11n
118
(K11n
118
)
A knot diagram
1
Linearized knot diagam
5 1 8 10 2 9 10 5 1 5 7
Solving Sequence
1,5
2 3
6,9
7 10 4 8 11
c
1
c
2
c
5
c
6
c
9
c
4
c
8
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
8
− 6u
7
+ u
6
+ 3u
5
+ 15u
4
− 13u
3
− 10u
2
+ b + 3u + 3, u
8
− u
7
− 5u
6
+ 3u
5
+ 7u
4
+ 7u
3
− 15u
2
+ 2a + 2,
u
9
− 5u
8
+ 7u
7
− u
6
+ 5u
5
− 21u
4
+ 11u
3
+ 8u
2
− 2u − 2i
I
u
2
= h−u
3
− u
2
+ b + u + 1, −u
4
− 2u
3
+ u
2
+ 2a + u, u
5
+ 2u
4
− u
3
− 3u
2
+ 2i
I
u
3
= h−u
2
a − au + u
2
+ b + u − 1, u
2
a + a
2
− 5u
2
− 3a − 2u + 11, u
3
+ u
2
− 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 20 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
= h2u
8
− 6u
7
+ · · · + b + 3, u
8
− u
7
+ · · · + 2a + 2, u
9
− 5u
8
+ · · · − 2u − 2i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
−u
2
+ 1
u
2
a
6
=
−u
−u
3
+ u
a
9
=
−
1
2
u
8
+
1
2
u
7
+ ··· +
15
2
u
2
− 1
−2u
8
+ 6u
7
− u
6
− 3u
5
− 15u
4
+ 13u
3
+ 10u
2
− 3u − 3
a
7
=
−
3
2
u
8
+
11
2
u
7
+ ··· − 3u − 2
−u
8
+ 4u
7
− 3u
6
− u
5
− 8u
4
+ 12u
3
+ u
2
− u − 1
a
10
=
3
2
u
8
−
11
2
u
7
+ ··· + 3u + 2
−2u
8
+ 6u
7
− u
6
− 3u
5
− 15u
4
+ 13u
3
+ 10u
2
− 3u − 3
a
4
=
1
2
u
8
−
3
2
u
7
+ ··· + u + 1
−u
8
+ 3u
7
− u
6
− u
5
− 7u
4
+ 7u
3
+ 4u
2
− u − 1
a
8
=
−
1
2
u
8
+
1
2
u
7
+ ··· +
15
2
u
2
− 1
2u
8
− 6u
7
+ 2u
6
+ u
5
+ 14u
4
− 13u
3
− 4u
2
+ 2u + 1
a
11
=
3
2
u
8
−
11
2
u
7
+ ··· + 3u + 2
u
8
− 5u
7
+ 5u
6
+ 2u
5
+ 8u
4
− 18u
3
− u
2
+ 4u + 1
a
11
=
3
2
u
8
−
11
2
u
7
+ ··· + 3u + 2
u
8
− 5u
7
+ 5u
6
+ 2u
5
+ 8u
4
− 18u
3
− u
2
+ 4u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −3u
8
+ 13u
7
− 14u
6
+ u
5
− 20u
4
+ 46u
3
− 14u
2
− 12u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
9
+ 5u
8
+ 7u
7
+ u
6
+ 5u
5
+ 21u
4
+ 11u
3
− 8u
2
− 2u + 2
c
2
u
9
+ 11u
8
+ ··· + 36u + 4
c
3
, c
4
, c
10
u
9
+ 7u
7
+ 2u
6
+ 18u
5
+ 8u
4
+ 16u
3
+ 7u
2
+ 2u + 1
c
6
, c
9
u
9
− 2u
8
− 3u
7
+ 8u
6
+ 6u
5
− 10u
4
− 10u
3
+ 3u
2
+ 8u + 1
c
7
u
9
+ 6u
8
+ 19u
7
+ 38u
6
+ 54u
5
+ 56u
4
+ 49u
3
+ 36u
2
+ 16u + 2
c
8
u
9
+ u
8
− 8u
7
− 4u
6
+ 21u
5
+ 3u
4
− 3u
3
− 5u
2
+ u + 1
c
11
u
9
− 8u
8
+ 30u
7
− 69u
6
+ 106u
5
− 109u
4
+ 69u
3
− 18u
2
− 8u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
9
− 11y
8
+ ··· + 36y −4
c
2
y
9
− 23y
8
+ ··· − 240y −16
c
3
, c
4
, c
10
y
9
+ 14y
8
+ 85y
7
+ 280y
6
+ 520y
5
+ 512y
4
+ 212y
3
− y
2
− 10y −1
c
6
, c
9
y
9
− 10y
8
+ ··· + 58y −1
c
7
y
9
+ 2y
8
+ 13y
7
+ 34y
6
+ 122y
5
+ 4y
4
− 55y
3
+ 48y
2
+ 112y −4
c
8
y
9
− 17y
8
+ ··· + 11y −1
c
11
y
9
− 4y
8
+ 8y
7
− 7y
6
+ 30y
5
− 89y
4
+ 245y
3
+ 316y
2
+ 352y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.827217 + 1.065600I
a = 0.127211 + 0.403713I
b = 1.189760 − 0.208029I
4.50943 + 3.60395I −7.94742 − 3.61538I
u = −0.827217 − 1.065600I
a = 0.127211 − 0.403713I
b = 1.189760 + 0.208029I
4.50943 − 3.60395I −7.94742 + 3.61538I
u = 0.637971
a = 0.581775
b = −0.134369
−0.867730 −11.0840
u = −0.390331 + 0.211849I
a = −0.15178 − 1.44152I
b = −0.619342 − 0.660345I
−0.58699 − 1.71933I −2.65828 + 4.51037I
u = −0.390331 − 0.211849I
a = −0.15178 + 1.44152I
b = −0.619342 + 0.660345I
−0.58699 + 1.71933I −2.65828 − 4.51037I
u = 1.60275 + 0.27471I
a = −1.057300 + 0.502009I
b = −1.245760 − 0.193497I
−7.17964 − 0.81901I −9.63369 + 0.38923I
u = 1.60275 − 0.27471I
a = −1.057300 − 0.502009I
b = −1.245760 + 0.193497I
−7.17964 + 0.81901I −9.63369 − 0.38923I
u = 1.79582 + 0.27938I
a = 1.290980 + 0.002356I
b = 1.74253 + 0.93792I
−4.53360 − 8.88256I −8.21864 + 4.17646I
u = 1.79582 − 0.27938I
a = 1.290980 − 0.002356I
b = 1.74253 − 0.93792I
−4.53360 + 8.88256I −8.21864 − 4.17646I
5
II.
I
u
2
= h−u
3
− u
2
+ b + u + 1, −u
4
− 2u
3
+ u
2
+ 2a + u, u
5
+ 2u
4
− u
3
− 3u
2
+ 2i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
−u
2
+ 1
u
2
a
6
=
−u
−u
3
+ u
a
9
=
1
2
u
4
+ u
3
−
1
2
u
2
−
1
2
u
u
3
+ u
2
− u − 1
a
7
=
−
1
2
u
4
+
3
2
u
2
−
1
2
u − 1
u
2
+ u − 1
a
10
=
1
2
u
4
−
3
2
u
2
+
1
2
u + 1
u
3
+ u
2
− u − 1
a
4
=
1
2
u
4
+ u
3
−
3
2
u
2
−
3
2
u + 2
u
2
+ u − 1
a
8
=
1
2
u
4
+ u
3
−
1
2
u
2
−
1
2
u
u
2
− 1
a
11
=
1
2
u
4
−
3
2
u
2
+
1
2
u + 1
u
4
+ 2u
3
− u
2
− 2u + 1
a
11
=
1
2
u
4
−
3
2
u
2
+
1
2
u + 1
u
4
+ 2u
3
− u
2
− 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
4
− 2u
3
+ u
2
− 2u − 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
+ 2u
4
− u
3
− 3u
2
+ 2
c
2
u
5
+ 6u
4
+ 13u
3
+ 17u
2
+ 12u + 4
c
3
, c
10
u
5
+ 2u
3
+ u
2
− 3u + 1
c
4
u
5
+ 2u
3
− u
2
− 3u − 1
c
5
u
5
− 2u
4
− u
3
+ 3u
2
− 2
c
6
, c
9
u
5
− 2u
4
+ u
2
− u − 1
c
7
u
5
+ 3u
4
+ 2u
3
− u
2
− 2u − 2
c
8
u
5
− u
4
− u
3
− 4u
2
− 2u − 1
c
11
u
5
+ u
4
− u
3
+ 2u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
5
− 6y
4
+ 13y
3
− 17y
2
+ 12y −4
c
2
y
5
− 10y
4
− 11y
3
− 25y
2
+ 8y −16
c
3
, c
4
, c
10
y
5
+ 4y
4
− 2y
3
− 13y
2
+ 7y −1
c
6
, c
9
y
5
− 4y
4
+ 2y
3
− 5y
2
+ 3y −1
c
7
y
5
− 5y
4
+ 6y
3
+ 3y
2
− 4
c
8
y
5
− 3y
4
− 11y
3
− 14y
2
− 4y −1
c
11
y
5
− 3y
4
+ 5y
3
− 2y
2
+ 4y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.886428 + 0.266186I
a = −0.148382 + 0.576930I
b = −0.663438 + 0.814334I
−1.42879 + 1.52428I −12.65090 − 2.62716I
u = 0.886428 − 0.266186I
a = −0.148382 − 0.576930I
b = −0.663438 − 0.814334I
−1.42879 − 1.52428I −12.65090 + 2.62716I
u = −0.972160 + 0.575992I
a = −0.210793 + 1.027090I
b = 0.634295 − 0.253899I
6.00798 + 2.19755I −5.78391 − 2.40841I
u = −0.972160 − 0.575992I
a = −0.210793 − 1.027090I
b = 0.634295 + 0.253899I
6.00798 − 2.19755I −5.78391 + 2.40841I
u = −1.82854
a = −1.28165
b = −1.94171
−12.4482 −13.1300
9
III.
I
u
3
= h−u
2
a−au+u
2
+b+u−1, u
2
a+a
2
−5u
2
−3a−2u+11, u
3
+u
2
−2u−1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
−u
2
+ 1
u
2
a
6
=
−u
u
2
− u − 1
a
9
=
a
u
2
a + au − u
2
− u + 1
a
7
=
−u
2
a − au + u
2
+ a + u − 2
1
a
10
=
−u
2
a − au + u
2
+ a + u − 1
u
2
a + au − u
2
− u + 1
a
4
=
−au − 3u
2
− a − u + 7
au + u
2
+ u − 1
a
8
=
a
au − u
2
− u + 1
a
11
=
−u
2
a − au + u
2
+ a + u − 1
1
a
11
=
−u
2
a − au + u
2
+ a + u − 1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
7
(u
3
− u
2
− 2u + 1)
2
c
2
(u
3
+ 5u
2
+ 6u + 1)
2
c
3
, c
4
, c
10
u
6
− u
5
+ 2u
4
− 4u
3
− 2u
2
− 8u − 1
c
6
, c
9
u
6
− u
5
− 2u
4
+ 8u
3
− 14u
2
+ 14u − 7
c
8
u
6
+ u
5
− 4u
4
− 2u
2
− 12u − 13
c
11
(u + 1)
6
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
(y
3
− 5y
2
+ 6y −1)
2
c
2
(y
3
− 13y
2
+ 26y −1)
2
c
3
, c
4
, c
10
y
6
+ 3y
5
− 8y
4
− 42y
3
− 64y
2
− 60y + 1
c
6
, c
9
y
6
− 5y
5
− 8y
4
+ 6y
3
+ 49
c
8
y
6
− 9y
5
+ 12y
4
+ 14y
3
+ 108y
2
− 92y + 169
c
11
(y −1)
6
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.24698
a = 0.722521 + 0.457399I
b = 0.222521 + 1.281600I
0.234991 −6.00000
u = 1.24698
a = 0.722521 − 0.457399I
b = 0.222521 − 1.281600I
0.234991 −6.00000
u = −0.445042
a = 1.40097 + 2.98949I
b = 0.900969 − 0.738343I
5.87476 −6.00000
u = −0.445042
a = 1.40097 − 2.98949I
b = 0.900969 + 0.738343I
5.87476 −6.00000
u = −1.80194
a = 1.15958
b = 1.23060
−11.0446 −6.00000
u = −1.80194
a = −1.40656
b = −2.47758
−11.0446 −6.00000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
− 2u + 1)
2
(u
5
+ 2u
4
− u
3
− 3u
2
+ 2)
· (u
9
+ 5u
8
+ 7u
7
+ u
6
+ 5u
5
+ 21u
4
+ 11u
3
− 8u
2
− 2u + 2)
c
2
(u
3
+ 5u
2
+ 6u + 1)
2
(u
5
+ 6u
4
+ 13u
3
+ 17u
2
+ 12u + 4)
· (u
9
+ 11u
8
+ ··· + 36u + 4)
c
3
, c
10
(u
5
+ 2u
3
+ u
2
− 3u + 1)(u
6
− u
5
+ 2u
4
− 4u
3
− 2u
2
− 8u − 1)
· (u
9
+ 7u
7
+ 2u
6
+ 18u
5
+ 8u
4
+ 16u
3
+ 7u
2
+ 2u + 1)
c
4
(u
5
+ 2u
3
− u
2
− 3u − 1)(u
6
− u
5
+ 2u
4
− 4u
3
− 2u
2
− 8u − 1)
· (u
9
+ 7u
7
+ 2u
6
+ 18u
5
+ 8u
4
+ 16u
3
+ 7u
2
+ 2u + 1)
c
5
(u
3
− u
2
− 2u + 1)
2
(u
5
− 2u
4
− u
3
+ 3u
2
− 2)
· (u
9
+ 5u
8
+ 7u
7
+ u
6
+ 5u
5
+ 21u
4
+ 11u
3
− 8u
2
− 2u + 2)
c
6
, c
9
(u
5
− 2u
4
+ u
2
− u − 1)(u
6
− u
5
− 2u
4
+ 8u
3
− 14u
2
+ 14u − 7)
· (u
9
− 2u
8
− 3u
7
+ 8u
6
+ 6u
5
− 10u
4
− 10u
3
+ 3u
2
+ 8u + 1)
c
7
(u
3
− u
2
− 2u + 1)
2
(u
5
+ 3u
4
+ 2u
3
− u
2
− 2u − 2)
· (u
9
+ 6u
8
+ 19u
7
+ 38u
6
+ 54u
5
+ 56u
4
+ 49u
3
+ 36u
2
+ 16u + 2)
c
8
(u
5
− u
4
− u
3
− 4u
2
− 2u − 1)(u
6
+ u
5
− 4u
4
− 2u
2
− 12u − 13)
· (u
9
+ u
8
− 8u
7
− 4u
6
+ 21u
5
+ 3u
4
− 3u
3
− 5u
2
+ u + 1)
c
11
(u + 1)
6
(u
5
+ u
4
− u
3
+ 2u − 1)
· (u
9
− 8u
8
+ 30u
7
− 69u
6
+ 106u
5
− 109u
4
+ 69u
3
− 18u
2
− 8u + 8)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
3
− 5y
2
+ 6y −1)
2
(y
5
− 6y
4
+ 13y
3
− 17y
2
+ 12y −4)
· (y
9
− 11y
8
+ ··· + 36y −4)
c
2
(y
3
− 13y
2
+ 26y −1)
2
(y
5
− 10y
4
− 11y
3
− 25y
2
+ 8y −16)
· (y
9
− 23y
8
+ ··· − 240y −16)
c
3
, c
4
, c
10
(y
5
+ 4y
4
− 2y
3
− 13y
2
+ 7y −1)
· (y
6
+ 3y
5
− 8y
4
− 42y
3
− 64y
2
− 60y + 1)
· (y
9
+ 14y
8
+ 85y
7
+ 280y
6
+ 520y
5
+ 512y
4
+ 212y
3
− y
2
− 10y −1)
c
6
, c
9
(y
5
− 4y
4
+ 2y
3
− 5y
2
+ 3y −1)(y
6
− 5y
5
− 8y
4
+ 6y
3
+ 49)
· (y
9
− 10y
8
+ ··· + 58y −1)
c
7
(y
3
− 5y
2
+ 6y −1)
2
(y
5
− 5y
4
+ 6y
3
+ 3y
2
− 4)
· (y
9
+ 2y
8
+ 13y
7
+ 34y
6
+ 122y
5
+ 4y
4
− 55y
3
+ 48y
2
+ 112y −4)
c
8
(y
5
− 3y
4
− 11y
3
− 14y
2
− 4y −1)
· (y
6
− 9y
5
+ 12y
4
+ 14y
3
+ 108y
2
− 92y + 169)
· (y
9
− 17y
8
+ ··· + 11y −1)
c
11
(y −1)
6
(y
5
− 3y
4
+ 5y
3
− 2y
2
+ 4y −1)
· (y
9
− 4y
8
+ 8y
7
− 7y
6
+ 30y
5
− 89y
4
+ 245y
3
+ 316y
2
+ 352y −64)
15
