11n
28
(K11n
28
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 8 10 3 1 11 6 9
Solving Sequence
1,8 4,9
3 2 5 7 11 10 6
c
8
c
3
c
2
c
4
c
7
c
11
c
9
c
6
c
1
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−611u
13
− 377u
12
+ ··· + 3054b + 169, −3787u
13
− 7405u
12
+ ··· + 3054a − 21955,
u
14
+ 2u
13
+ ··· + 7u + 1i
I
u
2
= hb, −u
3
+ u
2
+ a − 3u + 2, u
4
− u
3
+ 3u
2
− 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 18 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−611u
13
− 377u
12
+ · · · + 3054b + 169, −3787u
13
− 7405u
12
+ · · · +
3054a − 21955, u
14
+ 2u
13
+ · · · + 7u + 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
1.24001u
13
+ 2.42469u
12
+ ··· + 26.9705u + 7.18893
0.200065u
13
+ 0.123445u
12
+ ··· + 1.85265u − 0.0553373
a
9
=
1
u
2
a
3
=
1.03995u
13
+ 2.30124u
12
+ ··· + 25.1179u + 7.24427
0.200065u
13
+ 0.123445u
12
+ ··· + 1.85265u − 0.0553373
a
2
=
1.03995u
13
+ 2.30124u
12
+ ··· + 25.1179u + 7.24427
0.399804u
13
+ 0.629666u
12
+ ··· + 4.44204u + 0.166012
a
5
=
0.599869u
13
+ 0.753111u
12
+ ··· + 5.29470u + 0.110675
−0.400458u
13
− 0.864113u
12
+ ··· − 4.96857u − 0.612639
a
7
=
0.612639u
13
+ 0.824820u
12
+ ··· + 2.56189u − 0.680092
−0.446627u
13
− 0.892600u
12
+ ··· − 4.08841u − 0.599869
a
11
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
+ 2u
2
a
6
=
1.00033u
13
+ 1.61722u
12
+ ··· + 10.2633u + 0.723314
−0.400458u
13
− 0.864113u
12
+ ··· − 4.96857u − 0.612639
a
6
=
1.00033u
13
+ 1.61722u
12
+ ··· + 10.2633u + 0.723314
−0.400458u
13
− 0.864113u
12
+ ··· − 4.96857u − 0.612639
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1595
509
u
13
+
2457
509
u
12
+ ··· +
11407
509
u −
1470
509
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
14
− 5u
13
+ ··· − 3u + 1
c
2
u
14
− u
13
+ ··· − 5u + 1
c
3
, c
7
u
14
+ u
13
+ ··· + 72u + 16
c
5
u
14
− 2u
13
+ ··· + 540u + 200
c
6
, c
10
u
14
− 2u
13
+ ··· − u + 1
c
8
, c
9
, c
11
u
14
+ 2u
13
+ ··· + 7u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
14
+ y
13
+ ··· + 5y + 1
c
2
y
14
+ 37y
13
+ ··· + 73y + 1
c
3
, c
7
y
14
− 27y
13
+ ··· − 832y + 256
c
5
y
14
+ 82y
13
+ ··· + 531600y + 40000
c
6
, c
10
y
14
+ 2y
13
+ ··· + 7y + 1
c
8
, c
9
, c
11
y
14
+ 22y
13
+ ··· + 7y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.571781 + 0.561972I
a = −0.501422 + 0.063559I
b = −0.607051 + 0.239027I
−0.35154 + 1.84409I −0.79789 − 4.83996I
u = −0.571781 − 0.561972I
a = −0.501422 − 0.063559I
b = −0.607051 − 0.239027I
−0.35154 − 1.84409I −0.79789 + 4.83996I
u = 0.191932 + 1.332820I
a = 0.818040 − 0.252861I
b = 1.48559 + 0.47442I
7.00688 + 0.55948I 2.27714 − 0.75874I
u = 0.191932 − 1.332820I
a = 0.818040 + 0.252861I
b = 1.48559 − 0.47442I
7.00688 − 0.55948I 2.27714 + 0.75874I
u = −0.004664 + 0.621250I
a = 0.877799 + 0.498919I
b = 0.331213 + 0.818885I
0.65784 + 1.53044I 1.45925 − 4.48215I
u = −0.004664 − 0.621250I
a = 0.877799 − 0.498919I
b = 0.331213 − 0.818885I
0.65784 − 1.53044I 1.45925 + 4.48215I
u = −0.38042 + 1.43966I
a = −0.689794 − 0.282260I
b = −1.401110 + 0.139917I
6.27413 + 5.41755I 1.11952 − 5.07443I
u = −0.38042 − 1.43966I
a = −0.689794 + 0.282260I
b = −1.401110 − 0.139917I
6.27413 − 5.41755I 1.11952 + 5.07443I
u = −0.206958 + 0.197769I
a = 2.96034 + 2.05739I
b = −0.386385 + 0.432449I
−1.89748 + 0.70166I −5.60702 + 2.76477I
u = −0.206958 − 0.197769I
a = 2.96034 − 2.05739I
b = −0.386385 − 0.432449I
−1.89748 − 0.70166I −5.60702 − 2.76477I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.07913 + 1.85059I
a = 0.707024 − 0.611419I
b = 2.31698 − 0.44656I
18.9281 − 1.0022I 1.174737 − 0.209171I
u = 0.07913 − 1.85059I
a = 0.707024 + 0.611419I
b = 2.31698 + 0.44656I
18.9281 + 1.0022I 1.174737 + 0.209171I
u = −0.10723 + 1.88534I
a = −0.671987 − 0.615446I
b = −2.23924 − 0.56690I
18.7301 + 8.0616I 0.87427 − 4.09385I
u = −0.10723 − 1.88534I
a = −0.671987 + 0.615446I
b = −2.23924 + 0.56690I
18.7301 − 8.0616I 0.87427 + 4.09385I
6
II. I
u
2
= hb, −u
3
+ u
2
+ a − 3u + 2, u
4
− u
3
+ 3u
2
− 2u + 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
u
3
− u
2
+ 3u − 2
0
a
9
=
1
u
2
a
3
=
u
3
− u
2
+ 3u − 2
0
a
2
=
u
3
− u
2
+ 3u − 2
u
a
5
=
0
−u
a
7
=
1
0
a
11
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
3
− u
2
+ 2u − 1
a
6
=
u
−u
a
6
=
u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
3
− 6u
2
+ 17u − 11
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
c
2
, c
4
(u + 1)
4
c
3
, c
7
u
4
c
5
, c
8
, c
9
u
4
− u
3
+ 3u
2
− 2u + 1
c
6
u
4
− u
3
+ u
2
+ 1
c
10
u
4
+ u
3
+ u
2
+ 1
c
11
u
4
+ u
3
+ 3u
2
+ 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
7
y
4
c
5
, c
8
, c
9
c
11
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
6
, c
10
y
4
+ y
3
+ 3y
2
+ 2y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.395123 + 0.506844I
a = −0.95668 + 1.22719I
b = 0
−1.85594 − 1.41510I −5.13523 + 6.85627I
u = 0.395123 − 0.506844I
a = −0.95668 − 1.22719I
b = 0
−1.85594 + 1.41510I −5.13523 − 6.85627I
u = 0.10488 + 1.55249I
a = −0.043315 + 0.641200I
b = 0
5.14581 − 3.16396I 0.63523 + 2.29471I
u = 0.10488 − 1.55249I
a = −0.043315 − 0.641200I
b = 0
5.14581 + 3.16396I 0.63523 − 2.29471I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
14
− 5u
13
+ ··· − 3u + 1)
c
2
((u + 1)
4
)(u
14
− u
13
+ ··· − 5u + 1)
c
3
, c
7
u
4
(u
14
+ u
13
+ ··· + 72u + 16)
c
4
((u + 1)
4
)(u
14
− 5u
13
+ ··· − 3u + 1)
c
5
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
14
− 2u
13
+ ··· + 540u + 200)
c
6
(u
4
− u
3
+ u
2
+ 1)(u
14
− 2u
13
+ ··· − u + 1)
c
8
, c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
14
+ 2u
13
+ ··· + 7u + 1)
c
10
(u
4
+ u
3
+ u
2
+ 1)(u
14
− 2u
13
+ ··· − u + 1)
c
11
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
14
+ 2u
13
+ ··· + 7u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
14
+ y
13
+ ··· + 5y + 1)
c
2
((y −1)
4
)(y
14
+ 37y
13
+ ··· + 73y + 1)
c
3
, c
7
y
4
(y
14
− 27y
13
+ ··· − 832y + 256)
c
5
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
14
+ 82y
13
+ ··· + 531600y + 40000)
c
6
, c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
14
+ 2y
13
+ ··· + 7y + 1)
c
8
, c
9
, c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
14
+ 22y
13
+ ··· + 7y + 1)
12
