11n
20
(K11n
20
)
A knot diagram
1
Linearized knot diagam
5 1 9 2 3 10 11 1 3 8 7
Solving Sequence
7,11
8
1,3
2 10 6 5 4 9
c
7
c
11
c
2
c
10
c
6
c
5
c
4
c
9
c
1
, c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
16
+ 2u
15
+ ··· + 2b − 1, −2u
16
+ 6u
15
+ ··· + 2a + 5, u
17
− 3u
16
+ ··· − 2u + 1i
I
u
2
= hu
2
a + b + a, −u
2
a + a
2
− au − a − u, u
3
+ u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 23 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−u
16
+2u
15
+· · ·+2b−1, −2u
16
+6u
15
+· · ·+2a+5, u
17
−3u
16
+· · ·−2u+1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
1
=
u
u
a
3
=
u
16
− 3u
15
+ ··· + 4u −
5
2
1
2
u
16
− u
15
+ ··· + 2u +
1
2
a
2
=
3
2
u
16
−
9
2
u
15
+ ··· +
11
2
u − 3
u
16
−
5
2
u
15
+ ··· + 12u
3
+
7
2
u
a
10
=
−u
u
3
+ u
a
6
=
−u
4
− u
2
+ 1
u
6
+ 2u
4
+ u
2
a
5
=
1
2
u
14
− u
13
+ ··· + 4u +
3
2
−
1
2
u
16
+ u
15
+ ··· − u +
1
2
a
4
=
−u
16
+ 5u
15
+ ··· − 5u +
9
2
−
3
2
u
16
+ 5u
15
+ ··· − 5u +
3
2
a
9
=
−u
4
− u
2
+ 1
−u
4
− 2u
2
a
9
=
−u
4
− u
2
+ 1
−u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
16
−
3
2
u
15
+
13
2
u
14
−
15
2
u
13
+
33
2
u
12
− 19u
11
+ 23u
10
−
67
2
u
9
+
24u
8
−
81
2
u
7
+ 22u
6
− 17u
5
+ 12u
4
+ 13u
3
+
19
2
u +
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
17
+ 4u
16
+ ··· + 3u + 1
c
2
u
17
+ 2u
16
+ ··· + 3u − 1
c
3
, c
9
u
17
+ u
16
+ ··· − 96u − 64
c
5
u
17
− 4u
16
+ ··· + 557u + 137
c
6
, c
8
u
17
− 3u
16
+ ··· + 2u − 1
c
7
, c
10
, c
11
u
17
+ 3u
16
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
17
+ 2y
16
+ ··· + 3y −1
c
2
y
17
+ 30y
16
+ ··· + 3y −1
c
3
, c
9
y
17
+ 35y
16
+ ··· + 9216y −4096
c
5
y
17
+ 58y
16
+ ··· − 518053y −18769
c
6
, c
8
y
17
− 31y
16
+ ··· − 14y −1
c
7
, c
10
, c
11
y
17
+ 13y
16
+ ··· − 14y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.219268 + 0.999289I
a = 1.80602 − 0.25255I
b = 0.77227 − 1.25221I
−1.31705 + 3.54605I 2.16487 − 2.95335I
u = 0.219268 − 0.999289I
a = 1.80602 + 0.25255I
b = 0.77227 + 1.25221I
−1.31705 − 3.54605I 2.16487 + 2.95335I
u = 1.052720 + 0.047013I
a = 0.092101 − 0.115408I
b = −0.02224 − 2.20267I
15.8539 + 3.8626I 6.39661 − 2.12816I
u = 1.052720 − 0.047013I
a = 0.092101 + 0.115408I
b = −0.02224 + 2.20267I
15.8539 − 3.8626I 6.39661 + 2.12816I
u = −0.095288 + 1.269800I
a = 0.670660 + 0.619159I
b = 0.689025 + 0.075156I
−4.44298 − 1.97657I −3.41444 + 3.62302I
u = −0.095288 − 1.269800I
a = 0.670660 − 0.619159I
b = 0.689025 − 0.075156I
−4.44298 + 1.97657I −3.41444 − 3.62302I
u = −0.228042 + 0.683004I
a = −1.181530 + 0.437561I
b = −0.295357 + 0.574121I
0.22550 − 1.43526I 4.15937 + 3.64291I
u = −0.228042 − 0.683004I
a = −1.181530 − 0.437561I
b = −0.295357 − 0.574121I
0.22550 + 1.43526I 4.15937 − 3.64291I
u = −0.710942
a = −0.587793
b = −0.244243
1.69761 6.54340
u = −0.281522 + 1.323870I
a = −0.413372 + 0.348973I
b = −0.121248 + 0.378439I
−2.51924 − 3.59257I 1.69678 + 1.62034I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.281522 − 1.323870I
a = −0.413372 − 0.348973I
b = −0.121248 − 0.378439I
−2.51924 + 3.59257I 1.69678 − 1.62034I
u = 0.55805 + 1.32231I
a = −1.54225 + 1.32872I
b = −0.28772 + 2.07576I
11.91540 + 1.84478I 3.96952 − 0.75367I
u = 0.55805 − 1.32231I
a = −1.54225 − 1.32872I
b = −0.28772 − 2.07576I
11.91540 − 1.84478I 3.96952 + 0.75367I
u = 0.50759 + 1.37481I
a = 1.54834 − 1.48310I
b = 0.23471 − 2.13882I
11.4057 + 9.4106I 3.33658 − 4.76975I
u = 0.50759 − 1.37481I
a = 1.54834 + 1.48310I
b = 0.23471 + 2.13882I
11.4057 − 9.4106I 3.33658 + 4.76975I
u = 0.122694 + 0.403191I
a = −1.68607 + 0.58270I
b = 0.152683 + 0.763557I
0.106087 − 1.407490I 0.91901 + 2.91397I
u = 0.122694 − 0.403191I
a = −1.68607 − 0.58270I
b = 0.152683 − 0.763557I
0.106087 + 1.407490I 0.91901 − 2.91397I
6
II. I
u
2
= hu
2
a + b + a, −u
2
a + a
2
− au − a − u, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
1
=
u
u
a
3
=
a
−u
2
a − a
a
2
=
−u
2
a − au
−2u
2
a − au − 2a
a
10
=
−u
−u
2
− u − 1
a
6
=
−u
−u
a
5
=
−u
2
+ a − 2u − 1
−u
2
a − a − u + 1
a
4
=
a
−u
2
a − a
a
9
=
−u
−u
2
− u − 1
a
9
=
−u
−u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
a − au + 3u
2
+ 5a + 3u + 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
3
c
3
, c
9
u
6
c
4
(u
2
− u + 1)
3
c
6
, c
8
(u
3
− u
2
+ 1)
2
c
7
(u
3
+ u
2
+ 2u + 1)
2
c
10
, c
11
(u
3
− u
2
+ 2u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
3
c
3
, c
9
y
6
c
6
, c
8
(y
3
− y
2
+ 2y −1)
2
c
7
, c
10
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = 0.206350 + 1.132320I
b = −0.500000 + 0.866025I
−3.02413 − 4.85801I −1.45566 + 6.64456I
u = −0.215080 + 1.307140I
a = −1.083790 − 0.387453I
b = −0.500000 − 0.866025I
−3.02413 − 0.79824I 2.09851 − 0.12339I
u = −0.215080 − 1.307140I
a = 0.206350 − 1.132320I
b = −0.500000 − 0.866025I
−3.02413 + 4.85801I −1.45566 − 6.64456I
u = −0.215080 − 1.307140I
a = −1.083790 + 0.387453I
b = −0.500000 + 0.866025I
−3.02413 + 0.79824I 2.09851 + 0.12339I
u = −0.569840
a = 0.377439 + 0.653743I
b = −0.500000 − 0.866025I
1.11345 − 2.02988I 5.85715 + 4.49037I
u = −0.569840
a = 0.377439 − 0.653743I
b = −0.500000 + 0.866025I
1.11345 + 2.02988I 5.85715 − 4.49037I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
17
+ 4u
16
+ ··· + 3u + 1)
c
2
((u
2
+ u + 1)
3
)(u
17
+ 2u
16
+ ··· + 3u − 1)
c
3
, c
9
u
6
(u
17
+ u
16
+ ··· − 96u − 64)
c
4
((u
2
− u + 1)
3
)(u
17
+ 4u
16
+ ··· + 3u + 1)
c
5
((u
2
+ u + 1)
3
)(u
17
− 4u
16
+ ··· + 557u + 137)
c
6
, c
8
((u
3
− u
2
+ 1)
2
)(u
17
− 3u
16
+ ··· + 2u − 1)
c
7
((u
3
+ u
2
+ 2u + 1)
2
)(u
17
+ 3u
16
+ ··· − 2u − 1)
c
10
, c
11
((u
3
− u
2
+ 2u − 1)
2
)(u
17
+ 3u
16
+ ··· − 2u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
3
)(y
17
+ 2y
16
+ ··· + 3y −1)
c
2
((y
2
+ y + 1)
3
)(y
17
+ 30y
16
+ ··· + 3y −1)
c
3
, c
9
y
6
(y
17
+ 35y
16
+ ··· + 9216y −4096)
c
5
((y
2
+ y + 1)
3
)(y
17
+ 58y
16
+ ··· − 518053y −18769)
c
6
, c
8
((y
3
− y
2
+ 2y −1)
2
)(y
17
− 31y
16
+ ··· − 14y −1)
c
7
, c
10
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
17
+ 13y
16
+ ··· − 14y −1)
12
