00630632u
19
+ 0.0334612u
18
+ ··· + 3.09219u + 1.21313
a
11
=
1.01016u
19
2.81974u
18
+ ··· 13.1176u + 2.12976
0.147188u
19
0.390662u
18
+ ··· 5.99436u 1.23581
a
7
=
1.20441u
19
+ 3.56367u
18
+ ··· + 29.2504u + 7.54341
0.127524u
19
0.273202u
18
+ ··· 3.26181u + 0.411467
a
10
=
1.15735u
19
3.21040u
18
+ ··· 19.1120u + 0.893945
0.147188u
19
0.390662u
18
+ ··· 5.99436u 1.23581
a
4
=
1.07688u
19
3.29047u
18
+ ··· 25.9886u 7.95487
0.115875u
19
+ 0.263182u
18
+ ··· + 4.66274u 0.262787
a
9
=
1.15735u
19
3.21040u
18
+ ··· 19.1120u + 0.893945
0.201060u
19
0.575464u
18
+ ··· 6.40693u 1.49747
a
9
=
1.15735u
19
3.21040u
18
+ ··· 19.1120u + 0.893945
0.201060u
19
0.575464u
18
+ ··· 6.40693u 1.49747
(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 = 0.93727 1.53815I
a = 0.097983 + 0.641269I
b = 1.281060 0.067311I
5.23226 + 2.45917I 1.69714 1.89268I
u = 0.93727 + 1.53815I
a = 0.097983 0.641269I
b = 1.281060 + 0.067311I
5.23226 2.45917I 1.69714 + 1.89268I
u = 0.562478 0.702926I
a = 0.164935 1.014006I
b = 0.932716 + 0.491902I
0.10443 + 4.15417I 5.96079 7.41844I
u = 0.562478 + 0.702926I
a = 0.164935 + 1.014006I
b = 0.932716 0.491902I
0.10443 4.15417I 5.96079 + 7.41844I
u = 0.546407 0.261165I
a = 0.016518 + 0.458675I
b = 0.503419 0.405661I
1.204445 + 0.232928I 9.01931 0.79005I
u = 0.546407 + 0.261165I
a = 0.016518 0.458675I
b = 0.503419 + 0.405661I
1.204445 0.232928I 9.01931 + 0.79005I
u = 0.37279 2.18713I
a = 0.266075 + 1.115659I
b = 1.40011 0.62778I
17.7144 + 10.2068I 1.00734 4.81283I
u = 0.37279 + 2.18713I
a = 0.266075 1.115659I
b = 1.40011 + 0.62778I
17.7144 10.2068I 1.00734 + 4.81283I
u = 0.345261 0.774594I
a = 0.178688 1.067022I
b = 0.262517 0.119217I
1.78458 + 2.08707I 0.67504 3.91538I
u = 0.345261 + 0.774594I
a = 0.178688 + 1.067022I
b = 0.262517 + 0.119217I
1.78458 2.08707I 0.67504 + 3.91538I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.140514 0.165365I
a = 1.96129 4.14993I
b = 0.715874 0.509667I
1.42072 + 2.15124I 1.64791 3.40317I
u = 0.140514 + 0.165365I
a = 1.96129 + 4.14993I
b = 0.715874 + 0.509667I
1.42072 2.15124I 1.64791 + 3.40317I
u = 0.00083 2.05851I
a = 0.027063 1.246065I
b = 0.076044 + 1.204783I
13.59473 + 3.72129I 0.72655 1.99965I
u = 0.00083 + 2.05851I
a = 0.027063 + 1.246065I
b = 0.076044 1.204783I
13.59473 3.72129I 0.72655 + 1.99965I
u = 0.368558 0.047969I
a = 1.78216 1.81771I
b = 1.053192 + 0.370537I
1.82190 1.34830I 1.59816 + 0.61194I
u = 0.368558 + 0.047969I
a = 1.78216 + 1.81771I
b = 1.053192 0.370537I
1.82190 + 1.34830I 1.59816 0.61194I
u = 0.48389 2.08874I
a = 0.342044 + 1.052762I
b = 1.48289 0.54439I
18.5347 2.5424I 1.82827 + 0.66047I
u = 0.48389 + 2.08874I
a = 0.342044 1.052762I
b = 1.48289 + 0.54439I
18.5347 + 2.5424I 1.82827 0.66047I
u = 0.55309 1.41617I
a = 0.152667 + 0.289034I
b = 1.369772 + 0.179262I
7.00299 + 3.90150I 2.54860 3.27736I
u = 0.55309 + 1.41617I
a = 0.152667 0.289034I
b = 1.369772 0.179262I
7.00299 3.90150I 2.54860 + 3.27736I
4
II. I
u
2
= hb
4
b
2
+ 1, b
2
+ u, b
3
+ a 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
b
2
a
2
=
1
b
2
+ 1
a
3
=
b
2
b
2
+ 1
a
5
=
b
2
b
2
+ 1
a
8
=
b
3
+ 1
b
a
11
=
b
2
b
b
2
a
7
=
b
2
+ 1
b
3
+ b
a
10
=
b
b
2
a
4
=
b
3
+ b
2
b 1
b
2
b
a
9
=
b
b
3
b
2
+ b
a
9
=
b
b
3
b
2
+ b
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 1.00000 + 1.00000I
b = 0.866025 0.500000I
1.64493 0
u = 0.500000 + 0.866025I
a = 1.00000 1.00000I
b = 0.866025 + 0.500000I
1.64493 0
u = 0.500000 + 0.866025I
a = 1.00000 + 1.00000I
b = 0.866025 0.500000I
1.64493 0
u = 0.500000 0.866025I
a = 1.00000 1.00000I
b = 0.866025 + 0.500000I
1.64493 0
6
III. I
u
3
= hb
4
b
2
+ 1, b
2
+ u + 1, b
3
b
2
+ a + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
b
2
1
a
2
=
1
b
2
a
3
=
b
2
+ 1
b
2
a
5
=
b
2
+ 1
b
2
a
8
=
b
3
+ b
2
1
b
a
11
=
b
3
+ b
2
+ b
b
2
a
7
=
b
2
b
3
+ b
a
10
=
b
3
+ b
b
2
a
4
=
b
3
b
2
b
b
3
+ 2b
2
a
9
=
b
3
+ b
b
2
b
a
9
=
b
3
+ b
b
2
b
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.50000 + 1.86603I
b = 0.866025 0.500000I
1.64493 4.05977I 6.92820I
u = 0.500000 0.866025I
a = 0.50000 1.86603I
b = 0.866025 + 0.500000I
1.64493 + 4.05977I 6.92820I
u = 0.500000 0.866025I
a = 0.500000 + 0.133975I
b = 0.866025 0.500000I
1.64493 + 4.05977I 6.92820I
u = 0.500000 + 0.866025I
a = 0.500000 0.133975I
b = 0.866025 + 0.500000I
1.64493 4.05977I 6.92820I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
4
(u
20
+ 3u
19
+ ··· + 6u + 1)
c
2
(u
2
+ u + 1)
4
(u
20
+ 33u
19
+ ··· 6u + 1)
c
3
(u
4
2u
3
+ 2u
2
+ 2u + 1)(u
4
2u
3
+ 5u
2
4u + 1)
(u
20
+ u
19
+ ··· + 1264u + 517)
c
4
(u
4
2u
3
+ 2u
2
+ 2u + 1)(u
4
+ 4u
3
+ 5u
2
+ 2u + 1)
(u
20
+ u
19
+ ··· + 1876u + 647)
c
5
(u
2
u + 1)
4
(u
20
+ 3u
19
+ ··· + 6u + 1)
c
6
, c
9
(u
2
+ 1)
4
(u
20
+ u
19
+ ··· + 12u + 4)
c
7
, c
11
(u
4
u
2
+ 1)
2
(u
20
+ u
19
+ ··· + 6u + 1)
c
8
(u
4
u
2
+ 1)
2
(u
20
+ u
19
+ ··· + 4u + 1)
c
10
(u
2
u + 1)
4
(u
20
+ 15u
19
+ ··· + 22u + 1)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
4
(y
20
+ 33y
19
+ ··· 6y + 1)
c
2
(y
2
+ y + 1)
4
(y
20
87y
19
+ ··· + 770y + 1)
c
3
(y
4
+ 14y
2
+ 1)(y
4
+ 6y
3
+ 11y
2
6y + 1)
(y
20
+ 27y
19
+ ··· + 842544y + 267289)
c
4
(y
4
+ 14y
2
+ 1)(y
4
6y
3
+ 11y
2
+ 6y + 1)
(y
20
+ 55y
19
+ ··· 892556y + 418609)
c
6
, c
9
(y + 1)
8
(y
20
+ 33y
19
+ ··· 120y + 16)
c
7
, c
11
(y
2
y + 1)
4
(y
20
15y
19
+ ··· 22y + 1)
c
8
(y
2
y + 1)
4
(y
20
3y
19
+ ··· 6y + 1)
c
10
(y
2
+ y + 1)
4
(y
20
15y
19
+ ··· + 50y + 1)
10