11n
19
(K11n
19
)
A knot diagram
1
Linearized knot diagam
5 1 9 2 3 10 11 1 3 7 8
Solving Sequence
1,8 3,9
4 2 5 11 7 10 6
c
8
c
3
c
2
c
4
c
11
c
7
c
10
c
6
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
5
+ u
4
− 3u
3
− 3u
2
+ 2b − 3u − 1, u
4
− 5u
2
+ 2a + 3, u
6
+ 3u
5
− 2u
4
− 11u
3
− 6u
2
− u − 1i
I
u
2
= hau + b + a, a
2
+ au − a − u + 2, u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 10 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
5
+ u
4
− 3u
3
− 3u
2
+ 2b − 3u − 1, u
4
− 5u
2
+ 2a + 3, u
6
+ 3u
5
−
2u
4
− 11u
3
− 6u
2
− u − 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
3
=
−
1
2
u
4
+
5
2
u
2
−
3
2
−
1
2
u
5
−
1
2
u
4
+ ··· +
3
2
u +
1
2
a
9
=
1
−u
2
a
4
=
−u
5
−
3
2
u
4
+ 4u
3
+
11
2
u
2
− u −
3
2
7
2
u
5
+
7
2
u
4
+ ··· +
1
2
u −
3
2
a
2
=
−
1
2
u
4
+
5
2
u
2
−
3
2
−2u
5
− 2u
4
+ 7u
3
+ 6u
2
+ 2u + 1
a
5
=
1
2
u
4
+ u
3
−
3
2
u
2
− 4u −
3
2
1
2
u
5
+
1
2
u
4
+ ··· +
1
2
u −
1
2
a
11
=
−u
u
a
7
=
−u
2
+ 1
u
2
a
10
=
u
3
− 2u
−u
3
+ u
a
6
=
u
4
− 3u
2
+ 1
−u
4
+ 2u
2
a
6
=
u
4
− 3u
2
+ 1
−u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
3
2
u
5
− 5u
4
+
7
2
u
3
+ 20u
2
+
15
2
u + 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
6
+ 3u
5
+ 4u
4
+ u
3
+ 3u + 1
c
2
u
6
− u
5
+ 10u
4
− 17u
3
+ 2u
2
− 9u + 1
c
3
, c
9
u
6
+ 6u
5
+ 28u
4
+ 60u
3
+ 20u
2
− 16u − 16
c
5
u
6
− 3u
5
+ 12u
4
+ 127u
3
+ 52u
2
+ 113u + 41
c
6
, c
7
, c
8
c
10
, c
11
u
6
− 3u
5
− 2u
4
+ 11u
3
− 6u
2
+ u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
6
− y
5
+ 10y
4
− 17y
3
+ 2y
2
− 9y + 1
c
2
y
6
+ 19y
5
+ 70y
4
− 265y
3
− 282y
2
− 77y + 1
c
3
, c
9
y
6
+ 20y
5
+ 104y
4
− 2320y
3
+ 1424y
2
− 896y + 256
c
5
y
6
+ 15y
5
+ 1010y
4
− 14121y
3
− 25014y
2
− 8505y + 1681
c
6
, c
7
, c
8
c
10
, c
11
y
6
− 13y
5
+ 58y
4
− 93y
3
+ 18y
2
+ 11y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.787648
a = −0.141467
b = −0.524726
1.36678 7.32050
u = 0.049860 + 0.377590I
a = −1.85932 + 0.09941I
b = 0.321306 + 0.548438I
0.088081 − 1.387970I 1.39961 + 3.44965I
u = 0.049860 − 0.377590I
a = −1.85932 − 0.09941I
b = 0.321306 − 0.548438I
0.088081 + 1.387970I 1.39961 − 3.44965I
u = 1.93055
a = 0.872199
b = −0.574576
11.1902 8.52400
u = −2.12131 + 0.18327I
a = −0.00604 + 1.52892I
b = 0.22835 − 2.85610I
−11.30140 − 4.76989I 7.67813 + 1.77109I
u = −2.12131 − 0.18327I
a = −0.00604 − 1.52892I
b = 0.22835 + 2.85610I
−11.30140 + 4.76989I 7.67813 − 1.77109I
5
II. I
u
2
= hau + b + a, a
2
+ au − a − u + 2, u
2
− u − 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
3
=
a
−au − a
a
9
=
1
−u − 1
a
4
=
a
−au − a
a
2
=
a
−2au − 2a
a
5
=
a + u − 1
−au − a − 2u
a
11
=
−u
u
a
7
=
−u
u + 1
a
10
=
1
−u − 1
a
6
=
0
−u
a
6
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3au + 2a + u + 4
6
(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
9
u
4
c
4
(u
2
− u + 1)
2
c
6
, c
7
, c
8
(u
2
− u − 1)
2
c
10
, c
11
(u
2
+ u − 1)
2
7
(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
9
y
4
c
6
, c
7
, c
8
c
10
, c
11
(y
2
− 3y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.80902 + 1.40126I
b = −0.309017 − 0.535233I
0.98696 − 2.02988I 6.50000 + 5.40059I
u = −0.618034
a = 0.80902 − 1.40126I
b = −0.309017 + 0.535233I
0.98696 + 2.02988I 6.50000 − 5.40059I
u = 1.61803
a = −0.309017 + 0.535233I
b = 0.80902 − 1.40126I
8.88264 + 2.02988I 6.50000 − 1.52761I
u = 1.61803
a = −0.309017 − 0.535233I
b = 0.80902 + 1.40126I
8.88264 − 2.02988I 6.50000 + 1.52761I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
+ u + 1)
2
(u
6
+ 3u
5
+ 4u
4
+ u
3
+ 3u + 1)
c
2
(u
2
+ u + 1)
2
(u
6
− u
5
+ 10u
4
− 17u
3
+ 2u
2
− 9u + 1)
c
3
, c
9
u
4
(u
6
+ 6u
5
+ 28u
4
+ 60u
3
+ 20u
2
− 16u − 16)
c
4
(u
2
− u + 1)
2
(u
6
+ 3u
5
+ 4u
4
+ u
3
+ 3u + 1)
c
5
(u
2
+ u + 1)
2
(u
6
− 3u
5
+ 12u
4
+ 127u
3
+ 52u
2
+ 113u + 41)
c
6
, c
7
, c
8
(u
2
− u − 1)
2
(u
6
− 3u
5
− 2u
4
+ 11u
3
− 6u
2
+ u − 1)
c
10
, c
11
(u
2
+ u − 1)
2
(u
6
− 3u
5
− 2u
4
+ 11u
3
− 6u
2
+ u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
2
+ y + 1)
2
(y
6
− y
5
+ 10y
4
− 17y
3
+ 2y
2
− 9y + 1)
c
2
(y
2
+ y + 1)
2
(y
6
+ 19y
5
+ 70y
4
− 265y
3
− 282y
2
− 77y + 1)
c
3
, c
9
y
4
(y
6
+ 20y
5
+ 104y
4
− 2320y
3
+ 1424y
2
− 896y + 256)
c
5
(y
2
+ y + 1)
2
· (y
6
+ 15y
5
+ 1010y
4
− 14121y
3
− 25014y
2
− 8505y + 1681)
c
6
, c
7
, c
8
c
10
, c
11
(y
2
− 3y + 1)
2
(y
6
− 13y
5
+ 58y
4
− 93y
3
+ 18y
2
+ 11y + 1)
11
