11n
106
(K11n
106
)
A knot diagram
1
Linearized knot diagam
5 1 7 8 2 11 9 5 4 6 7
Solving Sequence
5,8 2,9
1 4 7 3 11 6 10
c
8
c
1
c
4
c
7
c
3
c
11
c
6
c
10
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
16
− 3u
15
+ ··· + 4b + 4,
u
16
− 5u
14
+ 2u
13
+ 11u
12
− 8u
11
− 8u
10
+ 14u
9
− 7u
8
− 6u
7
+ 18u
6
− 8u
5
− 8u
4
+ 12u
3
− 2u
2
+ 4a − 2u + 2,
u
17
+ 2u
16
− 3u
15
− 7u
14
+ 5u
13
+ 12u
12
− u
10
− u
9
− 13u
8
+ 18u
6
+ 10u
5
+ 4u
4
+ 4u
3
− 2u
2
− 4u − 2i
I
u
2
= hu
3
− u
2
+ b − u + 1, −u
3
+ 2a + 2u, u
4
− 2u
2
+ 2i
I
u
3
= h−a
2
+ b + a, a
3
− a − 1, u − 1i
I
v
1
= ha, b + 1, v −1i
* 4 irreducible components of dim
C
= 0, with total 25 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−3u
16
−3u
15
+· · ·+4b+4, u
16
−5u
14
+· · ·+4a+2, u
17
+2u
16
+· · ·−4u−2i
(i) Arc colorings
a
5
=
0
u
a
8
=
1
0
a
2
=
−
1
4
u
16
+
5
4
u
14
+ ··· +
1
2
u −
1
2
3
4
u
16
+
3
4
u
15
+ ··· −
1
2
u − 1
a
9
=
1
−u
2
a
1
=
−
1
4
u
16
+
5
4
u
14
+ ··· +
1
2
u −
1
2
5
4
u
16
+
3
2
u
15
+ ··· − 2u − 2
a
4
=
−u
u
a
7
=
−u
2
+ 1
u
4
a
3
=
−u
7
+ 2u
5
− 2u
3
u
9
− u
7
+ u
5
+ u
a
11
=
−
1
2
u
10
+
3
2
u
8
+ ··· + u
2
− 1
1
4
u
15
− u
13
+ ··· +
1
2
u
2
+
1
2
u
a
6
=
−
1
4
u
15
+ u
13
+ ··· −
1
2
u + 1
1
4
u
15
− u
13
+ ··· +
1
2
u
2
+
1
2
u
a
10
=
−u
4
+ u
2
− 1
u
4
a
10
=
−u
4
+ u
2
− 1
u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
16
+ 10u
14
− 2u
13
− 22u
12
+ 8u
11
+ 16u
10
− 14u
9
+ 14u
8
+
10u
7
− 36u
6
+ 16u
4
− 6u
3
+ 4u
2
+ 10u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
17
+ 2u
16
+ ··· + 11u − 5
c
2
u
17
− 2u
16
+ ··· + 121u + 25
c
3
u
17
+ 4u
16
+ ··· − 7540u − 3866
c
4
, c
8
u
17
+ 2u
16
+ ··· − 4u − 2
c
6
, c
10
, c
11
u
17
− 2u
16
+ ··· − 17u − 5
c
7
u
17
− 10u
16
+ ··· + 8u − 4
c
9
u
17
− 3u
16
+ ··· + 32u − 46
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
17
+ 2y
16
+ ··· + 121y −25
c
2
y
17
+ 50y
16
+ ··· + 40441y −625
c
3
y
17
+ 70y
16
+ ··· + 33732920y −14945956
c
4
, c
8
y
17
− 10y
16
+ ··· + 8y −4
c
6
, c
10
, c
11
y
17
− 30y
16
+ ··· + 329y −25
c
7
y
17
− 6y
16
+ ··· − 96y −16
c
9
y
17
+ 31y
16
+ ··· + 14640y −2116
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.076903 + 1.006450I
a = −1.28833 + 1.08624I
b = −0.344784 − 0.561209I
13.5757 − 4.4662I 2.93957 + 1.91782I
u = 0.076903 − 1.006450I
a = −1.28833 − 1.08624I
b = −0.344784 + 0.561209I
13.5757 + 4.4662I 2.93957 − 1.91782I
u = −0.995392 + 0.405067I
a = 0.702879 + 0.603320I
b = −0.30791 − 1.68630I
0.05634 − 3.87007I −0.55814 + 7.00568I
u = −0.995392 − 0.405067I
a = 0.702879 − 0.603320I
b = −0.30791 + 1.68630I
0.05634 + 3.87007I −0.55814 − 7.00568I
u = 0.194679 + 0.752552I
a = 0.913683 + 0.406237I
b = 0.274898 + 0.378400I
2.46422 + 0.66350I 3.92785 − 1.28554I
u = 0.194679 − 0.752552I
a = 0.913683 − 0.406237I
b = 0.274898 − 0.378400I
2.46422 − 0.66350I 3.92785 + 1.28554I
u = 1.122330 + 0.557673I
a = 0.129300 + 0.669933I
b = −0.93908 − 1.59767I
5.04445 + 4.17066I 6.60682 − 3.70952I
u = 1.122330 − 0.557673I
a = 0.129300 − 0.669933I
b = −0.93908 + 1.59767I
5.04445 − 4.17066I 6.60682 + 3.70952I
u = 0.710570
a = −0.516138
b = 1.01848
1.26530 7.99450
u = −1.260410 + 0.354016I
a = −0.013562 − 0.903193I
b = 0.46322 + 2.04947I
6.78062 − 4.50780I 6.98768 + 3.92800I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.260410 − 0.354016I
a = −0.013562 + 0.903193I
b = 0.46322 − 2.04947I
6.78062 + 4.50780I 6.98768 − 3.92800I
u = −0.440513 + 0.412959I
a = −1.16799 − 0.87598I
b = −0.426652 + 0.425827I
−1.48592 + 0.26904I −6.62187 − 0.62877I
u = −0.440513 − 0.412959I
a = −1.16799 + 0.87598I
b = −0.426652 − 0.425827I
−1.48592 − 0.26904I −6.62187 + 0.62877I
u = 1.299750 + 0.543064I
a = 0.547862 − 1.121000I
b = −0.48688 + 2.87606I
17.3459 + 9.9963I 5.48062 − 4.80381I
u = 1.299750 − 0.543064I
a = 0.547862 + 1.121000I
b = −0.48688 − 2.87606I
17.3459 − 9.9963I 5.48062 + 4.80381I
u = −1.35263 + 0.45076I
a = −1.065780 + 0.664446I
b = 1.25795 − 1.54860I
18.0935 − 0.6930I 6.24024 + 0.75440I
u = −1.35263 − 0.45076I
a = −1.065780 − 0.664446I
b = 1.25795 + 1.54860I
18.0935 + 0.6930I 6.24024 − 0.75440I
6
II. I
u
2
= hu
3
− u
2
+ b − u + 1, −u
3
+ 2a + 2u, u
4
− 2u
2
+ 2i
(i) Arc colorings
a
5
=
0
u
a
8
=
1
0
a
2
=
1
2
u
3
− u
−u
3
+ u
2
+ u − 1
a
9
=
1
−u
2
a
1
=
1
2
u
3
− u
−u
3
+ u
2
+ 2u − 1
a
4
=
−u
u
a
7
=
−u
2
+ 1
2u
2
− 2
a
3
=
0
−u
a
11
=
1
2
u
3
+ u
2
− u − 1
−u
3
− u
2
+ 2u + 1
a
6
=
1
2
u
3
− u
−u
3
+ u
2
+ 2u − 1
a
10
=
u
2
− 1
−2u
2
+ 2
a
10
=
u
2
− 1
−2u
2
+ 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
+ 8
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
10
c
11
(u + 1)
4
c
3
, c
9
u
4
+ 2u
2
+ 2
c
4
, c
8
u
4
− 2u
2
+ 2
c
5
, c
6
(u − 1)
4
c
7
(u
2
+ 2u + 2)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
10
, c
11
(y −1)
4
c
3
, c
9
(y
2
+ 2y + 2)
2
c
4
, c
8
(y
2
− 2y + 2)
2
c
7
(y
2
+ 4)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.098680 + 0.455090I
a = −0.776887 + 0.321797I
b = 0.455090 − 0.098684I
2.46740 + 3.66386I 4.00000 − 4.00000I
u = 1.098680 − 0.455090I
a = −0.776887 − 0.321797I
b = 0.455090 + 0.098684I
2.46740 − 3.66386I 4.00000 + 4.00000I
u = −1.098680 + 0.455090I
a = 0.776887 + 0.321797I
b = −0.45509 − 2.09868I
2.46740 − 3.66386I 4.00000 + 4.00000I
u = −1.098680 − 0.455090I
a = 0.776887 − 0.321797I
b = −0.45509 + 2.09868I
2.46740 + 3.66386I 4.00000 − 4.00000I
10
III. I
u
3
= h−a
2
+ b + a, a
3
− a − 1, u − 1i
(i) Arc colorings
a
5
=
0
1
a
8
=
1
0
a
2
=
a
a
2
− a
a
9
=
1
−1
a
1
=
a
a
2
− 2a
a
4
=
−1
1
a
7
=
0
1
a
3
=
−1
2
a
11
=
a
a
2
− a
a
6
=
−a
2
a
2
− a
a
10
=
−1
1
a
10
=
−1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
10
, c
11
u
3
− u + 1
c
2
u
3
+ 2u
2
+ u + 1
c
3
, c
4
, c
7
c
8
(u − 1)
3
c
9
u
3
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
10
, c
11
y
3
− 2y
2
+ y −1
c
2
y
3
− 2y
2
− 3y −1
c
3
, c
4
, c
7
c
8
(y −1)
3
c
9
y
3
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.662359 + 0.562280I
b = 0.78492 − 1.30714I
1.64493 6.00000
u = 1.00000
a = −0.662359 − 0.562280I
b = 0.78492 + 1.30714I
1.64493 6.00000
u = 1.00000
a = 1.32472
b = 0.430160
1.64493 6.00000
14
IV. I
v
1
= ha, b + 1, v − 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
1
0
a
2
=
0
−1
a
9
=
1
0
a
1
=
−1
−1
a
4
=
1
0
a
7
=
1
0
a
3
=
1
0
a
11
=
0
−1
a
6
=
1
1
a
10
=
1
0
a
10
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
, c
11
u − 1
c
2
, c
5
, c
6
u + 1
c
3
, c
4
, c
7
c
8
, c
9
u
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
10
, c
11
y −1
c
3
, c
4
, c
7
c
8
, c
9
y
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
0 0
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u + 1)
4
(u
3
− u + 1)(u
17
+ 2u
16
+ ··· + 11u − 5)
c
2
((u + 1)
5
)(u
3
+ 2u
2
+ u + 1)(u
17
− 2u
16
+ ··· + 121u + 25)
c
3
u(u − 1)
3
(u
4
+ 2u
2
+ 2)(u
17
+ 4u
16
+ ··· − 7540u − 3866)
c
4
, c
8
u(u − 1)
3
(u
4
− 2u
2
+ 2)(u
17
+ 2u
16
+ ··· − 4u − 2)
c
5
((u − 1)
4
)(u + 1)(u
3
− u + 1)(u
17
+ 2u
16
+ ··· + 11u − 5)
c
6
((u − 1)
4
)(u + 1)(u
3
− u + 1)(u
17
− 2u
16
+ ··· − 17u − 5)
c
7
u(u − 1)
3
(u
2
+ 2u + 2)
2
(u
17
− 10u
16
+ ··· + 8u − 4)
c
9
u
4
(u
4
+ 2u
2
+ 2)(u
17
− 3u
16
+ ··· + 32u − 46)
c
10
, c
11
(u − 1)(u + 1)
4
(u
3
− u + 1)(u
17
− 2u
16
+ ··· − 17u − 5)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
((y −1)
5
)(y
3
− 2y
2
+ y −1)(y
17
+ 2y
16
+ ··· + 121y −25)
c
2
((y −1)
5
)(y
3
− 2y
2
− 3y −1)(y
17
+ 50y
16
+ ··· + 40441y −625)
c
3
y(y −1)
3
(y
2
+ 2y + 2)
2
· (y
17
+ 70y
16
+ ··· + 33732920y −14945956)
c
4
, c
8
y(y −1)
3
(y
2
− 2y + 2)
2
(y
17
− 10y
16
+ ··· + 8y −4)
c
6
, c
10
, c
11
((y −1)
5
)(y
3
− 2y
2
+ y −1)(y
17
− 30y
16
+ ··· + 329y −25)
c
7
y(y −1)
3
(y
2
+ 4)
2
(y
17
− 6y
16
+ ··· − 96y −16)
c
9
y
4
(y
2
+ 2y + 2)
2
(y
17
+ 31y
16
+ ··· + 14640y −2116)
20
