11n
45
(K11n
45
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 11 10 3 6 1 8 9
Solving Sequence
1,4
2
5,9
10 11 6 7 8 3
c
1
c
4
c
9
c
11
c
5
c
6
c
8
c
3
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, a − u − 2, u
12
+ 5u
11
+ 9u
10
− 21u
8
− 22u
7
+ 10u
6
+ 26u
5
+ 4u
4
− 11u
3
− 3u
2
+ 2u + 1i
I
u
2
= hb + 1, u
5
+ u
4
− u
3
− 2u
2
+ a + 1, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
I
u
3
= hb
6
− b
5
− b
4
+ 2b
3
− b + 1, a − 1, u − 1i
I
u
4
= h−u
11
− 2u
10
− 6u
9
− u
8
− 7u
7
+ 15u
6
− 14u
5
+ 28u
4
− 50u
3
+ 41u
2
+ 32b − 66u + 31,
u
11
+ 3u
10
+ 8u
9
+ 7u
8
+ 8u
7
− 8u
6
− u
5
− 14u
4
+ 22u
3
+ 9u
2
+ a + 25u + 3,
u
12
+ 3u
11
+ 8u
10
+ 7u
9
+ 8u
8
− 8u
7
− u
6
− 14u
5
+ 22u
4
+ 9u
3
+ 25u
2
+ 3u + 1i
* 4 irreducible components of dim
C
= 0, with total 36 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
= hb + u, a − u − 2, u
12
+ 5u
11
+ · · · + 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
9
=
u + 2
−u
a
10
=
2u + 2
−u
a
11
=
u
2
+ 2u + 1
−u
2
a
6
=
u
7
+ 4u
6
+ 6u
5
+ 2u
4
− 4u
3
− 4u
2
− 2u
−u
7
− 2u
6
− u
5
+ 2u
4
+ u
a
7
=
−2u
9
− 6u
8
− 3u
7
+ 12u
6
+ 14u
5
− 6u
4
− 12u
3
+ 2u
u
9
+ 2u
8
− u
7
− 6u
6
− u
5
+ 6u
4
− 2u
2
+ u
a
8
=
u
10
+ 4u
9
+ 5u
8
− 4u
7
− 14u
6
− 6u
5
+ 11u
4
+ 8u
3
− 3u
2
− 2u + 1
−u
11
− 5u
10
+ ··· − 2u − 1
a
3
=
−u
2
+ 1
u
2
a
3
=
−u
2
+ 1
u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
11
− 16u
10
− 24u
9
+ 24u
7
− 32u
5
+ 16u
4
+ 36u
3
− 16u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
9
c
11
u
12
− 5u
11
+ ··· − 2u + 1
c
2
u
12
+ 7u
11
+ ··· + 10u + 1
c
3
, c
7
, c
10
u
12
+ u
11
+ ··· + 2u + 1
c
5
u
12
− 3u
11
+ ··· − 14u + 4
c
6
u
12
− u
11
+ ··· + 44u + 23
c
8
u
12
+ u
11
+ u
10
+ 5u
8
− 4u
6
− 8u
5
+ 6u
4
+ 3u
3
+ 3u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
9
c
11
y
12
− 7y
11
+ ··· − 10y + 1
c
2
y
12
+ 29y
11
+ ··· + 22y + 1
c
3
, c
7
, c
10
y
12
− 15y
11
+ ··· − 2y + 1
c
5
y
12
+ 5y
11
+ ··· + 68y + 16
c
6
y
12
− 23y
11
+ ··· − 4098y + 529
c
8
y
12
+ y
11
+ ··· + 6y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.017000 + 0.101771I
a = 3.01700 + 0.10177I
b = −1.017000 − 0.101771I
−3.52730 − 0.57280I −2.7091 − 26.6989I
u = 1.017000 − 0.101771I
a = 3.01700 − 0.10177I
b = −1.017000 + 0.101771I
−3.52730 + 0.57280I −2.7091 + 26.6989I
u = −0.997809 + 0.382742I
a = 1.002190 + 0.382742I
b = 0.997809 − 0.382742I
−1.70690 + 6.65526I −0.69156 − 12.28500I
u = −0.997809 − 0.382742I
a = 1.002190 − 0.382742I
b = 0.997809 + 0.382742I
−1.70690 − 6.65526I −0.69156 + 12.28500I
u = 0.568808 + 0.252332I
a = 2.56881 + 0.25233I
b = −0.568808 − 0.252332I
−1.61529 − 1.35793I −3.64822 + 4.51645I
u = 0.568808 − 0.252332I
a = 2.56881 − 0.25233I
b = −0.568808 + 0.252332I
−1.61529 + 1.35793I −3.64822 − 4.51645I
u = −0.417930 + 0.278210I
a = 1.58207 + 0.27821I
b = 0.417930 − 0.278210I
1.46216 − 0.16286I 7.96188 − 1.03516I
u = −0.417930 − 0.278210I
a = 1.58207 − 0.27821I
b = 0.417930 + 0.278210I
1.46216 + 0.16286I 7.96188 + 1.03516I
u = −1.29679 + 1.06566I
a = 0.703205 + 1.065660I
b = 1.29679 − 1.06566I
12.72390 + 5.46645I −0.22295 − 2.11548I
u = −1.29679 − 1.06566I
a = 0.703205 − 1.065660I
b = 1.29679 + 1.06566I
12.72390 − 5.46645I −0.22295 + 2.11548I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.37328 + 1.07803I
a = 0.626724 + 1.078030I
b = 1.37328 − 1.07803I
12.4026 + 12.7511I −0.69002 − 5.94531I
u = −1.37328 − 1.07803I
a = 0.626724 − 1.078030I
b = 1.37328 + 1.07803I
12.4026 − 12.7511I −0.69002 + 5.94531I
6
II. I
u
2
= hb + 1, u
5
+ u
4
− u
3
− 2u
2
+ a + 1, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
9
=
−u
5
− u
4
+ u
3
+ 2u
2
− 1
−1
a
10
=
−u
5
− u
4
+ u
3
+ 2u
2
−1
a
11
=
−u
5
− u
4
+ u
3
+ 2u
2
−1
a
6
=
u
5
+ u
4
−2u
3
− u
2
+ u + 1
a
7
=
−u
−u
3
+ u
a
8
=
−u
4
+ u
2
− 1
u
5
+ u
4
− 2u
3
− u
2
+ u + 1
a
3
=
−u
2
+ 1
u
2
a
3
=
−u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
5
+ 7u
4
+ u
3
− 6u
2
− 5u − 1
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
2
u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1
c
3
, c
4
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
5
, c
6
u
6
+ u
5
+ 2u
4
+ 4u
3
+ 5u
2
+ 3u + 1
c
8
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
c
9
(u − 1)
6
c
10
u
6
c
11
(u + 1)
6
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
c
2
, c
8
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
c
5
, c
6
y
6
+ 3y
5
+ 6y
4
+ 5y
2
+ y + 1
c
9
, c
11
(y −1)
6
c
10
y
6
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002190 + 0.295542I
a = 0.917982 − 0.270708I
b = −1.00000
−3.53554 − 0.92430I −6.82874 + 7.13914I
u = 1.002190 − 0.295542I
a = 0.917982 + 0.270708I
b = −1.00000
−3.53554 + 0.92430I −6.82874 − 7.13914I
u = −0.428243 + 0.664531I
a = −0.685196 − 1.063260I
b = −1.00000
0.245672 − 0.924305I 1.12292 + 1.33143I
u = −0.428243 − 0.664531I
a = −0.685196 + 1.063260I
b = −1.00000
0.245672 + 0.924305I 1.12292 − 1.33143I
u = −1.073950 + 0.558752I
a = −0.732786 − 0.381252I
b = −1.00000
−1.64493 + 5.69302I −0.29418 − 2.69056I
u = −1.073950 − 0.558752I
a = −0.732786 + 0.381252I
b = −1.00000
−1.64493 − 5.69302I −0.29418 + 2.69056I
10
III. I
u
3
= hb
6
− b
5
− b
4
+ 2b
3
− b + 1, a − 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
9
=
1
b
a
10
=
−b + 1
b
a
11
=
−b + 1
−b
2
a
6
=
−b
3
+ b
2
− 1
−b
4
a
7
=
0
b
5
− b
4
− 2b
3
+ b
2
+ b − 1
a
8
=
0
b
5
− b
4
− 2b
3
+ b
2
+ b − 1
a
3
=
0
1
a
3
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3b
5
+ 7b
4
− b
3
− 6b
2
+ 5b − 1
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
6
c
2
, c
4
(u + 1)
6
c
3
, c
7
u
6
c
5
, c
8
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
c
6
, c
11
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
9
, c
10
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
7
y
6
c
5
, c
8
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
c
6
, c
9
, c
10
c
11
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = −1.002190 + 0.295542I
−3.53554 + 0.92430I −6.82874 − 7.13914I
u = 1.00000
a = 1.00000
b = −1.002190 − 0.295542I
−3.53554 − 0.92430I −6.82874 + 7.13914I
u = 1.00000
a = 1.00000
b = 0.428243 + 0.664531I
0.245672 + 0.924305I 1.12292 − 1.33143I
u = 1.00000
a = 1.00000
b = 0.428243 − 0.664531I
0.245672 − 0.924305I 1.12292 + 1.33143I
u = 1.00000
a = 1.00000
b = 1.073950 + 0.558752I
−1.64493 − 5.69302I −0.29418 + 2.69056I
u = 1.00000
a = 1.00000
b = 1.073950 − 0.558752I
−1.64493 + 5.69302I −0.29418 − 2.69056I
14
IV.
I
u
4
= h−u
11
−2u
10
+· · ·+32b+31, u
11
+3u
10
+· · ·+a+3, u
12
+3u
11
+· · ·+3u+1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
9
=
−u
11
− 3u
10
+ ··· − 25u − 3
0.0312500u
11
+ 0.0625000u
10
+ ··· + 2.06250u − 0.968750
a
10
=
−1.03125u
11
− 3.06250u
10
+ ··· − 27.0625u − 2.03125
0.0312500u
11
+ 0.0625000u
10
+ ··· + 2.06250u − 0.968750
a
11
=
−0.968750u
11
− 2.93750u
10
+ ··· − 22.9375u − 3.96875
7
32
u
11
+
1
2
u
10
+ ··· +
9
2
u −
23
32
a
6
=
0.531250u
11
+ 1.75000u
10
+ ··· + 13.7500u + 5.09375
−0.156250u
11
− 0.562500u
10
+ ··· − 5.31250u − 0.0312500
a
7
=
1
4
u
10
+
3
2
u
9
+ ··· +
35
4
u + 2
0.218750u
11
+ 0.437500u
10
+ ··· − 1.81250u + 0.218750
a
8
=
−
1
4
u
11
−
3
4
u
10
+ ··· −
29
4
u −
9
4
0.0312500u
11
+ 0.0625000u
10
+ ··· + 1.81250u − 0.218750
a
3
=
−u
2
+ 1
u
2
a
3
=
−u
2
+ 1
u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
3
16
u
11
−
11
16
u
10
−
33
16
u
9
−3u
8
−
55
16
u
7
+
1
2
u
6
+
11
4
u
5
+
21
4
u
4
−
21
8
u
3
−
83
16
u
2
−
183
16
u −
23
8
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
9
c
11
u
12
− 3u
11
+ ··· − 3u + 1
c
2
u
12
− 7u
11
+ ··· − 41u + 1
c
3
, c
7
, c
10
u
12
+ u
11
+ ··· + 320u + 64
c
5
u
12
− 2u
11
+ ··· + 144u + 121
c
6
u
12
− 14u
10
+ ··· + 120u + 77
c
8
(u
6
+ u
5
+ u
4
+ 2u
2
+ u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
9
c
11
y
12
+ 7y
11
+ ··· + 41y + 1
c
2
y
12
+ 27y
11
+ ··· − 451y + 1
c
3
, c
7
, c
10
y
12
− 27y
11
+ ··· − 12288y + 4096
c
5
y
12
+ 24y
11
+ ··· + 28148y + 14641
c
6
y
12
− 28y
11
+ ··· + 53360y + 5929
c
8
(y
6
+ y
5
+ 5y
4
+ 4y
3
+ 6y
2
+ 3y + 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.282006 + 0.991713I
a = −0.265287 − 0.932918I
b = 0.042043 + 1.323160I
2.99789 + 2.65597I 1.54637 − 3.55162I
u = −0.282006 − 0.991713I
a = −0.265287 + 0.932918I
b = 0.042043 − 1.323160I
2.99789 − 2.65597I 1.54637 + 3.55162I
u = 1.032840 + 0.430283I
a = 0.825019 − 0.343706I
b = 0.058341 + 0.199318I
−1.90302 − 1.10871I −2.03402 + 2.13465I
u = 1.032840 − 0.430283I
a = 0.825019 + 0.343706I
b = 0.058341 − 0.199318I
−1.90302 + 1.10871I −2.03402 − 2.13465I
u = −0.042043 + 1.323160I
a = −0.023990 − 0.755006I
b = 0.282006 + 0.991713I
2.99789 − 2.65597I 1.54637 + 3.55162I
u = −0.042043 − 1.323160I
a = −0.023990 + 0.755006I
b = 0.282006 − 0.991713I
2.99789 + 2.65597I 1.54637 − 3.55162I
u = −1.07187 + 1.35065I
a = −0.360515 − 0.454280I
b = 1.07857 + 1.47659I
13.70950 + 3.42721I 0.48765 − 2.36550I
u = −1.07187 − 1.35065I
a = −0.360515 + 0.454280I
b = 1.07857 − 1.47659I
13.70950 − 3.42721I 0.48765 + 2.36550I
u = −0.058341 + 0.199318I
a = −1.35265 − 4.62119I
b = −1.032840 + 0.430283I
−1.90302 + 1.10871I −2.03402 − 2.13465I
u = −0.058341 − 0.199318I
a = −1.35265 + 4.62119I
b = −1.032840 − 0.430283I
−1.90302 − 1.10871I −2.03402 + 2.13465I
18
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.07857 + 1.47659I
a = −0.322576 − 0.441612I
b = 1.07187 + 1.35065I
13.70950 − 3.42721I 0.48765 + 2.36550I
u = −1.07857 − 1.47659I
a = −0.322576 + 0.441612I
b = 1.07187 − 1.35065I
13.70950 + 3.42721I 0.48765 − 2.36550I
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
9
((u − 1)
6
)(u
6
+ u
5
+ ··· + u + 1)(u
12
− 5u
11
+ ··· − 2u + 1)
· (u
12
− 3u
11
+ ··· − 3u + 1)
c
2
(u + 1)
6
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
· (u
12
− 7u
11
+ ··· − 41u + 1)(u
12
+ 7u
11
+ ··· + 10u + 1)
c
3
u
6
(u
6
− u
5
+ ··· − u + 1)(u
12
+ u
11
+ ··· + 320u + 64)
· (u
12
+ u
11
+ ··· + 2u + 1)
c
4
, c
11
((u + 1)
6
)(u
6
− u
5
+ ··· − u + 1)(u
12
− 5u
11
+ ··· − 2u + 1)
· (u
12
− 3u
11
+ ··· − 3u + 1)
c
5
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 4u
3
+ 5u
2
+ 3u + 1)
· (u
12
− 3u
11
+ ··· − 14u + 4)(u
12
− 2u
11
+ ··· + 144u + 121)
c
6
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
6
+ u
5
+ 2u
4
+ 4u
3
+ 5u
2
+ 3u + 1)
· (u
12
− 14u
10
+ ··· + 120u + 77)(u
12
− u
11
+ ··· + 44u + 23)
c
7
, c
10
u
6
(u
6
+ u
5
+ ··· + u + 1)(u
12
+ u
11
+ ··· + 320u + 64)
· (u
12
+ u
11
+ ··· + 2u + 1)
c
8
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
(u
6
+ u
5
+ u
4
+ 2u
2
+ u + 1)
2
· (u
12
+ u
11
+ u
10
+ 5u
8
− 4u
6
− 8u
5
+ 6u
4
+ 3u
3
+ 3u
2
+ 1)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
9
c
11
(y −1)
6
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
· (y
12
− 7y
11
+ ··· − 10y + 1)(y
12
+ 7y
11
+ ··· + 41y + 1)
c
2
((y −1)
6
)(y
6
+ y
5
+ ··· + 3y + 1)(y
12
+ 27y
11
+ ··· − 451y + 1)
· (y
12
+ 29y
11
+ ··· + 22y + 1)
c
3
, c
7
, c
10
y
6
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
· (y
12
− 27y
11
+ ··· − 12288y + 4096)(y
12
− 15y
11
+ ··· − 2y + 1)
c
5
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
6
+ 3y
5
+ 6y
4
+ 5y
2
+ y + 1)
· (y
12
+ 5y
11
+ ··· + 68y + 16)(y
12
+ 24y
11
+ ··· + 28148y + 14641)
c
6
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
6
+ 3y
5
+ 6y
4
+ 5y
2
+ y + 1)
· (y
12
− 28y
11
+ ··· + 53360y + 5929)
· (y
12
− 23y
11
+ ··· − 4098y + 529)
c
8
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
(y
6
+ y
5
+ 5y
4
+ 4y
3
+ 6y
2
+ 3y + 1)
2
· (y
12
+ y
11
+ ··· + 6y + 1)
21
