11n
64
(K11n
64
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 9 11 10 4 1 7 6
Solving Sequence
7,11
6
1,4
2 5 10 8 3 9
c
6
c
11
c
1
c
4
c
10
c
7
c
3
c
9
c
2
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
− 2u
10
+ 8u
9
− 12u
8
+ 22u
7
− 23u
6
+ 24u
5
− 15u
4
+ 9u
3
− 4u
2
+ b + 2u − 1,
− u
13
+ 2u
12
− 11u
11
+ 18u
10
− 45u
9
+ 58u
8
− 84u
7
+ 79u
6
− 70u
5
+ 43u
4
− 23u
3
+ 14u
2
+ a − 5u + 3,
u
14
− 2u
13
+ 11u
12
− 18u
11
+ 46u
10
− 60u
9
+ 91u
8
− 90u
7
+ 86u
6
− 61u
5
+ 36u
4
− 22u
3
+ 8u
2
− 5u + 1i
I
u
2
= hu
2
+ b + u + 1, −u
3
− u
2
+ a − 3u − 1, u
4
+ u
3
+ 3u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 18 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
11
−2u
10
+· · ·+b−1, −u
13
+2u
12
+· · ·+a+3, u
14
−2u
13
+· · ·−5u+1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
6
=
1
u
2
a
1
=
u
u
3
+ u
a
4
=
u
13
− 2u
12
+ ··· + 5u − 3
−u
11
+ 2u
10
+ ··· − 2u + 1
a
2
=
u
13
− 2u
12
+ ··· + 4u − 2
u
12
− 2u
11
+ ··· − 2u + 1
a
5
=
u
10
+ 5u
8
+ 6u
6
− u
4
− u
2
− 1
u
12
+ 6u
10
+ 12u
8
+ 10u
6
+ 5u
4
a
10
=
−u
u
a
8
=
u
2
+ 1
−u
2
a
3
=
u
13
− 2u
12
+ ··· + 3u − 2
−u
13
+ 2u
12
+ ··· − 3u + 1
a
9
=
u
5
+ 2u
3
− u
u
7
+ 3u
5
+ 2u
3
+ u
a
9
=
u
5
+ 2u
3
− u
u
7
+ 3u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −u
13
+2u
12
−11u
11
+17u
10
−42u
9
+48u
8
−62u
7
+44u
6
−17u
5
−7u
4
+23u
3
−11u
2
+6u−9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
14
− 5u
13
+ ··· + 7u − 1
c
2
u
14
+ 23u
13
+ ··· + 3u + 1
c
3
, c
8
u
14
− u
13
+ ··· − 24u − 16
c
5
u
14
+ 2u
13
+ ··· + 3u + 1
c
6
, c
7
, c
10
c
11
u
14
+ 2u
13
+ ··· + 5u + 1
c
9
u
14
− 6u
13
+ ··· − 117u + 19
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
14
− 23y
13
+ ··· − 3y + 1
c
2
y
14
− 59y
13
+ ··· + 681y + 1
c
3
, c
8
y
14
− 27y
13
+ ··· + 960y + 256
c
5
y
14
− 30y
13
+ ··· − 9y + 1
c
6
, c
7
, c
10
c
11
y
14
+ 18y
13
+ ··· − 9y + 1
c
9
y
14
− 18y
13
+ ··· − 22277y + 361
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.550866 + 0.900632I
a = −1.08064 − 1.46517I
b = 0.098170 − 0.182990I
−14.5979 + 4.4309I −8.77759 − 3.45380I
u = 0.550866 − 0.900632I
a = −1.08064 + 1.46517I
b = 0.098170 + 0.182990I
−14.5979 − 4.4309I −8.77759 + 3.45380I
u = 0.131850 + 0.795140I
a = 1.42680 + 0.95287I
b = 0.002648 − 0.749310I
−3.41989 + 1.31906I −10.67824 − 1.83447I
u = 0.131850 − 0.795140I
a = 1.42680 − 0.95287I
b = 0.002648 + 0.749310I
−3.41989 − 1.31906I −10.67824 + 1.83447I
u = 0.778815
a = 0.532114
b = −1.55087
−11.8742 −5.71440
u = −0.310969 + 0.512101I
a = −0.546661 + 0.069487I
b = −0.054284 + 0.327429I
−0.044354 − 1.170560I −0.70219 + 5.58030I
u = −0.310969 − 0.512101I
a = −0.546661 − 0.069487I
b = −0.054284 − 0.327429I
−0.044354 + 1.170560I −0.70219 − 5.58030I
u = −0.07909 + 1.57522I
a = −0.593424 + 0.223836I
b = 1.203460 + 0.007835I
−7.25996 − 2.49887I −4.67922 + 1.75896I
u = −0.07909 − 1.57522I
a = −0.593424 − 0.223836I
b = 1.203460 − 0.007835I
−7.25996 + 2.49887I −4.67922 − 1.75896I
u = 0.03110 + 1.66209I
a = 1.73343 + 0.06583I
b = −3.48457 − 0.73242I
−12.09880 + 1.91262I −10.51406 − 1.13289I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.03110 − 1.66209I
a = 1.73343 − 0.06583I
b = −3.48457 + 0.73242I
−12.09880 − 1.91262I −10.51406 + 1.13289I
u = 0.16129 + 1.68407I
a = −2.01585 − 0.88397I
b = 4.18614 + 1.67352I
15.9841 + 7.2397I −10.36154 − 2.69654I
u = 0.16129 − 1.68407I
a = −2.01585 + 0.88397I
b = 4.18614 − 1.67352I
15.9841 − 7.2397I −10.36154 + 2.69654I
u = 0.251089
a = −2.37942
b = 0.647742
−1.17986 −7.85990
6
II. I
u
2
= hu
2
+ b + u + 1, −u
3
− u
2
+ a − 3u − 1, u
4
+ u
3
+ 3u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
6
=
1
u
2
a
1
=
u
u
3
+ u
a
4
=
u
3
+ u
2
+ 3u + 1
−u
2
− u − 1
a
2
=
u
3
+ u
2
+ 4u + 1
u
3
− u
2
− 1
a
5
=
−u
−u
3
− u
a
10
=
−u
u
a
8
=
u
2
+ 1
−u
2
a
3
=
u
3
+ u
2
+ 3u + 1
−u
2
− u − 1
a
9
=
u
2
+ 1
−u
2
a
9
=
u
2
+ 1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
3
+ 3u
2
+ 10u − 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
c
2
, c
4
(u + 1)
4
c
3
, c
8
u
4
c
5
, c
9
u
4
+ u
3
+ u
2
+ 1
c
6
, c
7
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
10
, c
11
u
4
− u
3
+ 3u
2
− 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
8
y
4
c
5
, c
9
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
6
, c
7
, c
10
c
11
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.395123 + 0.506844I
a = −0.043315 + 1.227190I
b = −0.504108 − 0.106312I
−1.43393 − 1.41510I −7.52507 + 4.18840I
u = −0.395123 − 0.506844I
a = −0.043315 − 1.227190I
b = −0.504108 + 0.106312I
−1.43393 + 1.41510I −7.52507 − 4.18840I
u = −0.10488 + 1.55249I
a = −0.956685 + 0.641200I
b = 1.50411 − 1.22685I
−8.43568 − 3.16396I −9.97493 + 3.47609I
u = −0.10488 − 1.55249I
a = −0.956685 − 0.641200I
b = 1.50411 + 1.22685I
−8.43568 + 3.16396I −9.97493 − 3.47609I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
14
− 5u
13
+ ··· + 7u − 1)
c
2
((u + 1)
4
)(u
14
+ 23u
13
+ ··· + 3u + 1)
c
3
, c
8
u
4
(u
14
− u
13
+ ··· − 24u − 16)
c
4
((u + 1)
4
)(u
14
− 5u
13
+ ··· + 7u − 1)
c
5
(u
4
+ u
3
+ u
2
+ 1)(u
14
+ 2u
13
+ ··· + 3u + 1)
c
6
, c
7
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
14
+ 2u
13
+ ··· + 5u + 1)
c
9
(u
4
+ u
3
+ u
2
+ 1)(u
14
− 6u
13
+ ··· − 117u + 19)
c
10
, c
11
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
14
+ 2u
13
+ ··· + 5u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
14
− 23y
13
+ ··· − 3y + 1)
c
2
((y −1)
4
)(y
14
− 59y
13
+ ··· + 681y + 1)
c
3
, c
8
y
4
(y
14
− 27y
13
+ ··· + 960y + 256)
c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
14
− 30y
13
+ ··· − 9y + 1)
c
6
, c
7
, c
10
c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
14
+ 18y
13
+ ··· − 9y + 1)
c
9
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
14
− 18y
13
+ ··· − 22277y + 361)
12
