12a
0143
(K12a
0143
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 10 11 12 4 1 6 7 8
Solving Sequence
5,10
6 11 7
3,12
2 1 4 9 8
c
5
c
10
c
6
c
11
c
2
c
1
c
4
c
9
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
44
− u
43
+ ··· + b + 1, −u
26
+ 19u
24
+ ··· + a − 1, u
45
+ 2u
44
+ ··· + u − 1i
I
u
2
= hb + 1, a + 1, u
3
− u
2
− 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 48 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
44
−u
43
+· · ·+b+1, −u
26
+19u
24
+· · ·+a−1, u
45
+2u
44
+· · ·+u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
11
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
u
26
− 19u
24
+ ··· + 2u + 1
u
44
+ u
43
+ ··· + u − 1
a
12
=
u
3
− 2u
u
5
− 3u
3
+ u
a
2
=
u
44
+ u
43
+ ··· − 19u
3
+ 3u
u
44
+ u
43
+ ··· + u − 1
a
1
=
u
5
− 4u
3
+ 3u
u
7
− 5u
5
+ 6u
3
− u
a
4
=
2u
44
+ 2u
43
+ ··· + 4u
2
+ 3u
u
44
+ u
43
+ ··· + 2u − 1
a
9
=
u
11
− 8u
9
+ 22u
7
− 24u
5
+ 9u
3
u
13
− 9u
11
+ 29u
9
− 40u
7
+ 22u
5
− 3u
3
+ u
a
8
=
u
4
− 3u
2
+ 1
u
6
− 4u
4
+ 3u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 5u
44
+ 4u
43
+ ··· − 4u − 19
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
45
+ 22u
44
+ ··· + 74u + 1
c
2
, c
4
u
45
− 4u
44
+ ··· + 6u + 1
c
3
, c
8
u
45
− u
44
+ ··· + 12u + 8
c
5
, c
6
, c
7
c
10
, c
11
, c
12
u
45
− 2u
44
+ ··· + u + 1
c
9
u
45
+ 8u
44
+ ··· + 409u + 55
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
45
+ 6y
44
+ ··· + 4358y −1
c
2
, c
4
y
45
− 22y
44
+ ··· + 74y −1
c
3
, c
8
y
45
+ 21y
44
+ ··· + 16y −64
c
5
, c
6
, c
7
c
10
, c
11
, c
12
y
45
− 64y
44
+ ··· + 13y −1
c
9
y
45
− 4y
44
+ ··· − 43479y −3025
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.962969 + 0.169603I
a = 0.440533 − 0.958645I
b = 0.692370 + 0.677447I
−0.28984 − 2.31607I −13.48731 + 4.06907I
u = 0.962969 − 0.169603I
a = 0.440533 + 0.958645I
b = 0.692370 − 0.677447I
−0.28984 + 2.31607I −13.48731 − 4.06907I
u = −1.148590 + 0.071392I
a = 0.801348 + 0.494679I
b = −0.427212 − 0.486489I
−5.22276 + 0.37204I 0
u = −1.148590 − 0.071392I
a = 0.801348 − 0.494679I
b = −0.427212 + 0.486489I
−5.22276 − 0.37204I 0
u = 1.176560 + 0.211087I
a = −0.653062 + 0.188078I
b = 0.348834 − 0.855887I
−2.36170 − 5.17327I 0
u = 1.176560 − 0.211087I
a = −0.653062 − 0.188078I
b = 0.348834 + 0.855887I
−2.36170 + 5.17327I 0
u = 0.763709 + 0.251528I
a = −0.0103561 − 0.1259090I
b = 0.955005 − 0.546675I
−1.03136 + 2.41021I −15.6897 − 1.6774I
u = 0.763709 − 0.251528I
a = −0.0103561 + 0.1259090I
b = 0.955005 + 0.546675I
−1.03136 − 2.41021I −15.6897 + 1.6774I
u = 1.191250 + 0.124036I
a = −1.094580 + 0.310585I
b = −1.252440 + 0.198803I
−7.65311 − 1.99196I 0
u = 1.191250 − 0.124036I
a = −1.094580 − 0.310585I
b = −1.252440 − 0.198803I
−7.65311 + 1.99196I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.191140 + 0.163730I
a = −0.78548 − 1.96254I
b = −1.029290 + 0.504098I
−6.91071 + 4.51785I 0
u = −1.191140 − 0.163730I
a = −0.78548 + 1.96254I
b = −1.029290 − 0.504098I
−6.91071 − 4.51785I 0
u = 1.219810 + 0.229016I
a = 1.09073 − 1.61980I
b = 1.150280 + 0.609142I
−4.75327 − 10.58690I 0
u = 1.219810 − 0.229016I
a = 1.09073 + 1.61980I
b = 1.150280 − 0.609142I
−4.75327 + 10.58690I 0
u = −1.316520 + 0.069582I
a = 0.845369 + 0.138843I
b = 0.949980 + 0.382113I
−7.86567 − 1.43041I 0
u = −1.316520 − 0.069582I
a = 0.845369 − 0.138843I
b = 0.949980 − 0.382113I
−7.86567 + 1.43041I 0
u = −0.503453 + 0.459470I
a = 0.37795 + 2.46430I
b = 1.101250 − 0.610224I
0.78058 + 8.20264I −13.8836 − 9.3349I
u = −0.503453 − 0.459470I
a = 0.37795 − 2.46430I
b = 1.101250 + 0.610224I
0.78058 − 8.20264I −13.8836 + 9.3349I
u = −0.432497 + 0.443637I
a = −1.29853 − 0.73079I
b = 0.428503 + 0.789342I
2.77707 + 2.93987I −10.12479 − 5.18879I
u = −0.432497 − 0.443637I
a = −1.29853 + 0.73079I
b = 0.428503 − 0.789342I
2.77707 − 2.93987I −10.12479 + 5.18879I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.442889 + 0.349610I
a = 0.03028 + 3.06490I
b = −0.954541 − 0.430846I
−1.64546 − 2.76575I −16.0685 + 7.3874I
u = 0.442889 − 0.349610I
a = 0.03028 − 3.06490I
b = −0.954541 + 0.430846I
−1.64546 + 2.76575I −16.0685 − 7.3874I
u = −0.147862 + 0.512639I
a = −1.34459 − 1.27617I
b = 1.043660 + 0.604880I
1.83508 − 4.99628I −10.28872 + 3.31837I
u = −0.147862 − 0.512639I
a = −1.34459 + 1.27617I
b = 1.043660 − 0.604880I
1.83508 + 4.99628I −10.28872 − 3.31837I
u = −0.229805 + 0.471597I
a = −0.25152 + 1.93643I
b = 0.527284 − 0.732325I
3.37526 + 0.10760I −7.61315 − 2.95177I
u = −0.229805 − 0.471597I
a = −0.25152 − 1.93643I
b = 0.527284 + 0.732325I
3.37526 − 0.10760I −7.61315 + 2.95177I
u = −0.426938 + 0.242547I
a = −0.516634 − 0.830332I
b = −1.145690 − 0.133170I
−2.39909 + 0.70824I −15.1495 − 9.1471I
u = −0.426938 − 0.242547I
a = −0.516634 + 0.830332I
b = −1.145690 + 0.133170I
−2.39909 − 0.70824I −15.1495 + 9.1471I
u = 0.181432 + 0.311155I
a = 2.35597 − 1.25070I
b = −0.825100 + 0.293052I
−0.905303 + 0.433735I −12.18552 + 1.89292I
u = 0.181432 − 0.311155I
a = 2.35597 + 1.25070I
b = −0.825100 − 0.293052I
−0.905303 − 0.433735I −12.18552 − 1.89292I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.346739
a = 1.01677
b = −0.122928
−0.548092 −17.9540
u = −1.73488 + 0.01556I
a = 0.165858 + 0.798942I
b = 0.805113 − 0.767877I
−10.02540 + 2.83788I 0
u = −1.73488 − 0.01556I
a = 0.165858 − 0.798942I
b = 0.805113 + 0.767877I
−10.02540 − 2.83788I 0
u = 1.77511 + 0.02023I
a = 0.581848 − 0.527304I
b = −0.425874 + 0.628346I
−15.9251 − 0.7872I 0
u = 1.77511 − 0.02023I
a = 0.581848 + 0.527304I
b = −0.425874 − 0.628346I
−15.9251 + 0.7872I 0
u = −1.77793 + 0.05224I
a = −0.414828 − 0.211707I
b = 0.314583 + 0.909747I
−13.1145 + 6.3153I 0
u = −1.77793 − 0.05224I
a = −0.414828 + 0.211707I
b = 0.314583 − 0.909747I
−13.1145 − 6.3153I 0
u = 1.78253 + 0.04085I
a = −0.78346 + 1.57093I
b = −1.063730 − 0.545729I
−17.7770 − 5.4165I 0
u = 1.78253 − 0.04085I
a = −0.78346 − 1.57093I
b = −1.063730 + 0.545729I
−17.7770 + 5.4165I 0
u = −1.78292 + 0.03166I
a = −1.096720 − 0.115819I
b = −1.302410 − 0.221345I
−18.5397 + 2.6826I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.78292 − 0.03166I
a = −1.096720 + 0.115819I
b = −1.302410 + 0.221345I
−18.5397 − 2.6826I 0
u = −1.78906 + 0.05870I
a = 1.08096 + 1.26242I
b = 1.181650 − 0.611600I
−15.7314 + 11.8740I 0
u = −1.78906 − 0.05870I
a = 1.08096 − 1.26242I
b = 1.181650 + 0.611600I
−15.7314 − 11.8740I 0
u = 1.81198 + 0.01590I
a = 0.970531 − 0.091202I
b = 0.989237 − 0.314096I
−19.4518 + 1.0384I 0
u = 1.81198 − 0.01590I
a = 0.970531 + 0.091202I
b = 0.989237 + 0.314096I
−19.4518 − 1.0384I 0
9
II. I
u
2
= hb + 1, a + 1, u
3
− u
2
− 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
11
=
−u
−u
2
− u + 1
a
7
=
−u
2
+ 1
−u
2
− u + 1
a
3
=
−1
−1
a
12
=
u
2
− 1
u
2
a
2
=
−2
−1
a
1
=
−1
0
a
4
=
−1
−1
a
9
=
u
u
a
8
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
+ u − 17
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
7
c
9
u
3
− u
2
− 2u + 1
c
10
, c
11
, c
12
u
3
+ u
2
− 2u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
8
y
3
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = −1.00000
b = −1.00000
−7.98968 −19.8020
u = 0.445042
a = −1.00000
b = −1.00000
−2.34991 −16.7530
u = 1.80194
a = −1.00000
b = −1.00000
−19.2692 −18.4450
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
45
+ 22u
44
+ ··· + 74u + 1)
c
2
((u − 1)
3
)(u
45
− 4u
44
+ ··· + 6u + 1)
c
3
, c
8
u
3
(u
45
− u
44
+ ··· + 12u + 8)
c
4
((u + 1)
3
)(u
45
− 4u
44
+ ··· + 6u + 1)
c
5
, c
6
, c
7
(u
3
− u
2
− 2u + 1)(u
45
− 2u
44
+ ··· + u + 1)
c
9
(u
3
− u
2
− 2u + 1)(u
45
+ 8u
44
+ ··· + 409u + 55)
c
10
, c
11
, c
12
(u
3
+ u
2
− 2u − 1)(u
45
− 2u
44
+ ··· + u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
3
)(y
45
+ 6y
44
+ ··· + 4358y −1)
c
2
, c
4
((y −1)
3
)(y
45
− 22y
44
+ ··· + 74y −1)
c
3
, c
8
y
3
(y
45
+ 21y
44
+ ··· + 16y −64)
c
5
, c
6
, c
7
c
10
, c
11
, c
12
(y
3
− 5y
2
+ 6y −1)(y
45
− 64y
44
+ ··· + 13y −1)
c
9
(y
3
− 5y
2
+ 6y −1)(y
45
− 4y
44
+ ··· − 43479y −3025)
15
