9
41
(K9a
29
)
A knot diagram
1
Linearized knot diagam
8 7 9 2 1 3 5 4 6
Solving Sequence
3,9 4,6
7 1 2 5 8
c
3
c
6
c
9
c
2
c
5
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, a − 1, u
4
− 2u
3
+ 4u
2
− 3u + 1i
I
u
2
= hb + u, a
2
+ au + 2u
2
+ 2a + 2u + 4, u
3
+ u
2
+ 2u + 1i
I
u
3
= h−3u
5
+ 11u
4
− 26u
3
+ 35u
2
+ 4b − 28u + 12, 3u
5
− 9u
4
+ 20u
3
− 23u
2
+ 8a + 14u − 4,
u
6
− 5u
5
+ 14u
4
− 25u
3
+ 28u
2
− 20u + 8i
I
u
4
= h−u
2
b + b
2
− 2bu − 2u, a − 1, u
3
+ u
2
+ 2u + 1i
I
u
5
= hb + u, a + 1, u
4
+ 2u
2
− u + 1i
I
u
6
= h−au + b − u − 1, −u
2
a + a
2
− au + 3u
2
− a + u + 5, u
3
+ u
2
+ 2u + 1i
* 6 irreducible components of dim
C
= 0, with total 32 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
= hb + u, a − 1, u
4
− 2u
3
+ 4u
2
− 3u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
1
−u
a
7
=
−u + 1
−u
a
1
=
u
−u
2
+ u
a
2
=
u
2
− u + 1
u
2
a
5
=
u
2
+ 1
−u
3
+ u
2
− u
a
8
=
−u
u
3
+ u
a
8
=
−u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −3u
3
+ 9u
2
− 18u + 15
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
7
u
4
− 3u
3
+ 4u
2
− 2u + 1
c
2
, c
3
, c
5
c
6
, c
8
, c
9
u
4
− 2u
3
+ 4u
2
− 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
y
4
− y
3
+ 6y
2
+ 4y + 1
c
2
, c
3
, c
5
c
6
, c
8
, c
9
y
4
+ 4y
3
+ 6y
2
− y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.363271I
a = 1.00000
b = −0.500000 − 0.363271I
0.986960 + 0.735995I 7.28115 − 3.94298I
u = 0.500000 − 0.363271I
a = 1.00000
b = −0.500000 + 0.363271I
0.986960 − 0.735995I 7.28115 + 3.94298I
u = 0.50000 + 1.53884I
a = 1.00000
b = −0.50000 − 1.53884I
−10.8566 + 12.0989I −2.78115 − 6.37988I
u = 0.50000 − 1.53884I
a = 1.00000
b = −0.50000 + 1.53884I
−10.8566 − 12.0989I −2.78115 + 6.37988I
5
II. I
u
2
= hb + u, a
2
+ au + 2u
2
+ 2a + 2u + 4, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
a
−u
a
7
=
a − u
−u
a
1
=
−u
2
a − 2au + 2
−u
2
a + u
a
2
=
−au + u
2
+ 1
u
2
a
5
=
−u
2
a − 5au − 2u
2
− 2a − 4u
−2u
2
a − 3au − 2u
2
− a − u
a
8
=
−u
−u
2
− u − 1
a
8
=
−u
−u
2
− u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4au + 8u
2
+ 4a + 12u + 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
3
+ u
2
− 1)
2
c
2
, c
3
, c
6
c
8
(u
3
+ u
2
+ 2u + 1)
2
c
4
u
6
− 7u
5
+ 24u
4
− 47u
3
+ 54u
2
− 32u + 8
c
5
, c
9
u
6
− 5u
5
+ 14u
4
− 25u
3
+ 28u
2
− 20u + 8
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
3
− y
2
+ 2y −1)
2
c
2
, c
3
, c
6
c
8
(y
3
+ 3y
2
+ 2y −1)
2
c
4
y
6
− y
5
+ 26y
4
− 49y
3
+ 292y
2
− 160y + 64
c
5
, c
9
y
6
+ 3y
5
+ 2y
4
− 25y
3
+ 8y
2
+ 48y + 64
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −0.407481 − 0.986732I
b = 0.215080 − 1.307140I
−4.93480 − 5.65624I −2.00000 + 5.95889I
u = −0.215080 + 1.307140I
a = −1.37744 − 0.32041I
b = 0.215080 − 1.307140I
−9.07239 − 2.82812I −8.52927 + 2.97945I
u = −0.215080 − 1.307140I
a = −0.407481 + 0.986732I
b = 0.215080 + 1.307140I
−4.93480 + 5.65624I −2.00000 − 5.95889I
u = −0.215080 − 1.307140I
a = −1.37744 + 0.32041I
b = 0.215080 + 1.307140I
−9.07239 + 2.82812I −8.52927 − 2.97945I
u = −0.569840
a = −0.71508 + 1.73159I
b = 0.569840
−0.79722 − 2.82812I 4.52927 + 2.97945I
u = −0.569840
a = −0.71508 − 1.73159I
b = 0.569840
−0.79722 + 2.82812I 4.52927 − 2.97945I
9
III. I
u
3
= h−3u
5
+ 11u
4
+ · · · + 4b + 12, 3u
5
− 9u
4
+ 20u
3
− 23u
2
+ 8a +
14u − 4, u
6
− 5u
5
+ 14u
4
− 25u
3
+ 28u
2
− 20u + 8i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
−
3
8
u
5
+
9
8
u
4
+ ··· −
7
4
u +
1
2
3
4
u
5
−
11
4
u
4
+ ··· + 7u − 3
a
7
=
3
8
u
5
−
13
8
u
4
+ ··· +
21
4
u −
5
2
3
4
u
5
−
11
4
u
4
+ ··· + 7u − 3
a
1
=
5
8
u
5
−
19
8
u
4
+ ··· +
23
4
u − 3
−
3
4
u
5
+
13
4
u
4
+ ··· −
17
2
u + 5
a
2
=
1
8
u
5
−
7
8
u
4
+ ··· +
15
4
u − 1
3
4
u
5
−
13
4
u
4
+ ··· +
19
2
u − 5
a
5
=
−
1
8
u
5
+
5
8
u
4
+ ··· +
13
8
u
2
−
1
2
u
−u
5
+
7
2
u
4
−
17
2
u
3
+ 11u
2
−
19
2
u + 5
a
8
=
−u
u
3
+ u
a
8
=
−u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
5
+ u
4
− 2u
3
+ u
2
− 4u + 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
3
+ u
2
− 1)
2
c
2
, c
5
, c
6
c
9
(u
3
+ u
2
+ 2u + 1)
2
c
3
, c
8
u
6
− 5u
5
+ 14u
4
− 25u
3
+ 28u
2
− 20u + 8
c
7
u
6
− 7u
5
+ 24u
4
− 47u
3
+ 54u
2
− 32u + 8
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
3
− y
2
+ 2y −1)
2
c
2
, c
5
, c
6
c
9
(y
3
+ 3y
2
+ 2y −1)
2
c
3
, c
8
y
6
+ 3y
5
+ 2y
4
− 25y
3
+ 8y
2
+ 48y + 64
c
7
y
6
− y
5
+ 26y
4
− 49y
3
+ 292y
2
− 160y + 64
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.407481 + 0.986732I
a = −0.203741 + 0.493366I
b = 0.569840
−0.79722 + 2.82812I 4.52927 − 2.97945I
u = 0.407481 − 0.986732I
a = −0.203741 − 0.493366I
b = 0.569840
−0.79722 − 2.82812I 4.52927 + 2.97945I
u = 1.37744 + 0.32041I
a = −0.357540 − 0.865797I
b = 0.215080 + 1.307140I
−4.93480 + 5.65624I −2.00000 − 5.95889I
u = 1.37744 − 0.32041I
a = −0.357540 + 0.865797I
b = 0.215080 − 1.307140I
−4.93480 − 5.65624I −2.00000 + 5.95889I
u = 0.71508 + 1.73159I
a = −0.688719 − 0.160205I
b = 0.215080 + 1.307140I
−9.07239 + 2.82812I −8.52927 − 2.97945I
u = 0.71508 − 1.73159I
a = −0.688719 + 0.160205I
b = 0.215080 − 1.307140I
−9.07239 − 2.82812I −8.52927 + 2.97945I
13
IV. I
u
4
= h−u
2
b + b
2
− 2bu − 2u, a − 1, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
1
b
a
7
=
b + 1
b
a
1
=
u
bu + u
a
2
=
u
2
b + 2bu + b + 2u + 1
u
2
b + 2bu + 2u
a
5
=
u
2
+ 1
u
2
b + u
2
+ b
a
8
=
−u
−u
2
− u − 1
a
8
=
−u
−u
2
− u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
2
b + 4bu + 8u
2
+ 4b + 12u + 10
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
− 7u
5
+ 24u
4
− 47u
3
+ 54u
2
− 32u + 8
c
2
, c
6
u
6
− 5u
5
+ 14u
4
− 25u
3
+ 28u
2
− 20u + 8
c
3
, c
5
, c
8
c
9
(u
3
+ u
2
+ 2u + 1)
2
c
4
, c
7
(u
3
+ u
2
− 1)
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− y
5
+ 26y
4
− 49y
3
+ 292y
2
− 160y + 64
c
2
, c
6
y
6
+ 3y
5
+ 2y
4
− 25y
3
+ 8y
2
+ 48y + 64
c
3
, c
5
, c
8
c
9
(y
3
+ 3y
2
+ 2y −1)
2
c
4
, c
7
(y
3
− y
2
+ 2y −1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = 1.00000
b = −1.37744 + 0.32041I
−4.93480 − 5.65624I −2.00000 + 5.95889I
u = −0.215080 + 1.307140I
a = 1.00000
b = −0.71508 + 1.73159I
−9.07239 − 2.82812I −8.52927 + 2.97945I
u = −0.215080 − 1.307140I
a = 1.00000
b = −1.37744 − 0.32041I
−4.93480 + 5.65624I −2.00000 − 5.95889I
u = −0.215080 − 1.307140I
a = 1.00000
b = −0.71508 − 1.73159I
−9.07239 + 2.82812I −8.52927 − 2.97945I
u = −0.569840
a = 1.00000
b = −0.407481 + 0.986732I
−0.79722 − 2.82812I 4.52927 + 2.97945I
u = −0.569840
a = 1.00000
b = −0.407481 − 0.986732I
−0.79722 + 2.82812I 4.52927 − 2.97945I
17
V. I
u
5
= hb + u, a + 1, u
4
+ 2u
2
− u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
−1
−u
a
7
=
−u − 1
−u
a
1
=
u
u
2
+ u
a
2
=
u
2
+ u + 1
u
2
a
5
=
−u
2
− 1
−u
3
− u
2
− u
a
8
=
−u
u
3
+ u
a
8
=
−u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
3
− 3u
2
− 6u − 3
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
7
u
4
+ u
3
+ 1
c
2
, c
5
, c
8
u
4
+ 2u
2
+ u + 1
c
3
, c
6
, c
9
u
4
+ 2u
2
− u + 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
y
4
− y
3
+ 2y
2
+ 1
c
2
, c
3
, c
5
c
6
, c
8
, c
9
y
4
+ 4y
3
+ 6y
2
+ 3y + 1
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.343815 + 0.625358I
a = −1.00000
b = −0.343815 − 0.625358I
−2.15173 + 3.38562I −3.15611 − 4.97381I
u = 0.343815 − 0.625358I
a = −1.00000
b = −0.343815 + 0.625358I
−2.15173 − 3.38562I −3.15611 + 4.97381I
u = −0.343815 + 1.358440I
a = −1.00000
b = 0.343815 − 1.358440I
−7.71788 − 2.37936I −1.34389 + 0.72682I
u = −0.343815 − 1.358440I
a = −1.00000
b = 0.343815 + 1.358440I
−7.71788 + 2.37936I −1.34389 − 0.72682I
21
VI.
I
u
6
= h−au + b − u − 1, −u
2
a + a
2
− au + 3u
2
− a + u + 5, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
a
au + u + 1
a
7
=
au + a + u + 1
au + u + 1
a
1
=
−au + 2u
2
− a + u + 3
−u
2
− 2
a
2
=
u
2
a + au + 2u
2
+ 2u + 3
u
2
a + au + u − 1
a
5
=
au − u
2
− 1
u
2
a + au + a + u + 1
a
8
=
−u
−u
2
− u − 1
a
8
=
−u
−u
2
− u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
7
(u
3
+ u
2
− 1)
2
c
2
, c
3
, c
5
c
6
, c
8
, c
9
(u
3
+ u
2
+ 2u + 1)
2
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
(y
3
− y
2
+ 2y − 1)
2
c
2
, c
3
, c
5
c
6
, c
8
, c
9
(y
3
+ 3y
2
+ 2y − 1)
2
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −0.947279 + 0.320410I
b = 0.569840
−4.93480 −2.00000
u = −0.215080 + 1.307140I
a = 0.069840 + 0.424452I
b = 0.215080 + 1.307140I
−4.93480 −2.00000
u = −0.215080 − 1.307140I
a = −0.947279 − 0.320410I
b = 0.569840
−4.93480 −2.00000
u = −0.215080 − 1.307140I
a = 0.069840 − 0.424452I
b = 0.215080 − 1.307140I
−4.93480 −2.00000
u = −0.569840
a = 0.37744 + 2.29387I
b = 0.215080 − 1.307140I
−4.93480 −2.00000
u = −0.569840
a = 0.37744 − 2.29387I
b = 0.215080 + 1.307140I
−4.93480 −2.00000
25
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
7
(u
3
+ u
2
− 1)
6
(u
4
− 3u
3
+ 4u
2
− 2u + 1)(u
4
+ u
3
+ 1)
· (u
6
− 7u
5
+ 24u
4
− 47u
3
+ 54u
2
− 32u + 8)
c
2
, c
5
, c
8
(u
3
+ u
2
+ 2u + 1)
6
(u
4
+ 2u
2
+ u + 1)(u
4
− 2u
3
+ 4u
2
− 3u + 1)
· (u
6
− 5u
5
+ 14u
4
− 25u
3
+ 28u
2
− 20u + 8)
c
3
, c
6
, c
9
(u
3
+ u
2
+ 2u + 1)
6
(u
4
+ 2u
2
− u + 1)(u
4
− 2u
3
+ 4u
2
− 3u + 1)
· (u
6
− 5u
5
+ 14u
4
− 25u
3
+ 28u
2
− 20u + 8)
26
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
(y
3
− y
2
+ 2y − 1)
6
(y
4
− y
3
+ 2y
2
+ 1)(y
4
− y
3
+ 6y
2
+ 4y + 1)
· (y
6
− y
5
+ 26y
4
− 49y
3
+ 292y
2
− 160y + 64)
c
2
, c
3
, c
5
c
6
, c
8
, c
9
((y
3
+ 3y
2
+ 2y − 1)
6
)(y
4
+ 4y
3
+ 6y
2
− y + 1)(y
4
+ 4y
3
+ ··· + 3y + 1)
· (y
6
+ 3y
5
+ 2y
4
− 25y
3
+ 8y
2
+ 48y + 64)
27
