11n
48
(K11n
48
)
A knot diagram
1
Linearized knot diagam
6 1 7 9 2 10 11 1 6 4 8
Solving Sequence
1,6
2
3,9
10 7 5 4 8 11
c
1
c
2
c
9
c
6
c
5
c
4
c
8
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.14153 × 10
15
u
19
− 4.26909 × 10
15
u
18
+ ··· + 8.56391 × 10
15
b + 1.10053 × 10
16
,
6.37825 × 10
15
u
19
+ 2.69128 × 10
16
u
18
+ ··· + 8.56391 × 10
15
a + 3.19918 × 10
16
, u
20
+ 4u
19
+ ··· + 2u + 1i
I
u
2
= hb
2
− 2, a + u + 1, u
2
+ u + 1i
I
u
3
= hb, a − u − 1, u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 26 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−1.14×10
15
u
19
−4.27×10
15
u
18
+· · ·+8.56×10
15
b+1.10×10
16
, 6.38×
10
15
u
19
+2.69×10
16
u
18
+· · ·+8.56×10
15
a+3.20×10
16
, u
20
+4u
19
+· · ·+2u+1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
3
=
u
2
+ 1
−u
2
a
9
=
−0.744782u
19
− 3.14258u
18
+ ··· − 36.0811u − 3.73565
0.133295u
19
+ 0.498498u
18
+ ··· + 1.29523u − 1.28508
a
10
=
−0.744782u
19
− 3.14258u
18
+ ··· − 36.0811u − 3.73565
0.169440u
19
+ 0.644676u
18
+ ··· + 2.36692u − 1.12163
a
7
=
0.408694u
19
+ 1.81167u
18
+ ··· + 31.0989u + 4.13465
−0.233980u
19
− 0.931452u
18
+ ··· − 3.19070u + 0.665806
a
5
=
−u
u
3
+ u
a
4
=
−0.174714u
19
− 0.880217u
18
+ ··· − 27.9082u − 4.80045
0.258642u
19
+ 1.02897u
18
+ ··· + 6.32631u − 0.537037
a
8
=
−0.878077u
19
− 3.64108u
18
+ ··· − 37.3764u − 2.45056
0.133295u
19
+ 0.498498u
18
+ ··· + 1.29523u − 1.28508
a
11
=
1.12163u
19
+ 4.65596u
18
+ ··· + 50.9286u + 4.61018
−0.278931u
19
− 1.06850u
18
+ ··· − 2.73563u + 1.50344
a
11
=
1.12163u
19
+ 4.65596u
18
+ ··· + 50.9286u + 4.61018
−0.278931u
19
− 1.06850u
18
+ ··· − 2.73563u + 1.50344
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
1557563512851975
8563912733265916
u
19
−
8797950809282545
8563912733265916
u
18
+ ··· −
62067062106158523
8563912733265916
u −
1550791485072395
2140978183316479
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
20
− 4u
19
+ ··· − 2u + 1
c
2
u
20
+ 28u
19
+ ··· + 74u + 1
c
3
u
20
+ 8u
18
+ ··· − 16u + 41
c
4
u
20
− 2u
19
+ ··· + 2204u + 839
c
6
, c
9
u
20
+ 3u
19
+ ··· − 19u + 7
c
7
, c
8
, c
11
u
20
− u
19
+ ··· − 12u + 4
c
10
u
20
+ 2u
19
+ ··· − 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
20
+ 28y
19
+ ··· + 74y + 1
c
2
y
20
− 68y
19
+ ··· − 1654y + 1
c
3
y
20
+ 16y
19
+ ··· + 10814y + 1681
c
4
y
20
− 52y
19
+ ··· + 5336234y + 703921
c
6
, c
9
y
20
− 29y
19
+ ··· + 297y + 49
c
7
, c
8
, c
11
y
20
− 15y
19
+ ··· − 48y + 16
c
10
y
20
+ 4y
19
+ ··· + 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.729618 + 0.601963I
a = −0.002660 − 0.502436I
b = 1.343650 − 0.072587I
5.16928 − 2.57908I 8.86572 + 4.96809I
u = −0.729618 − 0.601963I
a = −0.002660 + 0.502436I
b = 1.343650 + 0.072587I
5.16928 + 2.57908I 8.86572 − 4.96809I
u = 0.337333 + 0.681420I
a = −1.15996 + 1.26131I
b = 0.515876 + 0.585061I
−3.04083 − 0.89466I −2.94359 − 0.44261I
u = 0.337333 − 0.681420I
a = −1.15996 − 1.26131I
b = 0.515876 − 0.585061I
−3.04083 + 0.89466I −2.94359 + 0.44261I
u = −0.342865 + 1.301010I
a = −0.567886 − 0.523820I
b = −0.778134 − 0.346045I
−1.166150 − 0.686038I 1.05440 − 1.40687I
u = −0.342865 − 1.301010I
a = −0.567886 + 0.523820I
b = −0.778134 + 0.346045I
−1.166150 + 0.686038I 1.05440 + 1.40687I
u = −0.007874 + 1.394550I
a = 0.585444 − 0.602506I
b = 1.027390 − 0.480106I
−1.53618 − 5.13397I 0.62229 + 5.85487I
u = −0.007874 − 1.394550I
a = 0.585444 + 0.602506I
b = 1.027390 + 0.480106I
−1.53618 + 5.13397I 0.62229 − 5.85487I
u = −1.10251 + 0.94498I
a = 0.757275 + 0.607433I
b = −0.965691 + 0.331710I
−0.59440 − 3.60439I 2.12405 + 4.48047I
u = −1.10251 − 0.94498I
a = 0.757275 − 0.607433I
b = −0.965691 − 0.331710I
−0.59440 + 3.60439I 2.12405 − 4.48047I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.195248 + 0.372946I
a = −0.774007 + 0.603593I
b = −0.277579 + 0.390975I
0.131319 − 1.058260I 2.00315 + 6.24655I
u = −0.195248 − 0.372946I
a = −0.774007 − 0.603593I
b = −0.277579 − 0.390975I
0.131319 + 1.058260I 2.00315 − 6.24655I
u = 0.36556 + 1.75156I
a = −0.101116 + 1.030470I
b = 1.39118 + 0.66555I
−11.13930 + 2.93869I −0.922503 − 0.819894I
u = 0.36556 − 1.75156I
a = −0.101116 − 1.030470I
b = 1.39118 − 0.66555I
−11.13930 − 2.93869I −0.922503 + 0.819894I
u = 0.030822 + 0.170366I
a = −0.29189 − 5.45429I
b = −1.370380 + 0.067940I
3.14870 − 0.11294I 1.81708 − 1.16761I
u = 0.030822 − 0.170366I
a = −0.29189 + 5.45429I
b = −1.370380 − 0.067940I
3.14870 + 0.11294I 1.81708 + 1.16761I
u = −0.36649 + 1.92873I
a = 0.003517 + 0.916631I
b = −1.46300 + 0.57660I
−10.3499 − 9.9960I 0.04826 + 5.02986I
u = −0.36649 − 1.92873I
a = 0.003517 − 0.916631I
b = −1.46300 − 0.57660I
−10.3499 + 9.9960I 0.04826 − 5.02986I
u = 0.01089 + 1.99331I
a = 0.051284 − 0.957095I
b = 0.076693 − 1.208720I
−15.1661 − 3.6593I −2.66886 + 2.23636I
u = 0.01089 − 1.99331I
a = 0.051284 + 0.957095I
b = 0.076693 + 1.208720I
−15.1661 + 3.6593I −2.66886 − 2.23636I
6
II. I
u
2
= hb
2
− 2, a + u + 1, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
9
=
−u − 1
b
a
10
=
−u − 1
b + u
a
7
=
u + 1
−b
a
5
=
−u
u + 1
a
4
=
b − u − 1
−bu + 3u + 1
a
8
=
−b − u − 1
b
a
11
=
−bu − b − 1
2
a
11
=
−bu − b − 1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
10
(u
2
+ u + 1)
2
c
3
u
4
− 2u
3
+ 5u
2
+ 2u + 1
c
4
u
4
+ 2u
3
+ 5u
2
− 2u + 1
c
5
(u
2
− u + 1)
2
c
6
(u + 1)
4
c
7
, c
8
, c
11
(u
2
− 2)
2
c
9
(u − 1)
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
10
(y
2
+ y + 1)
2
c
3
, c
4
y
4
+ 6y
3
+ 35y
2
+ 6y + 1
c
6
, c
9
(y −1)
4
c
7
, c
8
, c
11
(y −2)
4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.500000 − 0.866025I
b = 1.41421
3.28987 − 2.02988I 2.00000 + 3.46410I
u = −0.500000 + 0.866025I
a = −0.500000 − 0.866025I
b = −1.41421
3.28987 − 2.02988I 2.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = −0.500000 + 0.866025I
b = 1.41421
3.28987 + 2.02988I 2.00000 − 3.46410I
u = −0.500000 − 0.866025I
a = −0.500000 + 0.866025I
b = −1.41421
3.28987 + 2.02988I 2.00000 − 3.46410I
10
III. I
u
3
= hb, a − u − 1, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
9
=
u + 1
0
a
10
=
u + 1
−u
a
7
=
u + 1
0
a
5
=
−u
u + 1
a
4
=
−u − 1
u + 1
a
8
=
u + 1
0
a
11
=
1
0
a
11
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 2
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
u
2
+ u + 1
c
3
, c
4
, c
5
c
10
u
2
− u + 1
c
6
(u − 1)
2
c
7
, c
8
, c
11
u
2
c
9
(u + 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
10
y
2
+ y + 1
c
6
, c
9
(y −1)
2
c
7
, c
8
, c
11
y
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0
−1.64493 − 2.02988I 0. + 3.46410I
u = −0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = 0
−1.64493 + 2.02988I 0. − 3.46410I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
20
− 4u
19
+ ··· − 2u + 1)
c
2
((u
2
+ u + 1)
3
)(u
20
+ 28u
19
+ ··· + 74u + 1)
c
3
(u
2
− u + 1)(u
4
− 2u
3
+ ··· + 2u + 1)(u
20
+ 8u
18
+ ··· − 16u + 41)
c
4
(u
2
− u + 1)(u
4
+ 2u
3
+ ··· − 2u + 1)(u
20
− 2u
19
+ ··· + 2204u + 839)
c
5
((u
2
− u + 1)
3
)(u
20
− 4u
19
+ ··· − 2u + 1)
c
6
((u − 1)
2
)(u + 1)
4
(u
20
+ 3u
19
+ ··· − 19u + 7)
c
7
, c
8
, c
11
u
2
(u
2
− 2)
2
(u
20
− u
19
+ ··· − 12u + 4)
c
9
((u − 1)
4
)(u + 1)
2
(u
20
+ 3u
19
+ ··· − 19u + 7)
c
10
(u
2
− u + 1)(u
2
+ u + 1)
2
(u
20
+ 2u
19
+ ··· − 2u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
((y
2
+ y + 1)
3
)(y
20
+ 28y
19
+ ··· + 74y + 1)
c
2
((y
2
+ y + 1)
3
)(y
20
− 68y
19
+ ··· − 1654y + 1)
c
3
(y
2
+ y + 1)(y
4
+ 6y
3
+ 35y
2
+ 6y + 1)
· (y
20
+ 16y
19
+ ··· + 10814y + 1681)
c
4
(y
2
+ y + 1)(y
4
+ 6y
3
+ 35y
2
+ 6y + 1)
· (y
20
− 52y
19
+ ··· + 5336234y + 703921)
c
6
, c
9
((y −1)
6
)(y
20
− 29y
19
+ ··· + 297y + 49)
c
7
, c
8
, c
11
y
2
(y −2)
4
(y
20
− 15y
19
+ ··· − 48y + 16)
c
10
((y
2
+ y + 1)
3
)(y
20
+ 4y
19
+ ··· + 2y + 1)
16
