9
23
(K9a
16
)
A knot diagram
1
Linearized knot diagam
8 6 2 9 7 3 4 1 5
Solving Sequence
1,5
9 4 8 2 3 7 6
c
9
c
4
c
8
c
1
c
3
c
7
c
5
c
2
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
5
− u
4
+ 2u − 1i
I
u
2
= hu
16
− u
15
− 2u
14
+ 3u
13
+ 4u
12
− 7u
11
− 3u
10
+ 10u
9
− 9u
7
+ 3u
6
+ 5u
5
− 4u
4
+ 2u
2
− 2u + 1i
I
u
3
= hu + 1i
* 3 irreducible components of dim
C
= 0, with total 22 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
5
− u
4
+ 2u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
9
=
1
−u
2
a
4
=
u
−u
3
+ u
a
8
=
−u
2
+ 1
−u
2
a
2
=
u
4
− u
2
+ 1
u
4
a
3
=
−u
2
+ 1
u
4
− u
3
− 2u
2
+ 1
a
7
=
u
−u
4
− u
3
+ u
2
+ u − 1
a
6
=
−u
3
u
4
− u
3
− u
2
− 2u + 2
a
6
=
−u
3
u
4
− u
3
− u
2
− 2u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
4
+ 4u
3
+ 4u
2
− 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
8
u
5
+ u
4
+ 4u
3
+ 2u
2
+ 4u + 1
c
2
, c
4
, c
6
c
9
u
5
+ u
4
+ 2u + 1
c
7
u
5
− 4u
4
+ 9u
3
− 9u
2
+ 4u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
8
y
5
+ 7y
4
+ 20y
3
+ 26y
2
+ 12y −1
c
2
, c
4
, c
6
c
9
y
5
− y
4
+ 4y
3
− 2y
2
+ 4y −1
c
7
y
5
+ 2y
4
+ 17y
3
+ 23y
2
+ 88y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.760506 + 0.815892I
6.30195 + 1.13825I −3.90398 − 2.34058I
u = −0.760506 − 0.815892I
6.30195 − 1.13825I −3.90398 + 2.34058I
u = 1.001870 + 0.741764I
4.78344 − 10.61130I −6.76481 + 7.85454I
u = 1.001870 − 0.741764I
4.78344 + 10.61130I −6.76481 − 7.85454I
u = 0.517281
−0.786636 −12.6620
5
II. I
u
2
= hu
16
− u
15
− 2u
14
+ 3u
13
+ 4u
12
− 7u
11
− 3u
10
+ 10u
9
− 9u
7
+ 3u
6
+
5u
5
− 4u
4
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
9
=
1
−u
2
a
4
=
u
−u
3
+ u
a
8
=
−u
2
+ 1
−u
2
a
2
=
u
4
− u
2
+ 1
u
4
a
3
=
−u
11
+ 2u
9
− 4u
7
+ 4u
5
− 3u
3
+ 2u
−u
11
+ u
9
− 2u
7
+ u
5
− u
3
+ u
a
7
=
−u
6
+ u
4
− 2u
2
+ 1
u
8
− 2u
6
+ 2u
4
− 2u
2
a
6
=
−u
13
+ 2u
11
− 5u
9
+ 6u
7
− 6u
5
+ 4u
3
− u
u
15
− 3u
13
+ 6u
11
− 9u
9
+ 8u
7
− 6u
5
+ 2u
3
+ u
a
6
=
−u
13
+ 2u
11
− 5u
9
+ 6u
7
− 6u
5
+ 4u
3
− u
u
15
− 3u
13
+ 6u
11
− 9u
9
+ 8u
7
− 6u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
12
− 8u
10
+ 16u
8
− 4u
7
− 16u
6
+ 8u
5
+ 12u
4
− 8u
3
− 4u
2
+ 4u − 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
8
u
16
+ 5u
15
+ ··· − 4u
2
+ 1
c
2
, c
4
, c
6
c
9
u
16
+ u
15
+ ··· + 2u + 1
c
7
(u
8
+ 2u
7
+ 3u
6
+ u
4
+ 2u
2
+ 2u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
8
y
16
+ 11y
15
+ ··· − 8y + 1
c
2
, c
4
, c
6
c
9
y
16
− 5y
15
+ ··· − 4y
2
+ 1
c
7
(y
8
+ 2y
7
+ 11y
6
+ 10y
5
+ 7y
4
+ 10y
3
+ 6y
2
+ 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.017320 + 0.191091I
−1.08130 + 5.29622I −12.10789 − 6.28296I
u = −1.017320 − 0.191091I
−1.08130 − 5.29622I −12.10789 + 6.28296I
u = 0.908738 + 0.252477I
−0.328380 − 0.252703I −10.38985 + 0.96511I
u = 0.908738 − 0.252477I
−0.328380 + 0.252703I −10.38985 − 0.96511I
u = 0.708362 + 0.611401I
−0.328380 + 0.252703I −10.38985 − 0.96511I
u = 0.708362 − 0.611401I
−0.328380 − 0.252703I −10.38985 + 0.96511I
u = 0.724199 + 0.826388I
5.63436 + 4.73566I −5.11364 − 2.91588I
u = 0.724199 − 0.826388I
5.63436 − 4.73566I −5.11364 + 2.91588I
u = −0.866890 + 0.696274I
2.35506 + 2.67607I −4.38861 − 3.32415I
u = −0.866890 − 0.696274I
2.35506 − 2.67607I −4.38861 + 3.32415I
u = 0.960503 + 0.654282I
−1.08130 − 5.29622I −12.10789 + 6.28296I
u = 0.960503 − 0.654282I
−1.08130 + 5.29622I −12.10789 − 6.28296I
u = −0.977539 + 0.749941I
5.63436 + 4.73566I −5.11364 − 2.91588I
u = −0.977539 − 0.749941I
5.63436 − 4.73566I −5.11364 + 2.91588I
u = 0.059947 + 0.622852I
2.35506 − 2.67607I −4.38861 + 3.32415I
u = 0.059947 − 0.622852I
2.35506 + 2.67607I −4.38861 − 3.32415I
9
III. I
u
3
= hu + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
−1
a
9
=
1
−1
a
4
=
−1
0
a
8
=
0
−1
a
2
=
1
1
a
3
=
0
1
a
7
=
−1
−1
a
6
=
1
0
a
6
=
1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = −18
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
8
u + 1
c
2
, c
4
, c
6
c
7
, c
9
u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
−4.93480 −18.0000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
8
(u + 1)(u
5
+ u
4
+ ··· + 4u + 1)(u
16
+ 5u
15
+ ··· − 4u
2
+ 1)
c
2
, c
4
, c
6
c
9
(u − 1)(u
5
+ u
4
+ 2u + 1)(u
16
+ u
15
+ ··· + 2u + 1)
c
7
(u − 1)(u
5
− 4u
4
+ 9u
3
− 9u
2
+ 4u + 4)
· (u
8
+ 2u
7
+ 3u
6
+ u
4
+ 2u
2
+ 2u + 1)
2
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
8
(y −1)(y
5
+ 7y
4
+ ··· + 12y −1)(y
16
+ 11y
15
+ ··· − 8y + 1)
c
2
, c
4
, c
6
c
9
(y −1)(y
5
− y
4
+ ··· + 4y −1)(y
16
− 5y
15
+ ··· − 4y
2
+ 1)
c
7
(y −1)(y
5
+ 2y
4
+ 17y
3
+ 23y
2
+ 88y −16)
· (y
8
+ 2y
7
+ 11y
6
+ 10y
5
+ 7y
4
+ 10y
3
+ 6y
2
+ 1)
2
15
