9
25
(K9a
4
)
A knot diagram
1
Linearized knot diagam
5 9 6 2 7 4 1 3 8
Solving Sequence
3,6
4
7,9
2 5 1 8
c
3
c
6
c
2
c
5
c
1
c
8
c
4
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
24
− 2u
23
+ ··· + 2b + 1, −u
24
− 2u
23
+ ··· + a + 5u, u
25
+ 3u
24
+ ··· − 4u − 1i
I
u
2
= h2b − a − 1, a
2
+ 3, u − 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
=
h−u
24
−2u
23
+· · ·+2b+1, −u
24
−2u
23
+· · ·+a+5u, u
25
+3u
24
+· · ·−4u−1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
4
=
1
u
2
a
7
=
−u
−u
3
+ u
a
9
=
u
24
+ 2u
23
+ ··· − 5u
2
− 5u
1
2
u
24
+ u
23
+ ··· −
7
2
u −
1
2
a
2
=
7
2
u
24
+ 9u
23
+ ··· −
33
2
u −
9
2
3
2
u
24
+ 4u
23
+ ··· −
13
2
u −
5
2
a
5
=
u
3
u
5
− u
3
+ u
a
1
=
3
2
u
24
+ 3u
23
+ ··· −
15
2
u −
3
2
5
2
u
24
+ 6u
23
+ ··· −
21
2
u −
7
2
a
8
=
1
2
u
24
+ u
23
+ ··· −
3
2
u +
1
2
1
2
u
24
+ u
23
+ ··· −
7
2
u −
1
2
a
8
=
1
2
u
24
+ u
23
+ ··· −
3
2
u +
1
2
1
2
u
24
+ u
23
+ ··· −
7
2
u −
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
24
+ 7u
23
−5u
22
−42u
21
−25u
20
+ 108u
19
+ 144u
18
−128u
17
−
357u
16
+ 2u
15
+ 526u
14
+ 286u
13
− 498u
12
− 538u
11
+ 238u
10
+ 584u
9
+ 35u
8
− 389u
7
−
165u
6
+ 164u
5
+ 119u
4
− 22u
3
− 38u
2
− 6u − 5
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
25
− u
24
+ ··· + 4u + 4
c
2
, c
8
u
25
+ 2u
24
+ ··· + 3u + 1
c
3
, c
6
u
25
− 3u
24
+ ··· − 4u + 1
c
5
u
25
+ 11u
24
+ ··· − 2u + 1
c
7
, c
9
u
25
+ 8u
24
+ ··· + 11u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
25
+ 15y
24
+ ··· − 88y −16
c
2
, c
8
y
25
+ 8y
24
+ ··· + 11y −1
c
3
, c
6
y
25
− 11y
24
+ ··· − 2y −1
c
5
y
25
+ 9y
24
+ ··· − 2y −1
c
7
, c
9
y
25
+ 20y
24
+ ··· + 251y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.781818 + 0.585895I
a = −0.235890 − 0.629868I
b = 0.734813 + 0.804167I
1.52493 + 0.43356I −3.08804 + 0.04506I
u = 0.781818 − 0.585895I
a = −0.235890 + 0.629868I
b = 0.734813 − 0.804167I
1.52493 − 0.43356I −3.08804 − 0.04506I
u = −0.840318 + 0.621070I
a = 0.114344 + 0.489930I
b = 0.723797 + 0.117969I
2.10182 + 2.44039I −0.16599 − 3.61173I
u = −0.840318 − 0.621070I
a = 0.114344 − 0.489930I
b = 0.723797 − 0.117969I
2.10182 − 2.44039I −0.16599 + 3.61173I
u = −0.479273 + 0.936834I
a = 0.544317 + 0.502084I
b = −0.776571 + 0.974090I
6.34798 − 5.44271I −0.49829 + 3.51350I
u = −0.479273 − 0.936834I
a = 0.544317 − 0.502084I
b = −0.776571 − 0.974090I
6.34798 + 5.44271I −0.49829 − 3.51350I
u = −0.563663 + 0.911236I
a = 0.646213 − 0.436873I
b = −0.842489 − 0.787076I
6.92874 + 0.59688I 0.46758 − 1.80507I
u = −0.563663 − 0.911236I
a = 0.646213 + 0.436873I
b = −0.842489 + 0.787076I
6.92874 − 0.59688I 0.46758 + 1.80507I
u = 0.903290 + 0.591334I
a = −1.72740 − 1.15219I
b = 0.719637 − 0.929655I
1.14086 − 5.11531I −4.18255 + 5.48464I
u = 0.903290 − 0.591334I
a = −1.72740 + 1.15219I
b = 0.719637 + 0.929655I
1.14086 + 5.11531I −4.18255 − 5.48464I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.073950 + 0.294320I
a = 1.15104 + 1.96262I
b = −0.071208 + 0.875733I
−3.46537 − 1.05922I −11.39395 + 0.37058I
u = 1.073950 − 0.294320I
a = 1.15104 − 1.96262I
b = −0.071208 − 0.875733I
−3.46537 + 1.05922I −11.39395 − 0.37058I
u = −1.012760 + 0.537221I
a = −0.77689 + 2.25052I
b = 0.204213 + 1.096690I
−1.91594 + 5.41987I −7.35697 − 6.54919I
u = −1.012760 − 0.537221I
a = −0.77689 − 2.25052I
b = 0.204213 − 1.096690I
−1.91594 − 5.41987I −7.35697 + 6.54919I
u = 0.819709
a = 0.530934
b = −0.251925
−1.19408 −8.44380
u = −0.706780 + 0.369020I
a = 0.42079 − 1.91115I
b = 0.427994 − 1.010940I
−0.62342 − 1.39976I −3.04278 + 0.06062I
u = −0.706780 − 0.369020I
a = 0.42079 + 1.91115I
b = 0.427994 + 1.010940I
−0.62342 + 1.39976I −3.04278 − 0.06062I
u = −1.089150 + 0.711472I
a = −0.490999 − 0.203095I
b = −0.865451 + 0.706038I
5.32382 + 5.36637I −1.53322 − 3.05337I
u = −1.089150 − 0.711472I
a = −0.490999 + 0.203095I
b = −0.865451 − 0.706038I
5.32382 − 5.36637I −1.53322 + 3.05337I
u = 1.306760 + 0.052319I
a = −0.27343 − 1.51011I
b = −0.691717 − 0.872891I
−0.20167 + 2.66172I −2.71477 − 3.57661I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.306760 − 0.052319I
a = −0.27343 + 1.51011I
b = −0.691717 + 0.872891I
−0.20167 − 2.66172I −2.71477 + 3.57661I
u = −1.139240 + 0.687767I
a = 1.14519 − 1.80727I
b = −0.753308 − 1.027550I
4.33274 + 11.39030I −3.28983 − 7.76664I
u = −1.139240 − 0.687767I
a = 1.14519 + 1.80727I
b = −0.753308 + 1.027550I
4.33274 − 11.39030I −3.28983 + 7.76664I
u = −0.144497 + 0.357570I
a = 1.21724 − 0.74670I
b = 0.316251 − 0.806276I
−0.33578 − 1.50728I −2.97928 + 4.31266I
u = −0.144497 − 0.357570I
a = 1.21724 + 0.74670I
b = 0.316251 + 0.806276I
−0.33578 + 1.50728I −2.97928 − 4.31266I
7
II. I
u
2
= h2b − a − 1, a
2
+ 3, u − 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
1
a
4
=
1
1
a
7
=
−1
0
a
9
=
a
1
2
a +
1
2
a
2
=
1
2
a −
1
2
1
2
a −
1
2
a
5
=
1
1
a
1
=
1
2
a −
1
2
1
2
a −
1
2
a
8
=
1
2
a −
1
2
1
2
a +
1
2
a
8
=
1
2
a −
1
2
1
2
a +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2a − 9
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
2
c
2
, c
7
u
2
− u + 1
c
3
, c
5
(u − 1)
2
c
6
(u + 1)
2
c
8
, c
9
u
2
+ u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
2
c
2
, c
7
, c
8
c
9
y
2
+ y + 1
c
3
, c
5
, c
6
(y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.73205I
b = 0.500000 + 0.866025I
−1.64493 + 2.02988I −9.00000 − 3.46410I
u = 1.00000
a = − 1.73205I
b = 0.500000 − 0.866025I
−1.64493 − 2.02988I −9.00000 + 3.46410I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
u
2
(u
25
− u
24
+ ··· + 4u + 4)
c
2
(u
2
− u + 1)(u
25
+ 2u
24
+ ··· + 3u + 1)
c
3
((u − 1)
2
)(u
25
− 3u
24
+ ··· − 4u + 1)
c
5
((u − 1)
2
)(u
25
+ 11u
24
+ ··· − 2u + 1)
c
6
((u + 1)
2
)(u
25
− 3u
24
+ ··· − 4u + 1)
c
7
(u
2
− u + 1)(u
25
+ 8u
24
+ ··· + 11u − 1)
c
8
(u
2
+ u + 1)(u
25
+ 2u
24
+ ··· + 3u + 1)
c
9
(u
2
+ u + 1)(u
25
+ 8u
24
+ ··· + 11u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
2
(y
25
+ 15y
24
+ ··· − 88y −16)
c
2
, c
8
(y
2
+ y + 1)(y
25
+ 8y
24
+ ··· + 11y −1)
c
3
, c
6
((y −1)
2
)(y
25
− 11y
24
+ ··· − 2y −1)
c
5
((y −1)
2
)(y
25
+ 9y
24
+ ··· − 2y −1)
c
7
, c
9
(y
2
+ y + 1)(y
25
+ 20y
24
+ ··· + 251y −1)
13
