9
28
(K9a
5
)
A knot diagram
1
Linearized knot diagam
9 8 2 7 4 1 5 3 6
Solving Sequence
3,8
9 2 4
1,6
5 7
c
8
c
2
c
3
c
1
c
5
c
7
c
4
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
3
+ b − u + 1, −2u
2
+ a − u + 1, u
4
+ u
3
− u
2
− u + 1i
I
u
2
= h−2u
15
+ 8u
13
+ 3u
12
− 14u
11
− 10u
10
+ 8u
9
+ 14u
8
+ 6u
7
− 6u
6
− 11u
5
− 3u
4
+ 3u
3
+ 2u
2
+ b + u + 2,
− 2u
15
+ 8u
13
+ 4u
12
− 14u
11
− 13u
10
+ 6u
9
+ 17u
8
+ 10u
7
− 4u
6
− 13u
5
− 7u
4
+ 3u
2
+ a + 3u + 3,
u
16
+ u
15
− 4u
14
− 6u
13
+ 5u
12
+ 13u
11
+ 3u
10
− 11u
9
− 12u
8
− 2u
7
+ 8u
6
+ 8u
5
+ 2u
4
− 2u
3
− 2u
2
− 2u − 1i
I
u
3
= hu
5
− u
3
+ b + u, u
2
+ a, u
6
− u
4
− u
3
+ u
2
+ u + 1i
I
u
4
= hb + 1, a + 1, u − 1i
* 4 irreducible components of dim
C
= 0, with total 27 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
3
+ b − u + 1, −2u
2
+ a − u + 1, u
4
+ u
3
− u
2
− u + 1i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
2
=
u
u
a
4
=
−u
3
−u
3
+ u
a
1
=
u
3
u
3
− u + 1
a
6
=
2u
2
+ u − 1
−u
3
+ u − 1
a
5
=
2u
3
+ 2u
2
− u
−u
a
7
=
u
2
+ 2u − 1
−u
2
a
7
=
u
2
+ 2u − 1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8u
2
+ 4u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
+ 2u
2
+ 3u + 1
c
2
, c
4
, c
7
c
8
u
4
− u
3
− u
2
+ u + 1
c
3
, c
5
u
4
+ 3u
3
+ 5u
2
+ 3u + 1
c
6
, c
9
u
4
− 2u
3
+ 2u
2
− u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
4
+ 4y
3
+ 6y
2
− 5y + 1
c
2
, c
4
, c
7
c
8
y
4
− 3y
3
+ 5y
2
− 3y + 1
c
3
, c
5
y
4
+ y
3
+ 9y
2
+ y + 1
c
6
, c
9
y
4
+ 2y
2
+ 3y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.692440 + 0.318148I
a = 0.448952 + 1.199340I
b = −0.429304 − 0.107280I
−1.07760 − 1.41376I −4.20419 + 4.79737I
u = 0.692440 − 0.318148I
a = 0.448952 − 1.199340I
b = −0.429304 + 0.107280I
−1.07760 + 1.41376I −4.20419 − 4.79737I
u = −1.192440 + 0.547877I
a = 0.05105 − 2.06537I
b = −1.57070 − 1.62477I
−3.85720 + 11.56320I −5.79581 − 8.26147I
u = −1.192440 − 0.547877I
a = 0.05105 + 2.06537I
b = −1.57070 + 1.62477I
−3.85720 − 11.56320I −5.79581 + 8.26147I
5
II.
I
u
2
= h−2u
15
+8u
13
+· · ·+b+2, −2u
15
+8u
13
+· · ·+a+3, u
16
+u
15
+· · ·−2u−1i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
2
=
u
u
a
4
=
−u
3
−u
3
+ u
a
1
=
u
3
u
5
− u
3
+ u
a
6
=
2u
15
− 8u
13
+ ··· − 3u − 3
2u
15
− 8u
13
+ ··· − u − 2
a
5
=
2u
15
+ u
14
+ ··· − 2u − 3
u
15
− 4u
13
− u
12
+ 7u
11
+ 3u
10
− 5u
9
− 4u
8
+ 2u
6
+ 2u
5
− 1
a
7
=
2u
15
− 9u
13
+ ··· − 3u − 3
u
15
− 5u
13
+ ··· − u − 2
a
7
=
2u
15
− 9u
13
+ ··· − 3u − 3
u
15
− 5u
13
+ ··· − u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
12
− 12u
10
− 4u
9
+ 16u
8
+ 8u
7
− 4u
6
− 8u
5
− 4u
4
+ 4u
2
− 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
2
c
2
, c
4
, c
7
c
8
u
16
− u
15
+ ··· + 2u − 1
c
3
, c
5
u
16
+ 9u
15
+ ··· − 8u
2
+ 1
c
6
, c
9
(u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
2
c
2
, c
4
, c
7
c
8
y
16
− 9y
15
+ ··· − 8y
2
+ 1
c
3
, c
5
y
16
− 5y
15
+ ··· − 16y + 1
c
6
, c
9
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.685501 + 0.640105I
a = 0.436635 − 0.582879I
b = 0.612928 − 0.418261I
3.21286 1.86404 + 0.I
u = −0.685501 − 0.640105I
a = 0.436635 + 0.582879I
b = 0.612928 + 0.418261I
3.21286 1.86404 + 0.I
u = −0.203747 + 0.848147I
a = −0.171437 − 0.597846I
b = −1.12222 + 1.11997I
−0.91019 − 6.44354I −2.57155 + 5.29417I
u = −0.203747 − 0.848147I
a = −0.171437 + 0.597846I
b = −1.12222 − 1.11997I
−0.91019 + 6.44354I −2.57155 − 5.29417I
u = 1.082580 + 0.348383I
a = 0.921772 + 0.891806I
b = −0.275134 + 0.901574I
−2.24921 − 1.13123I −4.58478 + 0.51079I
u = 1.082580 − 0.348383I
a = 0.921772 − 0.891806I
b = −0.275134 − 0.901574I
−2.24921 + 1.13123I −4.58478 − 0.51079I
u = 1.14767
a = 0.848070
b = 0.513726
−2.44483 −0.105540
u = −1.134620 + 0.424735I
a = 0.45794 − 2.18496I
b = −0.74376 − 2.19413I
−5.44928 + 2.57849I −7.72292 − 3.56796I
u = −1.134620 − 0.424735I
a = 0.45794 + 2.18496I
b = −0.74376 + 2.19413I
−5.44928 − 2.57849I −7.72292 + 3.56796I
u = −1.130780 + 0.529217I
a = −0.20737 + 1.95558I
b = 1.10166 + 1.54556I
−0.91019 + 6.44354I −2.57155 − 5.29417I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.130780 − 0.529217I
a = −0.20737 − 1.95558I
b = 1.10166 − 1.54556I
−0.91019 − 6.44354I −2.57155 + 5.29417I
u = 1.242710 + 0.322774I
a = −1.21486 − 0.76329I
b = −0.28199 − 1.40795I
−5.44928 + 2.57849I −7.72292 − 3.56796I
u = 1.242710 − 0.322774I
a = −1.21486 + 0.76329I
b = −0.28199 + 1.40795I
−5.44928 − 2.57849I −7.72292 + 3.56796I
u = −0.684028
a = −2.18804
b = −1.62708
−2.44483 −0.105540
u = 0.097535 + 0.616980I
a = −0.552685 − 1.087970I
b = −0.234797 + 1.067950I
−2.24921 + 1.13123I −4.58478 − 0.51079I
u = 0.097535 − 0.616980I
a = −0.552685 + 1.087970I
b = −0.234797 − 1.067950I
−2.24921 − 1.13123I −4.58478 + 0.51079I
10
III. I
u
3
= hu
5
− u
3
+ b + u, u
2
+ a, u
6
− u
4
− u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
2
=
u
u
a
4
=
−u
3
−u
3
+ u
a
1
=
u
3
u
5
− u
3
+ u
a
6
=
−u
2
−u
5
+ u
3
− u
a
5
=
u
5
− u
2
−u
a
7
=
u
4
− u
2
− u
−u
2
a
7
=
u
4
− u
2
− u
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
5
+ 4u
4
− 4u
3
− 2u
2
− 2u + 2
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
− 3u
5
+ 5u
4
− 7u
3
+ 9u
2
− 8u + 4
c
2
, c
4
, c
7
c
8
u
6
− u
4
+ u
3
+ u
2
− u + 1
c
3
, c
5
u
6
+ 2u
5
+ 3u
4
+ u
3
+ u
2
− u + 1
c
6
, c
9
u
6
− u
5
− u
4
+ 3u
3
− u
2
− 2u + 2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
+ y
5
+ y
4
+ y
3
+ 9y
2
+ 8y + 16
c
2
, c
4
, c
7
c
8
y
6
− 2y
5
+ 3y
4
− y
3
+ y
2
+ y + 1
c
3
, c
5
y
6
+ 2y
5
+ 7y
4
+ 11y
3
+ 9y
2
+ y + 1
c
6
, c
9
y
6
− 3y
5
+ 5y
4
− 7y
3
+ 9y
2
− 8y + 4
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.856601 + 0.623578I
a = −0.344917 + 1.068320I
b = −0.107958 + 0.512846I
2.72382 + 4.89103I 0.12173 − 6.59162I
u = −0.856601 − 0.623578I
a = −0.344917 − 1.068320I
b = −0.107958 − 0.512846I
2.72382 − 4.89103I 0.12173 + 6.59162I
u = 1.140590 + 0.471635I
a = −1.07851 − 1.07589I
b = 0.67021 − 1.38548I
−5.10856 − 5.32947I −7.48262 + 4.54389I
u = 1.140590 − 0.471635I
a = −1.07851 + 1.07589I
b = 0.67021 + 1.38548I
−5.10856 + 5.32947I −7.48262 − 4.54389I
u = −0.283992 + 0.709987I
a = 0.423430 + 0.403261I
b = 0.937752 − 0.810947I
1.56227 − 1.71504I 1.36090 + 1.32670I
u = −0.283992 − 0.709987I
a = 0.423430 − 0.403261I
b = 0.937752 + 0.810947I
1.56227 + 1.71504I 1.36090 − 1.32670I
14
IV. I
u
4
= hb + 1, a + 1, u − 1i
(i) Arc colorings
a
3
=
0
1
a
8
=
1
0
a
9
=
1
1
a
2
=
1
1
a
4
=
−1
0
a
1
=
1
1
a
6
=
−1
−1
a
5
=
−2
−1
a
7
=
−1
−1
a
7
=
−1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
9
u
c
2
, c
3
, c
5
c
7
u + 1
c
4
, c
8
u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
9
y
c
2
, c
3
, c
4
c
5
, c
7
, c
8
y −1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −1.00000
−3.28987 −12.0000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u(u
4
+ 2u
2
+ 3u + 1)(u
6
− 3u
5
+ 5u
4
− 7u
3
+ 9u
2
− 8u + 4)
· (u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
2
c
2
, c
7
(u + 1)(u
4
− u
3
− u
2
+ u + 1)(u
6
− u
4
+ u
3
+ u
2
− u + 1)
· (u
16
− u
15
+ ··· + 2u − 1)
c
3
, c
5
(u + 1)(u
4
+ 3u
3
+ 5u
2
+ 3u + 1)(u
6
+ 2u
5
+ 3u
4
+ u
3
+ u
2
− u + 1)
· (u
16
+ 9u
15
+ ··· − 8u
2
+ 1)
c
4
, c
8
(u − 1)(u
4
− u
3
− u
2
+ u + 1)(u
6
− u
4
+ u
3
+ u
2
− u + 1)
· (u
16
− u
15
+ ··· + 2u − 1)
c
6
, c
9
u(u
4
− 2u
3
+ 2u
2
− u + 1)(u
6
− u
5
− u
4
+ 3u
3
− u
2
− 2u + 2)
· (u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
2
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y(y
4
+ 4y
3
+ 6y
2
− 5y + 1)(y
6
+ y
5
+ y
4
+ y
3
+ 9y
2
+ 8y + 16)
· (y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
2
c
2
, c
4
, c
7
c
8
(y −1)(y
4
− 3y
3
+ 5y
2
− 3y + 1)(y
6
− 2y
5
+ 3y
4
− y
3
+ y
2
+ y + 1)
· (y
16
− 9y
15
+ ··· − 8y
2
+ 1)
c
3
, c
5
(y −1)(y
4
+ y
3
+ 9y
2
+ y + 1)(y
6
+ 2y
5
+ ··· + y + 1)
· (y
16
− 5y
15
+ ··· − 16y + 1)
c
6
, c
9
y(y
4
+ 2y
2
+ 3y + 1)(y
6
− 3y
5
+ 5y
4
− 7y
3
+ 9y
2
− 8y + 4)
· (y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
2
20
