11n
50
(K11n
50
)
A knot diagram
1
Linearized knot diagam
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
Ideals for irreducible components
2
of X
par
I
u
1
= h37910139708041u
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
, u
20
+ 3u
19
+ ··· + 6u + 1i
I
u
2
= h−au + b − u − 1, a
2
− 2au − u, u
2
+ u + 1i
I
u
3
= hau + b + a − u − 1, a
2
− 2a + 2, u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 28 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I.
I
u
1
= h3.79×10
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
, u
20
+3u
19
+· · ·+6u+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
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
=
4709825960466509
3005726050038688
u
19
+
3061022262562929
751431512509672
u
18
+ ··· +
62732012609244425
3005726050038688
u +
4116636408312339
751431512509672
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
20
− 3u
19
+ ··· − 6u + 1
c
2
u
20
+ 33u
19
+ ··· − 6u + 1
c
3
u
20
+ u
19
+ ··· + 1264u + 517
c
4
u
20
+ u
19
+ ··· + 1876u + 647
c
6
, c
9
u
20
+ u
19
+ ··· + 12u + 4
c
7
, c
11
u
20
+ u
19
+ ··· + 6u + 1
c
8
u
20
+ u
19
+ ··· + 4u + 1
c
10
u
20
+ 15u
19
+ ··· + 22u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
20
+ 33y
19
+ ··· − 6y + 1
c
2
y
20
− 87y
19
+ ··· + 770y + 1
c
3
y
20
+ 27y
19
+ ··· + 842544y + 267289
c
4
y
20
+ 55y
19
+ ··· − 892556y + 418609
c
6
, c
9
y
20
+ 33y
19
+ ··· − 120y + 16
c
7
, c
11
y
20
− 15y
19
+ ··· − 22y + 1
c
8
y
20
− 3y
19
+ ··· − 6y + 1
c
10
y
20
− 15y
19
+ ··· + 50y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.562478 + 0.702926I
a = −0.164935 + 1.014010I
b = −0.932716 − 0.491902I
0.10443 − 4.15417I 5.96079 + 7.41844I
u = −0.562478 − 0.702926I
a = −0.164935 − 1.014010I
b = −0.932716 + 0.491902I
0.10443 + 4.15417I 5.96079 − 7.41844I
u = −0.345261 + 0.774594I
a = −0.178688 + 1.067020I
b = 0.262517 + 0.119217I
−1.78458 − 2.08707I −0.67504 + 3.91538I
u = −0.345261 − 0.774594I
a = −0.178688 − 1.067020I
b = 0.262517 − 0.119217I
−1.78458 + 2.08707I −0.67504 − 3.91538I
u = −0.546407 + 0.261165I
a = −0.016518 − 0.458675I
b = −0.503419 + 0.405661I
1.204440 − 0.232928I 9.01931 + 0.79005I
u = −0.546407 − 0.261165I
a = −0.016518 + 0.458675I
b = −0.503419 − 0.405661I
1.204440 + 0.232928I 9.01931 − 0.79005I
u = 0.55309 + 1.41617I
a = 0.152667 − 0.289034I
b = −1.369770 − 0.179262I
−7.00299 − 3.90150I −2.54860 + 3.27736I
u = 0.55309 − 1.41617I
a = 0.152667 + 0.289034I
b = −1.369770 + 0.179262I
−7.00299 + 3.90150I −2.54860 − 3.27736I
u = 0.368558 + 0.047969I
a = −1.78216 + 1.81771I
b = 1.053190 − 0.370537I
−1.82190 + 1.34830I −1.59816 − 0.61194I
u = 0.368558 − 0.047969I
a = −1.78216 − 1.81771I
b = 1.053190 + 0.370537I
−1.82190 − 1.34830I −1.59816 + 0.61194I
5
Solutions 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 = −0.93727 + 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 = −0.00083 + 2.05851I
a = −0.027063 + 1.246070I
b = 0.076044 − 1.204780I
−13.59470 − 3.72129I 0.72655 + 1.99965I
u = −0.00083 − 2.05851I
a = −0.027063 − 1.246070I
b = 0.076044 + 1.204780I
−13.59470 + 3.72129I 0.72655 − 1.99965I
u = 0.48389 + 2.08874I
a = 0.342044 − 1.052760I
b = −1.48289 + 0.54439I
−18.5347 + 2.5424I −1.82827 + 0.I
u = 0.48389 − 2.08874I
a = 0.342044 + 1.052760I
b = −1.48289 − 0.54439I
−18.5347 − 2.5424I −1.82827 + 0.I
u = −0.37279 + 2.18713I
a = −0.266075 − 1.115660I
b = 1.40011 + 0.62778I
−17.7144 − 10.2068I 0. + 4.81283I
u = −0.37279 − 2.18713I
a = −0.266075 + 1.115660I
b = 1.40011 − 0.62778I
−17.7144 + 10.2068I 0. − 4.81283I
6
II. I
u
2
= h−au + b − u − 1, a
2
− 2au − u, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
5
=
−u
u + 1
a
8
=
a
au + u + 1
a
11
=
au + a + u + 2
−u − 1
a
7
=
u + 1
au + a + 1
a
10
=
au + a + 1
−u − 1
a
4
=
−au − a − u − 2
−a + 3u + 2
a
9
=
au + a + 1
−au − 2u − 2
a
9
=
au + a + 1
−au − 2u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u + 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u
2
+ u + 1)
2
c
3
u
4
− 2u
3
+ 5u
2
− 4u + 1
c
4
u
4
+ 4u
3
+ 5u
2
+ 2u + 1
c
5
, c
10
(u
2
− u + 1)
2
c
6
, c
9
(u
2
+ 1)
2
c
7
, c
8
, c
11
u
4
− u
2
+ 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
10
(y
2
+ y + 1)
2
c
3
y
4
+ 6y
3
+ 11y
2
− 6y + 1
c
4
y
4
− 6y
3
+ 11y
2
+ 6y + 1
c
6
, c
9
(y + 1)
4
c
7
, c
8
, c
11
(y
2
− y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.500000 − 0.133975I
b = 0.866025 + 0.500000I
−1.64493 − 4.05977I 0. + 6.92820I
u = −0.500000 + 0.866025I
a = −0.50000 + 1.86603I
b = −0.866025 − 0.500000I
−1.64493 − 4.05977I 0. + 6.92820I
u = −0.500000 − 0.866025I
a = −0.500000 + 0.133975I
b = 0.866025 − 0.500000I
−1.64493 + 4.05977I 0. − 6.92820I
u = −0.500000 − 0.866025I
a = −0.50000 − 1.86603I
b = −0.866025 + 0.500000I
−1.64493 + 4.05977I 0. − 6.92820I
10
III. I
u
3
= hau + b + a − u − 1, a
2
− 2a + 2, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
5
=
−u
u + 1
a
8
=
a
−au − a + u + 1
a
11
=
au + a − 2u − 1
u
a
7
=
u + 1
−au + u
a
10
=
au + a − u − 1
u
a
4
=
au − 2u − 1
au + a − 1
a
9
=
au + a − u − 1
−au + 2u
a
9
=
au + a − u − 1
−au + 2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u
2
+ u + 1)
2
c
3
, c
4
u
4
− 2u
3
+ 2u
2
+ 2u + 1
c
5
, c
10
(u
2
− u + 1)
2
c
6
, c
9
(u
2
+ 1)
2
c
7
, c
8
, c
11
u
4
− u
2
+ 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
10
(y
2
+ y + 1)
2
c
3
, c
4
y
4
+ 14y
2
+ 1
c
6
, c
9
(y + 1)
4
c
7
, c
8
, c
11
(y
2
− y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−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
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
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)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
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)
16
