11n
107
(K11n
107
)
A knot diagram
1
Linearized knot diagam
5 1 8 9 2 11 1 10 5 4 7
Solving Sequence
5,9
10 4 8
1,3
2 6 7 11
c
9
c
4
c
8
c
3
c
2
c
5
c
7
c
11
c
1
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
− u
8
− u
7
+ 3u
6
− u
5
− 2u
4
+ 3u
3
+ u
2
+ b − 2u + 1,
− u
10
+ u
9
+ 2u
8
− 5u
7
+ 6u
5
− 4u
4
− 4u
3
+ 4u
2
+ 2a − 2,
u
11
− 3u
10
+ 2u
9
+ 5u
8
− 10u
7
+ 4u
6
+ 8u
5
− 10u
4
+ 8u
2
− 6u + 2i
I
u
2
= hu
3
+ u
2
+ b − 2u − 1, −u
3
+ 2a + 2u, u
4
− 2u
2
+ 2i
I
u
3
= h−u
2
+ b + a − u, a
2
− au + u
2
− u − 1, u
3
+ u
2
− 1i
I
v
1
= ha, b − 1, v + 1i
* 4 irreducible components of dim
C
= 0, with total 22 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
= hu
9
− u
8
+ · · · + b + 1, −u
10
+ u
9
+ · · · + 2a − 2, u
11
− 3u
10
+ · · · − 6u + 2i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
4
=
−u
u
a
8
=
−u
2
+ 1
u
4
a
1
=
1
2
u
10
−
1
2
u
9
+ ··· − 2u
2
+ 1
−u
9
+ u
8
+ u
7
− 3u
6
+ u
5
+ 2u
4
− 3u
3
− u
2
+ 2u − 1
a
3
=
−u
7
+ 2u
5
− 2u
3
u
9
− u
7
+ u
5
+ u
a
2
=
1
2
u
10
−
1
2
u
9
+ ··· − 2u
2
+ 1
u
10
− 3u
9
+ u
8
+ 6u
7
− 9u
6
+ 10u
4
− 7u
3
− 5u
2
+ 7u − 3
a
6
=
1
2
u
10
−
3
2
u
9
+ ··· + 2 u − 1
−u
10
+ 2u
9
+ u
8
− 5u
7
+ 3u
6
+ 3u
5
− 5u
4
+ u
3
+ 3u
2
− 2u + 1
a
7
=
1
2
u
10
−
3
2
u
9
+ ··· − 3u
2
+ 2u
u
9
− u
8
− u
7
+ 3u
6
− u
5
− 2u
4
+ 3u
3
+ u
2
− 2u + 1
a
11
=
u
4
− u
2
+ 1
−u
4
a
11
=
u
4
− u
2
+ 1
−u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
10
+ 6u
8
− 6u
7
− 8u
6
+ 12u
5
− 4u
4
− 14u
3
+ 8u
2
+ 2u − 12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
u
11
+ u
10
− 10u
9
− 9u
8
+ 32u
7
+ 20u
6
− 30u
5
+ 8u
4
+ 7u
3
− 5u
2
+ 1
c
2
u
11
+ 21u
10
+ ··· + 10u + 1
c
3
u
11
− 3u
10
+ ··· − 38u − 26
c
4
, c
9
u
11
+ 3u
10
+ 2u
9
− 5u
8
− 10u
7
− 4u
6
+ 8u
5
+ 10u
4
− 8u
2
− 6u − 2
c
8
u
11
− 5u
10
+ ··· + 4u − 4
c
10
u
11
+ 9u
10
+ ··· + 82u + 22
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
y
11
− 21y
10
+ ··· + 10 y − 1
c
2
y
11
− 77y
10
+ ··· + 18 y − 1
c
3
y
11
− 53y
10
+ ··· − 6252y − 676
c
4
, c
9
y
11
− 5y
10
+ ··· + 4 y − 4
c
8
y
11
+ 3y
10
+ ··· − 176y − 16
c
10
y
11
− y
10
+ ··· + 740y − 484
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.953935 + 0.430200I
a = 0.444004 − 0.346368I
b = −0.227630 + 0.840526I
1.43107 − 1.62893I −1.305997 + 0.384907I
u = −0.953935 − 0.430200I
a = 0.444004 + 0.346368I
b = −0.227630 − 0.840526I
1.43107 + 1.62893I −1.305997 − 0.384907I
u = 0.503404 + 1.011810I
a = −1.04436 + 1.58062I
b = 0.182852 + 0.486960I
19.6194 − 2.9792I −10.19163 + 0.32130I
u = 0.503404 − 1.011810I
a = −1.04436 − 1.58062I
b = 0.182852 − 0.486960I
19.6194 + 2.9792I −10.19163 − 0.32130I
u = 1.058610 + 0.489604I
a = −0.342079 + 0.374312I
b = 0.360191 − 1.128510I
0.85115 + 4.56323I −3.37160 − 8.19390I
u = 1.058610 − 0.489604I
a = −0.342079 − 0.374312I
b = 0.360191 + 1.128510I
0.85115 − 4.56323I −3.37160 + 8.19390I
u = −1.34731
a = −1.44877
b = 3.46146
−12.7233 −6.15860
u = 0.391067 + 0.508377I
a = 0.568725 − 0.454716I
b = 0.228216 + 0.200046I
−1.063060 − 0.421255I −8.79110 + 2.32258I
u = 0.391067 − 0.508377I
a = 0.568725 + 0.454716I
b = 0.228216 − 0.200046I
−1.063060 + 0.421255I −8.79110 − 2.32258I
u = 1.174510 + 0.719102I
a = 1.09810 − 1.00915I
b = −2.77436 + 1.78824I
−17.7667 + 9.2729I −8.26036 − 4.31721I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.174510 − 0.719102I
a = 1.09810 + 1.00915I
b = −2.77436 − 1.78824I
−17.7667 − 9.2729I −8.26036 + 4.31721I
6
II. I
u
2
= hu
3
+ u
2
+ b − 2u − 1, −u
3
+ 2a + 2u, u
4
− 2u
2
+ 2i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
4
=
−u
u
a
8
=
−u
2
+ 1
2u
2
− 2
a
1
=
1
2
u
3
− u
−u
3
− u
2
+ 2u + 1
a
3
=
0
−u
a
2
=
1
2
u
3
− u
−u
3
− u
2
+ u + 1
a
6
=
1
2
u
3
− u
−u
3
− u
2
+ 2u + 1
a
7
=
1
2
u
3
− u
2
− u + 1
−u
3
+ u
2
+ 2u − 1
a
11
=
u
2
− 1
−2u
2
+ 2
a
11
=
u
2
− 1
−2u
2
+ 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
(u + 1)
4
c
3
, c
10
u
4
+ 2u
2
+ 2
c
4
, c
9
u
4
− 2u
2
+ 2
c
5
, c
11
(u − 1)
4
c
8
(u
2
+ 2u + 2)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
11
(y − 1)
4
c
3
, c
10
(y
2
+ 2y + 2)
2
c
4
, c
9
(y
2
− 2y + 2)
2
c
8
(y
2
+ 4)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.098680 + 0.455090I
a = −0.776887 + 0.321797I
b = 1.55377 − 1.64359I
−0.82247 + 3.66386I −8.00000 − 4.00000I
u = 1.098680 − 0.455090I
a = −0.776887 − 0.321797I
b = 1.55377 + 1.64359I
−0.82247 − 3.66386I −8.00000 + 4.00000I
u = −1.098680 + 0.455090I
a = 0.776887 + 0.321797I
b = −1.55377 + 0.35641I
−0.82247 − 3.66386I −8.00000 + 4.00000I
u = −1.098680 − 0.455090I
a = 0.776887 − 0.321797I
b = −1.55377 − 0.35641I
−0.82247 + 3.66386I −8.00000 − 4.00000I
10
III. I
u
3
= h−u
2
+ b + a − u, a
2
− au + u
2
− u − 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
4
=
−u
u
a
8
=
−u
2
+ 1
u
2
+ u − 1
a
1
=
a
u
2
− a + u
a
3
=
−2u + 1
−u
2
+ 2u
a
2
=
a
u
2
a + u
2
− a + u
a
6
=
−u
2
a − 2u
2
− u + 1
−au + 2u
2
a
7
=
u
2
+ u
−u
2
+ a − u
a
11
=
u
−u
2
− u + 1
a
11
=
u
−u
2
− u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u − 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
u
6
+ u
5
− 4u
4
− 2u
3
+ 10u
2
+ 4u − 5
c
2
u
6
+ 9u
5
+ 40u
4
+ 102u
3
+ 156u
2
+ 116u + 25
c
3
(u
3
+ u
2
+ 2u + 1)
2
c
4
, c
9
(u
3
− u
2
+ 1)
2
c
8
(u
3
− u
2
+ 2u − 1)
2
c
10
(u
3
− 3u
2
+ 2u + 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
y
6
− 9y
5
+ 40y
4
− 102y
3
+ 156y
2
− 116y + 25
c
2
y
6
− y
5
+ 76y
4
+ 38y
3
+ 2672y
2
− 5656y + 625
c
3
, c
8
(y
3
+ 3y
2
+ 2y − 1)
2
c
4
, c
9
(y
3
− y
2
+ 2y − 1)
2
c
10
(y
3
− 5y
2
+ 10y − 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.479677 + 1.311690I
b = −1.14204 − 1.87397I
−6.31400 − 2.82812I −9.50976 + 2.97945I
u = −0.877439 + 0.744862I
a = −1.35712 − 0.56682I
b = 0.694757 + 0.004545I
−6.31400 − 2.82812I −9.50976 + 2.97945I
u = −0.877439 − 0.744862I
a = 0.479677 − 1.311690I
b = −1.14204 + 1.87397I
−6.31400 + 2.82812I −9.50976 − 2.97945I
u = −0.877439 − 0.744862I
a = −1.35712 + 0.56682I
b = 0.694757 − 0.004545I
−6.31400 + 2.82812I −9.50976 − 2.97945I
u = 0.754878
a = −0.774732
b = 2.09945
−2.17641 −2.98050
u = 0.754878
a = 1.52961
b = −0.204892
−2.17641 −2.98050
14
IV. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
5
=
−1
0
a
9
=
1
0
a
10
=
1
0
a
4
=
−1
0
a
8
=
1
0
a
1
=
0
1
a
3
=
−1
0
a
2
=
−1
1
a
6
=
0
−1
a
7
=
1
−1
a
11
=
1
0
a
11
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
u − 1
c
2
, c
5
, c
11
u + 1
c
3
, c
4
, c
8
c
9
, c
10
u
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
11
y − 1
c
3
, c
4
, c
8
c
9
, c
10
y
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
(u − 1)(u + 1)
4
(u
6
+ u
5
− 4u
4
− 2u
3
+ 10u
2
+ 4u − 5)
· (u
11
+ u
10
− 10u
9
− 9u
8
+ 32u
7
+ 20u
6
− 30u
5
+ 8u
4
+ 7u
3
− 5u
2
+ 1)
c
2
(u + 1)
5
(u
6
+ 9u
5
+ 40u
4
+ 102u
3
+ 156u
2
+ 116u + 25)
· (u
11
+ 21u
10
+ ··· + 10u + 1)
c
3
u(u
3
+ u
2
+ 2u + 1)
2
(u
4
+ 2u
2
+ 2)(u
11
− 3u
10
+ ··· − 38u − 26)
c
4
, c
9
u(u
3
− u
2
+ 1)
2
(u
4
− 2u
2
+ 2)
· (u
11
+ 3u
10
+ 2u
9
− 5u
8
− 10u
7
− 4u
6
+ 8u
5
+ 10u
4
− 8u
2
− 6u − 2)
c
5
, c
11
(u − 1)
4
(u + 1)(u
6
+ u
5
− 4u
4
− 2u
3
+ 10u
2
+ 4u − 5)
· (u
11
+ u
10
− 10u
9
− 9u
8
+ 32u
7
+ 20u
6
− 30u
5
+ 8u
4
+ 7u
3
− 5u
2
+ 1)
c
8
u(u
2
+ 2u + 2)
2
(u
3
− u
2
+ 2u − 1)
2
(u
11
− 5u
10
+ ··· + 4 u − 4)
c
10
u(u
3
− 3u
2
+ 2u + 1)
2
(u
4
+ 2u
2
+ 2)(u
11
+ 9u
10
+ ··· + 82u + 22)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
(y − 1)
5
(y
6
− 9y
5
+ 40y
4
− 102y
3
+ 156y
2
− 116y + 25)
· (y
11
− 21y
10
+ ··· + 10 y − 1)
c
2
(y − 1)
5
(y
6
− y
5
+ 76y
4
+ 38y
3
+ 2672y
2
− 5656y + 625)
· (y
11
− 77y
10
+ ··· + 18 y − 1)
c
3
y(y
2
+ 2y + 2)
2
(y
3
+ 3y
2
+ 2y − 1)
2
· (y
11
− 53y
10
+ ··· − 6252y − 676)
c
4
, c
9
y(y
2
− 2y + 2)
2
(y
3
− y
2
+ 2y − 1)
2
(y
11
− 5y
10
+ ··· + 4 y − 4)
c
8
y(y
2
+ 4)
2
(y
3
+ 3y
2
+ 2y − 1)
2
(y
11
+ 3y
10
+ ··· − 176y − 16)
c
10
y(y
2
+ 2y + 2)
2
(y
3
− 5y
2
+ 10y − 1)
2
(y
11
− y
10
+ ··· + 740y − 484)
20
