11n
13
(K11n
13
)
A knot diagram
1
Linearized knot diagam
5 1 8 2 3 10 11 4 1 7 8
Solving Sequence
1,5
2 3
4,8
9 11 7 10 6
c
1
c
2
c
4
c
8
c
11
c
7
c
10
c
6
c
3
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
10
− 2u
9
+ 6u
8
− 8u
7
+ 13u
6
− 15u
5
+ 14u
4
− 15u
3
+ 7u
2
+ 2b − 7u + 2,
− 2u
10
+ 5u
9
− 16u
8
+ 26u
7
− 42u
6
+ 51u
5
− 49u
4
+ 50u
3
− 29u
2
+ 2a + 23u − 9,
u
11
− 3u
10
+ 9u
9
− 16u
8
+ 25u
7
− 32u
6
+ 32u
5
− 32u
4
+ 22u
3
− 15u
2
+ 8u − 1i
I
u
2
= h−au + b, a
2
+ au + a − u, u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 15 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
10
−2u
9
+· · ·+2b+2, −2u
10
+5u
9
+· · ·+2a−9, u
11
−3u
10
+· · ·+8u−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
4
=
−u
u
3
+ u
a
8
=
u
10
−
5
2
u
9
+ ··· −
23
2
u +
9
2
−
1
2
u
10
+ u
9
+ ··· +
7
2
u − 1
a
9
=
u
10
−
7
2
u
9
+ ··· −
25
2
u +
9
2
1
2
u
10
− 2u
9
+ ··· +
15
2
u
2
−
7
2
u
a
11
=
−
1
2
u
9
+ u
8
+ ··· −
3
2
u +
5
2
1
2
u
10
− u
9
+ ··· +
3
2
u
2
−
3
2
u
a
7
=
−
1
2
u
10
+
1
2
u
9
+ ··· + u +
3
2
1
2
u
10
− u
9
+ ··· +
5
2
u
2
−
5
2
u
a
10
=
1
2
u
10
−
3
2
u
9
+ ··· − 9u +
9
2
1
2
u
10
− 2u
9
+ ··· +
15
2
u
2
−
7
2
u
a
6
=
u
5
+ 2u
3
+ u
−u
5
− u
3
− u
a
6
=
u
5
+ 2u
3
+ u
−u
5
− u
3
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1
2
u
10
−2u
9
+ 5u
8
−9u
7
+
21
2
u
6
−
19
2
u
5
+ 4u
4
+
5
2
u
3
−
7
2
u
2
+
9
2
u −14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
11
+ 3u
10
+ ··· + 8u + 1
c
2
u
11
+ 9u
10
+ ··· + 34u − 1
c
3
, c
8
u
11
− u
10
+ ··· − 32u − 16
c
5
u
11
− 3u
10
+ ··· + 17u + 2
c
6
, c
7
, c
10
c
11
u
11
+ 3u
10
+ ··· + 2u − 1
c
9
u
11
− 13u
10
+ ··· − 2u + 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
11
+ 9y
10
+ ··· + 34y −1
c
2
y
11
− 11y
10
+ ··· + 1282y −1
c
3
, c
8
y
11
− 25y
10
+ ··· + 128y −256
c
5
y
11
− 31y
10
+ ··· + 109y −4
c
6
, c
7
, c
10
c
11
y
11
− 19y
10
+ ··· + 14y −1
c
9
y
11
− 79y
10
+ ··· − 38y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.417699 + 0.894239I
a = 0.258441 + 0.135782I
b = 0.229372 − 0.174392I
−0.35168 − 1.75940I −2.35906 + 1.98194I
u = −0.417699 − 0.894239I
a = 0.258441 − 0.135782I
b = 0.229372 + 0.174392I
−0.35168 + 1.75940I −2.35906 − 1.98194I
u = −0.053436 + 1.167960I
a = −0.409120 − 0.745046I
b = −0.892048 + 0.438025I
−3.73521 + 0.42312I −13.72245 − 1.16571I
u = −0.053436 − 1.167960I
a = −0.409120 + 0.745046I
b = −0.892048 − 0.438025I
−3.73521 − 0.42312I −13.72245 + 1.16571I
u = 1.23651
a = 1.53499
b = −1.89802
19.0799 −11.4300
u = 0.732319
a = −1.96295
b = 1.43751
−7.32923 −11.5330
u = 0.289180 + 1.380880I
a = 0.019963 + 1.125350I
b = 1.54820 − 0.35299I
−11.85310 + 3.71325I −14.2941 − 2.2784I
u = 0.289180 − 1.380880I
a = 0.019963 − 1.125350I
b = 1.54820 + 0.35299I
−11.85310 − 3.71325I −14.2941 + 2.2784I
u = 0.61390 + 1.45389I
a = 0.404068 − 1.146040I
b = −1.91428 + 0.11608I
14.5460 + 6.5663I −13.47746 − 2.65332I
u = 0.61390 − 1.45389I
a = 0.404068 + 1.146040I
b = −1.91428 − 0.11608I
14.5460 − 6.5663I −13.47746 + 2.65332I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.167281
a = 2.88126
b = −0.481979
−0.738036 −13.3310
6
II. I
u
2
= h−au + b, a
2
+ au + a − u, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
4
=
−u
u + 1
a
8
=
a
au
a
9
=
a
au
a
11
=
−a + u + 2
−au − 1
a
7
=
au − a + u + 1
−au − 1
a
10
=
−au + a
au
a
6
=
−1
0
a
6
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = au + 2a + 5u − 11
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
2
c
3
, c
8
u
4
c
4
(u
2
− u + 1)
2
c
6
, c
7
, c
9
(u
2
+ u − 1)
2
c
10
, c
11
(u
2
− u − 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)
2
c
3
, c
8
y
4
c
6
, c
7
, c
9
c
10
, c
11
(y
2
− 3y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.309017 + 0.535233I
b = −0.618034
−0.98696 − 2.02988I −13.5000 + 5.4006I
u = −0.500000 + 0.866025I
a = −0.80902 − 1.40126I
b = 1.61803
−8.88264 − 2.02988I −13.50000 + 1.52761I
u = −0.500000 − 0.866025I
a = 0.309017 − 0.535233I
b = −0.618034
−0.98696 + 2.02988I −13.5000 − 5.4006I
u = −0.500000 − 0.866025I
a = −0.80902 + 1.40126I
b = 1.61803
−8.88264 + 2.02988I −13.50000 − 1.52761I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
2
)(u
11
+ 3u
10
+ ··· + 8u + 1)
c
2
((u
2
+ u + 1)
2
)(u
11
+ 9u
10
+ ··· + 34u − 1)
c
3
, c
8
u
4
(u
11
− u
10
+ ··· − 32u − 16)
c
4
((u
2
− u + 1)
2
)(u
11
+ 3u
10
+ ··· + 8u + 1)
c
5
((u
2
+ u + 1)
2
)(u
11
− 3u
10
+ ··· + 17u + 2)
c
6
, c
7
((u
2
+ u − 1)
2
)(u
11
+ 3u
10
+ ··· + 2u − 1)
c
9
((u
2
+ u − 1)
2
)(u
11
− 13u
10
+ ··· − 2u + 7)
c
10
, c
11
((u
2
− u − 1)
2
)(u
11
+ 3u
10
+ ··· + 2u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
2
)(y
11
+ 9y
10
+ ··· + 34y −1)
c
2
((y
2
+ y + 1)
2
)(y
11
− 11y
10
+ ··· + 1282y −1)
c
3
, c
8
y
4
(y
11
− 25y
10
+ ··· + 128y −256)
c
5
((y
2
+ y + 1)
2
)(y
11
− 31y
10
+ ··· + 109y −4)
c
6
, c
7
, c
10
c
11
((y
2
− 3y + 1)
2
)(y
11
− 19y
10
+ ··· + 14y −1)
c
9
((y
2
− 3y + 1)
2
)(y
11
− 79y
10
+ ··· − 38y −49)
12
