11n
139
(K11n
139
)
A knot diagram
1
Linearized knot diagam
7 1 8 11 10 2 4 3 1 5 4
Solving Sequence
4,8
3
1,9
2 7 11 5 10 6
c
3
c
8
c
2
c
7
c
11
c
4
c
10
c
5
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−7u
7
− 28u
6
− 53u
5
− 79u
4
− 15u
3
+ 84u
2
+ 188b − 97u + 43,
21u
7
− 104u
6
+ 159u
5
− 703u
4
+ 797u
3
− 1380u
2
+ 940a + 1043u − 1069,
u
8
+ u
7
+ 9u
6
+ 2u
5
+ 22u
4
− 5u
3
+ 23u
2
+ 6u + 5i
I
u
2
= hb − a + 1, a
2
− au − 2a + u + 2, u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 12 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
= h−7u
7
− 28u
6
+ · · · + 188b + 43, 21u
7
− 104u
6
+ · · · + 940a −
1069, u
8
+ u
7
+ · · · + 6u + 5i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
3
=
1
−u
2
a
1
=
−0.0223404u
7
+ 0.110638u
6
+ ··· − 1.10957u + 1.13723
0.0372340u
7
+ 0.148936u
6
+ ··· + 0.515957u − 0.228723
a
9
=
−u
u
3
+ u
a
2
=
−0.0861702u
7
− 0.144681u
6
+ ··· − 1.27979u + 1.24362
−0.0265957u
7
− 0.106383u
6
+ ··· + 0.345745u − 0.122340
a
7
=
u
u
a
11
=
0.0148936u
7
+ 0.259574u
6
+ ··· − 0.593617u + 0.908511
0.0372340u
7
+ 0.148936u
6
+ ··· + 0.515957u − 0.228723
a
5
=
0.107447u
7
+ 0.229787u
6
+ ··· + 1.00319u + 1.05426
0.0425532u
7
+ 0.170213u
6
+ ··· − 0.0531915u + 0.0957447
a
10
=
0.0989362u
7
− 0.204255u
6
+ ··· − 0.586170u + 0.0351064
0.0744681u
7
− 0.202128u
6
+ ··· + 0.531915u − 0.457447
a
6
=
0.237234u
7
− 0.151064u
6
+ ··· + 1.11596u + 0.471277
0.0585106u
7
− 0.265957u
6
+ ··· − 0.760638u − 0.430851
a
6
=
0.237234u
7
− 0.151064u
6
+ ··· + 1.11596u + 0.471277
0.0585106u
7
− 0.265957u
6
+ ··· − 0.760638u − 0.430851
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
46
47
u
7
−
43
47
u
6
−
402
47
u
5
−
76
47
u
4
−
911
47
u
3
+
223
47
u
2
−
765
47
u −
362
47
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
8
+ u
7
− u
6
− 8u
5
+ 2u
4
+ 7u
3
+ 5u
2
+ 4u + 5
c
2
u
8
− 3u
7
+ 21u
6
− 72u
5
+ 108u
4
+ 25u
3
− 11u
2
+ 34u + 25
c
3
, c
7
, c
8
u
8
+ u
7
+ 9u
6
+ 2u
5
+ 22u
4
− 5u
3
+ 23u
2
+ 6u + 5
c
4
, c
5
, c
10
c
11
u
8
− u
7
+ 8u
6
− 4u
5
+ 18u
4
+ 11u
2
+ 5u + 2
c
9
u
8
+ 11u
7
+ 44u
6
+ 62u
5
+ 426u
4
− 1920u
3
+ 1693u
2
− 389u + 136
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
8
− 3y
7
+ 21y
6
− 72y
5
+ 108y
4
+ 25y
3
− 11y
2
+ 34y + 25
c
2
y
8
+ 33y
7
+ ··· − 1706y + 625
c
3
, c
7
, c
8
y
8
+ 17y
7
+ 121y
6
+ 448y
5
+ 916y
4
+ 1053y
3
+ 809y
2
+ 194y + 25
c
4
, c
5
, c
10
c
11
y
8
+ 15y
7
+ 92y
6
+ 294y
5
+ 514y
4
+ 468y
3
+ 193y
2
+ 19y + 4
c
9
y
8
− 33y
7
+ ··· + 309175y + 18496
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.611238 + 0.940914I
a = 0.190317 − 0.232859I
b = 0.234808 − 1.029490I
3.66920 − 1.06491I 1.31198 + 1.63429I
u = 0.611238 − 0.940914I
a = 0.190317 + 0.232859I
b = 0.234808 + 1.029490I
3.66920 + 1.06491I 1.31198 − 1.63429I
u = −0.187062 + 0.424849I
a = 1.051840 − 0.641967I
b = −0.258486 + 0.303432I
−0.482455 + 0.984921I −7.02443 − 7.03211I
u = −0.187062 − 0.424849I
a = 1.051840 + 0.641967I
b = −0.258486 − 0.303432I
−0.482455 − 0.984921I −7.02443 + 7.03211I
u = −0.45395 + 1.85746I
a = −0.045979 + 0.950045I
b = 0.36613 + 1.66771I
12.93020 − 1.89326I 1.23462 + 1.04722I
u = −0.45395 − 1.85746I
a = −0.045979 − 0.950045I
b = 0.36613 − 1.66771I
12.93020 + 1.89326I 1.23462 − 1.04722I
u = −0.47023 + 2.19541I
a = −0.396174 − 1.205290I
b = 0.15755 − 1.96154I
−12.82710 + 5.56972I 0.47783 − 1.89693I
u = −0.47023 − 2.19541I
a = −0.396174 + 1.205290I
b = 0.15755 + 1.96154I
−12.82710 − 5.56972I 0.47783 + 1.89693I
5
II. I
u
2
= hb − a + 1, a
2
− au − 2a + u + 2, u
2
+ 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
3
=
1
1
a
1
=
a
a − 1
a
9
=
−u
0
a
2
=
a + 1
a
a
7
=
u
u
a
11
=
2a − 1
a − 1
a
5
=
2au + a − 2u − 2
au − u − 1
a
10
=
au − a − 3u + 1
−a − u + 1
a
6
=
−au
−au + u
a
6
=
−au
−au + u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
(u
2
+ 1)
2
c
2
(u + 1)
4
c
4
, c
5
, c
10
c
11
u
4
+ 3u
2
+ 1
c
9
(u
2
− u − 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
(y + 1)
4
c
2
(y −1)
4
c
4
, c
5
, c
10
c
11
(y
2
+ 3y + 1)
2
c
9
(y
2
− 3y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 1.000000 − 0.618034I
b = − 0.618034I
0.986960 0
u = 1.000000I
a = 1.00000 + 1.61803I
b = 1.61803I
8.88264 0
u = − 1.000000I
a = 1.000000 + 0.618034I
b = 0.618034I
0.986960 0
u = − 1.000000I
a = 1.00000 − 1.61803I
b = − 1.61803I
8.88264 0
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
2
+ 1)
2
(u
8
+ u
7
− u
6
− 8u
5
+ 2u
4
+ 7u
3
+ 5u
2
+ 4u + 5)
c
2
((u + 1)
4
)(u
8
− 3u
7
+ ··· + 34u + 25)
c
3
, c
7
, c
8
(u
2
+ 1)
2
(u
8
+ u
7
+ 9u
6
+ 2u
5
+ 22u
4
− 5u
3
+ 23u
2
+ 6u + 5)
c
4
, c
5
, c
10
c
11
(u
4
+ 3u
2
+ 1)(u
8
− u
7
+ 8u
6
− 4u
5
+ 18u
4
+ 11u
2
+ 5u + 2)
c
9
(u
2
− u − 1)
2
· (u
8
+ 11u
7
+ 44u
6
+ 62u
5
+ 426u
4
− 1920u
3
+ 1693u
2
− 389u + 136)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
((y + 1)
4
)(y
8
− 3y
7
+ ··· + 34y + 25)
c
2
((y −1)
4
)(y
8
+ 33y
7
+ ··· − 1706y + 625)
c
3
, c
7
, c
8
(y + 1)
4
· (y
8
+ 17y
7
+ 121y
6
+ 448y
5
+ 916y
4
+ 1053y
3
+ 809y
2
+ 194y + 25)
c
4
, c
5
, c
10
c
11
(y
2
+ 3y + 1)
2
· (y
8
+ 15y
7
+ 92y
6
+ 294y
5
+ 514y
4
+ 468y
3
+ 193y
2
+ 19y + 4)
c
9
((y
2
− 3y + 1)
2
)(y
8
− 33y
7
+ ··· + 309175y + 18496)
11
