9
43
(K9n
3
)
A knot diagram
1
Linearized knot diagam
5 9 1 7 2 8 5 3 8
Solving Sequence
3,8
9 1
2,5
7 4 6
c
8
c
9
c
2
c
7
c
4
c
6
c
1
, c
3
, c
5
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
5
+ u
4
− 2u
3
+ u
2
+ b − u + 1, u
7
− 2u
6
+ 5u
5
− 5u
4
+ 6u
3
− 5u
2
+ a + 3u − 3,
u
8
− 2u
7
+ 5u
6
− 6u
5
+ 7u
4
− 7u
3
+ 4u
2
− 4u + 1i
I
u
2
= hb + 1, a − u − 1, u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 10 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−u
5
+ u
4
− 2u
3
+ u
2
+ b − u + 1, u
7
− 2u
6
+ 5u
5
− 5u
4
+ 6u
3
−
5u
2
+ a + 3u − 3, u
8
− 2u
7
+ · · · − 4u + 1i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
1
=
u
2
+ 1
−u
2
a
2
=
−u
u
3
+ u
a
5
=
−u
7
+ 2u
6
− 5u
5
+ 5u
4
− 6u
3
+ 5u
2
− 3u + 3
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
a
7
=
−u
7
+ 2u
6
− 4u
5
+ 5u
4
− 4u
3
+ 5u
2
− 2u + 3
−u
6
+ u
5
− 3u
4
+ 2u
3
− 2u
2
+ 2u − 1
a
4
=
u
5
+ 2u
3
+ u
−u
5
− u
3
− u
a
6
=
−u
7
+ u
6
− 3u
5
+ 2u
4
− 2u
3
+ 3u
2
+ 2
−u
6
+ u
5
− 3u
4
+ 2u
3
− 2u
2
+ 2u − 1
a
6
=
−u
7
+ u
6
− 3u
5
+ 2u
4
− 2u
3
+ 3u
2
+ 2
−u
6
+ u
5
− 3u
4
+ 2u
3
− 2u
2
+ 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
7
+ u
6
− u
5
− 2u
4
+ 6u
3
− 5u
2
+ 5u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
8
+ u
7
− 7u
6
− 4u
5
+ 16u
4
− 3u
3
− 9u
2
− 8u − 4
c
2
, c
8
u
8
+ 2u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 7u
3
+ 4u
2
+ 4u + 1
c
3
u
8
− 2u
7
− 7u
6
+ 12u
5
+ 5u
4
+ 3u
3
− 2u
2
+ 2u + 1
c
4
, c
7
u
8
− 3u
7
− 2u
6
+ 9u
5
+ 5u
4
− 13u
3
− 3u
2
+ 3u − 1
c
6
u
8
+ 13u
7
+ 68u
6
+ 185u
5
+ 287u
4
+ 249u
3
+ 77u
2
+ 3u + 1
c
9
u
8
+ 6u
7
+ 15u
6
+ 14u
5
− 9u
4
− 31u
3
− 26u
2
− 8u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
8
− 15y
7
+ 89y
6
− 252y
5
+ 366y
4
− 305y
3
− 95y
2
+ 8y + 16
c
2
, c
8
y
8
+ 6y
7
+ 15y
6
+ 14y
5
− 9y
4
− 31y
3
− 26y
2
− 8y + 1
c
3
y
8
− 18y
7
+ 107y
6
− 206y
5
− 9y
4
− 91y
3
+ 2y
2
− 8y + 1
c
4
, c
7
y
8
− 13y
7
+ 68y
6
− 185y
5
+ 287y
4
− 249y
3
+ 77y
2
− 3y + 1
c
6
y
8
− 33y
7
+ ··· + 145y + 1
c
9
y
8
− 6y
7
+ 39y
6
− 146y
5
+ 267y
4
− 239y
3
+ 162y
2
− 116y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.381025 + 0.877247I
a = 0.332599 + 0.127423I
b = 0.238510 − 0.243220I
−0.36340 − 1.66195I −2.61632 + 3.48117I
u = −0.381025 − 0.877247I
a = 0.332599 − 0.127423I
b = 0.238510 + 0.243220I
−0.36340 + 1.66195I −2.61632 − 3.48117I
u = 1.11498
a = −1.63389
b = 1.82176
−11.0713 −7.35940
u = 0.126694 + 1.193160I
a = −0.399095 − 1.030330I
b = −1.178780 + 0.606721I
−4.43209 + 1.62541I −10.58501 − 1.42555I
u = 0.126694 − 1.193160I
a = −0.399095 + 1.030330I
b = −1.178780 − 0.606721I
−4.43209 − 1.62541I −10.58501 + 1.42555I
u = 0.54402 + 1.39007I
a = −0.321827 + 1.239280I
b = 1.89776 − 0.22684I
−15.4360 + 5.9041I −9.72541 − 2.82977I
u = 0.54402 − 1.39007I
a = −0.321827 − 1.239280I
b = 1.89776 + 0.22684I
−15.4360 − 5.9041I −9.72541 + 2.82977I
u = 0.305633
a = 2.41054
b = −0.736738
−1.10361 −8.78710
5
II. I
u
2
= hb + 1, a − u − 1, u
2
+ u + 1i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
u + 1
a
1
=
−u
u + 1
a
2
=
−u
u + 1
a
5
=
u + 1
−1
a
7
=
u + 2
−1
a
4
=
−1
0
a
6
=
u + 1
−1
a
6
=
u + 1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u − 7
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
2
c
2
u
2
− u + 1
c
3
, c
8
, c
9
u
2
+ u + 1
c
4
, c
6
(u − 1)
2
c
7
(u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
2
c
2
, c
3
, c
8
c
9
y
2
+ y + 1
c
4
, c
6
, c
7
(y −1)
2
8
(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 = −1.00000
−1.64493 − 2.02988I −9.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = −1.00000
−1.64493 + 2.02988I −9.00000 − 3.46410I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
2
(u
8
+ u
7
− 7u
6
− 4u
5
+ 16u
4
− 3u
3
− 9u
2
− 8u − 4)
c
2
(u
2
− u + 1)(u
8
+ 2u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 7u
3
+ 4u
2
+ 4u + 1)
c
3
(u
2
+ u + 1)(u
8
− 2u
7
− 7u
6
+ 12u
5
+ 5u
4
+ 3u
3
− 2u
2
+ 2u + 1)
c
4
(u − 1)
2
(u
8
− 3u
7
− 2u
6
+ 9u
5
+ 5u
4
− 13u
3
− 3u
2
+ 3u − 1)
c
6
((u − 1)
2
)(u
8
+ 13u
7
+ ··· + 3u + 1)
c
7
(u + 1)
2
(u
8
− 3u
7
− 2u
6
+ 9u
5
+ 5u
4
− 13u
3
− 3u
2
+ 3u − 1)
c
8
(u
2
+ u + 1)(u
8
+ 2u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 7u
3
+ 4u
2
+ 4u + 1)
c
9
(u
2
+ u + 1)(u
8
+ 6u
7
+ 15u
6
+ 14u
5
− 9u
4
− 31u
3
− 26u
2
− 8u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
2
(y
8
− 15y
7
+ 89y
6
− 252y
5
+ 366y
4
− 305y
3
− 95y
2
+ 8y + 16)
c
2
, c
8
(y
2
+ y + 1)(y
8
+ 6y
7
+ 15y
6
+ 14y
5
− 9y
4
− 31y
3
− 26y
2
− 8y + 1)
c
3
(y
2
+ y + 1)(y
8
− 18y
7
+ ··· − 8y + 1)
c
4
, c
7
((y −1)
2
)(y
8
− 13y
7
+ ··· − 3y + 1)
c
6
((y −1)
2
)(y
8
− 33y
7
+ ··· + 145y + 1)
c
9
(y
2
+ y + 1)
· (y
8
− 6y
7
+ 39y
6
− 146y
5
+ 267y
4
− 239y
3
+ 162y
2
− 116y + 1)
11
