9
22
(K9a
2
)
A knot diagram
1
Linearized knot diagam
4 5 8 6 2 9 1 3 7
Solving Sequence
6,9
7
1,2
5 3 4 8
c
6
c
9
c
5
c
2
c
4
c
8
c
1
, c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
22
− 2u
21
+ ··· + 2b + 1, −u
6
+ 3u
4
− 2u
3
− 2u
2
+ a + 4u − 1, u
23
+ 3u
22
+ ··· − u − 1i
I
u
2
= hb
2
− b + 1, a + 1, u − 1i
* 2 irreducible components of dim
C
= 0, with total 25 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
22
− 2u
21
+ · · · + 2b + 1, −u
6
+ 3u
4
− 2u
3
− 2u
2
+ a + 4u −
1, u
23
+ 3u
22
+ · · · − u − 1i
(i) Arc colorings
a
6
=
1
0
a
9
=
0
u
a
7
=
1
u
2
a
1
=
−u
−u
3
+ u
a
2
=
u
6
− 3u
4
+ 2u
3
+ 2u
2
− 4u + 1
1
2
u
22
+ u
21
+ ··· + 2u
2
−
1
2
a
5
=
3
2
u
22
+ 3u
21
+ ··· − 2u
2
−
1
2
−
5
2
u
22
− 4u
21
+ ··· + u +
3
2
a
3
=
−2u
22
− 3u
21
+ ··· − u + 1
3
2
u
22
+ 2u
21
+ ··· − u
2
−
1
2
a
4
=
−u
22
− u
21
+ ··· + u + 1
−
5
2
u
22
− 4u
21
+ ··· + u +
3
2
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 3u
22
+ 3u
21
−32u
20
−19u
19
+ 155u
18
+ 15u
17
−432u
16
+ 194u
15
+ 690u
14
−758u
13
−
450u
12
+ 1221u
11
−359u
10
−839u
9
+ 820u
8
−2u
7
−401u
6
+ 227u
5
−22u
4
−21u
3
+ u
2
+ u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
23
− 2u
22
+ ··· + 18u − 9
c
2
, c
5
u
23
+ 2u
22
+ ··· − 2u − 1
c
3
, c
8
u
23
− u
22
+ ··· + 8u + 4
c
4
u
23
+ 12u
22
+ ··· − 2u − 1
c
6
, c
7
, c
9
u
23
− 3u
22
+ ··· − u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
23
− 12y
22
+ ··· − 450y −81
c
2
, c
5
y
23
+ 12y
22
+ ··· − 2y −1
c
3
, c
8
y
23
+ 15y
22
+ ··· − 40y −16
c
4
y
23
+ 24y
21
+ ··· + 10y −1
c
6
, c
7
, c
9
y
23
− 23y
22
+ ··· − 7y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.696926 + 0.678563I
a = −0.371551 − 0.457637I
b = 0.386982 + 1.120880I
−4.15124 + 1.33135I −7.15950 − 0.67575I
u = 0.696926 − 0.678563I
a = −0.371551 + 0.457637I
b = 0.386982 − 1.120880I
−4.15124 − 1.33135I −7.15950 + 0.67575I
u = 1.026370 + 0.230969I
a = −1.271710 − 0.069358I
b = 0.179248 − 0.701899I
−2.10210 − 0.88878I −6.39291 − 0.92577I
u = 1.026370 − 0.230969I
a = −1.271710 + 0.069358I
b = 0.179248 + 0.701899I
−2.10210 + 0.88878I −6.39291 + 0.92577I
u = 0.443194 + 0.830987I
a = −1.84438 + 0.30451I
b = 0.501837 − 1.137100I
−3.32060 − 6.47771I −4.77780 + 6.52194I
u = 0.443194 − 0.830987I
a = −1.84438 − 0.30451I
b = 0.501837 + 1.137100I
−3.32060 + 6.47771I −4.77780 − 6.52194I
u = 0.411789 + 0.657552I
a = −1.215710 − 0.639418I
b = 0.657802 + 0.201077I
−0.66432 − 2.00215I −1.23588 + 3.62705I
u = 0.411789 − 0.657552I
a = −1.215710 + 0.639418I
b = 0.657802 − 0.201077I
−0.66432 + 2.00215I −1.23588 − 3.62705I
u = 1.31043
a = −0.0893487
b = −0.616508
−2.78711 −2.32390
u = −1.349890 + 0.050765I
a = 1.185670 + 0.215112I
b = −0.730473 − 0.812317I
−3.41052 + 2.74438I −6.00137 − 3.42075I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.349890 − 0.050765I
a = 1.185670 − 0.215112I
b = −0.730473 + 0.812317I
−3.41052 − 2.74438I −6.00137 + 3.42075I
u = 1.42968 + 0.09520I
a = 0.89149 + 1.36719I
b = −0.449028 + 1.143790I
−5.84331 − 3.99588I −6.60901 + 3.49800I
u = 1.42968 − 0.09520I
a = 0.89149 − 1.36719I
b = −0.449028 − 1.143790I
−5.84331 + 3.99588I −6.60901 − 3.49800I
u = −1.48042 + 0.24817I
a = −0.537692 + 0.556573I
b = 0.868940 − 0.243856I
−6.80889 + 5.35900I −4.49542 − 3.06793I
u = −1.48042 − 0.24817I
a = −0.537692 − 0.556573I
b = 0.868940 + 0.243856I
−6.80889 − 5.35900I −4.49542 + 3.06793I
u = −1.51052 + 0.30516I
a = −1.54699 + 0.69863I
b = 0.565955 + 1.190510I
−9.6533 + 10.6207I −7.02627 − 6.45650I
u = −1.51052 − 0.30516I
a = −1.54699 − 0.69863I
b = 0.565955 − 1.190510I
−9.6533 − 10.6207I −7.02627 + 6.45650I
u = −1.55320 + 0.17815I
a = −0.002579 − 0.587301I
b = 0.282827 − 1.245840I
−11.61980 + 1.64388I −9.30470 − 0.40272I
u = −1.55320 − 0.17815I
a = −0.002579 + 0.587301I
b = 0.282827 + 1.245840I
−11.61980 − 1.64388I −9.30470 + 0.40272I
u = 0.008249 + 0.425434I
a = 0.49224 − 1.83322I
b = −0.476560 + 0.630579I
0.71923 − 1.37448I 2.70178 + 4.35124I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.008249 − 0.425434I
a = 0.49224 + 1.83322I
b = −0.476560 − 0.630579I
0.71923 + 1.37448I 2.70178 − 4.35124I
u = −0.277376 + 0.277332I
a = 2.26589 − 1.32800I
b = −0.479277 − 0.962679I
−0.27712 + 2.59653I 1.46303 − 3.78636I
u = −0.277376 − 0.277332I
a = 2.26589 + 1.32800I
b = −0.479277 + 0.962679I
−0.27712 − 2.59653I 1.46303 + 3.78636I
7
II. I
u
2
= hb
2
− b + 1, a + 1, u − 1i
(i) Arc colorings
a
6
=
1
0
a
9
=
0
1
a
7
=
1
1
a
1
=
−1
0
a
2
=
−1
b
a
5
=
−b + 1
b − 1
a
3
=
0
b − 1
a
4
=
0
b − 1
a
8
=
0
1
a
8
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4b − 1
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
u
2
− u + 1
c
2
u
2
+ u + 1
c
3
, c
8
u
2
c
6
, c
7
(u − 1)
2
c
9
(u + 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
y
2
+ y + 1
c
3
, c
8
y
2
c
6
, c
7
, c
9
(y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 0.500000 + 0.866025I
−1.64493 + 2.02988I −3.00000 − 3.46410I
u = 1.00000
a = −1.00000
b = 0.500000 − 0.866025I
−1.64493 − 2.02988I −3.00000 + 3.46410I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
23
− 2u
22
+ ··· + 18u − 9)
c
2
(u
2
+ u + 1)(u
23
+ 2u
22
+ ··· − 2u − 1)
c
3
, c
8
u
2
(u
23
− u
22
+ ··· + 8u + 4)
c
4
(u
2
− u + 1)(u
23
+ 12u
22
+ ··· − 2u − 1)
c
5
(u
2
− u + 1)(u
23
+ 2u
22
+ ··· − 2u − 1)
c
6
, c
7
((u − 1)
2
)(u
23
− 3u
22
+ ··· − u + 1)
c
9
((u + 1)
2
)(u
23
− 3u
22
+ ··· − u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)(y
23
− 12y
22
+ ··· − 450y −81)
c
2
, c
5
(y
2
+ y + 1)(y
23
+ 12y
22
+ ··· − 2y −1)
c
3
, c
8
y
2
(y
23
+ 15y
22
+ ··· − 40y −16)
c
4
(y
2
+ y + 1)(y
23
+ 24y
21
+ ··· + 10y −1)
c
6
, c
7
, c
9
((y −1)
2
)(y
23
− 23y
22
+ ··· − 7y −1)
13
