9
31
(K9a
13
)
A knot diagram
1
Linearized knot diagam
5 6 7 1 9 8 4 3 2
Solving Sequence
4,7
8 3 9 6 2 1 5
c
7
c
3
c
8
c
6
c
2
c
9
c
5
c
1
, c
4
Ideals for irreducible components
2
of X
par
I
u
1
= hu
7
− 2u
5
+ 2u
3
− u
2
+ 1i
I
u
2
= hu
20
+ u
19
+ ··· + 2u + 1i
* 2 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
7
− 2u
5
+ 2u
3
− u
2
+ 1i
(i) Arc colorings
a
4
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
u
u
a
9
=
u
4
− u
2
+ 1
u
4
a
6
=
−u
2
+ 1
−u
4
a
2
=
u
2
− 1
u
5
+ u
4
− 2u
3
+ u − 1
a
1
=
1
−u
6
+ u
4
+ u
3
+ 1
a
5
=
−u
u
5
− u
4
− 2u
3
+ u
2
− 1
a
5
=
−u
u
5
− u
4
− 2u
3
+ u
2
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
5
+ 4u
4
− 8u
3
− 4u
2
+ 4u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
u
7
− 2u
5
+ 2u
3
− u
2
+ 1
c
2
u
7
− 5u
6
+ 12u
5
− 17u
4
+ 15u
3
− 5u
2
− 4u + 4
c
5
, c
8
u
7
+ 2u
5
− 2u
4
+ 4u
3
− u
2
+ 2u + 1
c
6
, c
9
u
7
+ 4u
6
+ 8u
5
+ 8u
4
+ 4u
3
+ u
2
+ 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
y
7
− 4y
6
+ 8y
5
− 8y
4
+ 4y
3
− y
2
+ 2y −1
c
2
y
7
− y
6
+ 4y
5
+ 13y
4
− y
3
− 9y
2
+ 56y −16
c
5
, c
8
y
7
+ 4y
6
+ 12y
5
+ 16y
4
+ 20y
3
+ 19y
2
+ 6y −1
c
6
, c
9
y
7
+ 8y
5
− 4y
4
+ 24y
3
− y
2
+ 2y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.125110 + 0.343189I
−5.94607 − 3.76357I −10.60460 + 4.24459I
u = 1.125110 − 0.343189I
−5.94607 + 3.76357I −10.60460 − 4.24459I
u = 0.364544 + 0.701794I
1.82567 + 1.84683I 1.12815 − 1.09324I
u = 0.364544 − 0.701794I
1.82567 − 1.84683I 1.12815 + 1.09324I
u = −1.125830 + 0.566290I
−2.65707 + 11.68630I −5.70307 − 8.84509I
u = −1.125830 − 0.566290I
−2.65707 − 11.68630I −5.70307 + 8.84509I
u = −0.727635
−1.24946 −7.64100
5
II. I
u
2
= hu
20
+ u
19
− 4u
18
− 5u
17
+ 8u
16
+ 13u
15
− 7u
14
− 20u
13
− u
12
+
19u
11
+ 10u
10
− 10u
9
− 11u
8
+ 2u
7
+ 7u
6
+ u
5
− 3u
4
− u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
4
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
u
u
a
9
=
u
4
− u
2
+ 1
u
4
a
6
=
−u
2
+ 1
−u
4
a
2
=
u
7
− 2u
5
+ 2u
3
u
9
− u
7
+ u
5
+ u
a
1
=
−u
19
− u
18
+ ··· − 3u
2
− 2u
u
11
− 3u
9
+ 4u
7
− 3u
5
+ u
3
− u − 1
a
5
=
u
12
− 3u
10
+ 5u
8
− 4u
6
+ 2u
4
− u
2
+ 1
u
12
− 2u
10
+ 2u
8
− u
4
a
5
=
u
12
− 3u
10
+ 5u
8
− 4u
6
+ 2u
4
− u
2
+ 1
u
12
− 2u
10
+ 2u
8
− u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
18
−16u
16
+36u
14
−48u
12
+44u
10
−28u
8
−4u
7
+16u
6
+8u
5
−8u
4
−8u
3
+4u
2
+4u −2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
u
20
+ u
19
+ ··· + 2u + 1
c
2
(u
10
+ 2u
9
+ u
8
+ 4u
6
+ 6u
5
+ u
4
− 6u
3
− 5u
2
+ 1)
2
c
5
, c
8
u
20
+ 3u
19
+ ··· + 16u + 5
c
6
, c
9
u
20
+ 9u
19
+ ··· + 2u
2
+ 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
y
20
− 9y
19
+ ··· + 2y
2
+ 1
c
2
(y
10
− 2y
9
+ ··· − 10y + 1)
2
c
5
, c
8
y
20
+ 3y
19
+ ··· + 204y + 25
c
6
, c
9
y
20
+ 3y
19
+ ··· + 4y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.941429 + 0.547698I
0.197299 −2.26625 + 0.I
u = −0.941429 − 0.547698I
0.197299 −2.26625 + 0.I
u = −1.061040 + 0.273586I
−2.27340 + 0.51998I −5.71661 − 0.77505I
u = −1.061040 − 0.273586I
−2.27340 − 0.51998I −5.71661 + 0.77505I
u = −0.626658 + 0.633601I
1.11960 + 4.65452I −0.79654 − 6.04247I
u = −0.626658 − 0.633601I
1.11960 − 4.65452I −0.79654 + 6.04247I
u = 1.128770 + 0.240119I
−4.83313 + 3.92983I −9.04400 − 3.21471I
u = 1.128770 − 0.240119I
−4.83313 − 3.92983I −9.04400 + 3.21471I
u = 1.016360 + 0.552370I
1.11960 − 4.65452I −0.79654 + 6.04247I
u = 1.016360 − 0.552370I
1.11960 + 4.65452I −0.79654 − 6.04247I
u = −0.330984 + 0.758157I
−0.32496 − 6.68616I −2.49331 + 5.21994I
u = −0.330984 − 0.758157I
−0.32496 + 6.68616I −2.49331 − 5.21994I
u = 0.527984 + 0.630206I
2.55688 2.36717 + 0.I
u = 0.527984 − 0.630206I
2.55688 2.36717 + 0.I
u = −1.119570 + 0.508145I
−4.83313 + 3.92983I −9.04400 − 3.21471I
u = −1.119570 − 0.508145I
−4.83313 − 3.92983I −9.04400 + 3.21471I
u = 1.102100 + 0.557039I
−0.32496 − 6.68616I −2.49331 + 5.21994I
u = 1.102100 − 0.557039I
−0.32496 + 6.68616I −2.49331 − 5.21994I
u = −0.195538 + 0.653472I
−2.27340 + 0.51998I −5.71661 − 0.77505I
u = −0.195538 − 0.653472I
−2.27340 − 0.51998I −5.71661 + 0.77505I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
(u
7
− 2u
5
+ 2u
3
− u
2
+ 1)(u
20
+ u
19
+ ··· + 2u + 1)
c
2
(u
7
− 5u
6
+ 12u
5
− 17u
4
+ 15u
3
− 5u
2
− 4u + 4)
· (u
10
+ 2u
9
+ u
8
+ 4u
6
+ 6u
5
+ u
4
− 6u
3
− 5u
2
+ 1)
2
c
5
, c
8
(u
7
+ 2u
5
− 2u
4
+ 4u
3
− u
2
+ 2u + 1)(u
20
+ 3u
19
+ ··· + 16u + 5)
c
6
, c
9
(u
7
+ 4u
6
+ ··· + 2u + 1)(u
20
+ 9u
19
+ ··· + 2u
2
+ 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
(y
7
− 4y
6
+ ··· + 2y −1)(y
20
− 9y
19
+ ··· + 2y
2
+ 1)
c
2
(y
7
− y
6
+ 4y
5
+ 13y
4
− y
3
− 9y
2
+ 56y −16)
· (y
10
− 2y
9
+ ··· − 10y + 1)
2
c
5
, c
8
(y
7
+ 4y
6
+ 12y
5
+ 16y
4
+ 20y
3
+ 19y
2
+ 6y −1)
· (y
20
+ 3y
19
+ ··· + 204y + 25)
c
6
, c
9
(y
7
+ 8y
5
+ ··· + 2y −1)(y
20
+ 3y
19
+ ··· + 4y + 1)
11
