12a
0262
(K12a
0262
)
A knot diagram
1
Linearized knot diagam
3 6 7 11 9 2 12 5 1 8 4 10
Solving Sequence
3,6
2 7
1,9
10 5 8 11 4 12
c
2
c
6
c
1
c
9
c
5
c
8
c
10
c
4
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h8u
32
− 16u
31
+ ··· + b + 9, u
32
− 3u
31
+ ··· + a + 3, u
33
− 2u
32
+ ··· + 2u − 1i
* 1 irreducible components of dim
C
= 0, with total 33 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
= h8u
32
−16u
31
+· · ·+b+9, u
32
−3u
31
+· · ·+a+3, u
33
−2u
32
+· · ·+2u−1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
9
=
−u
32
+ 3u
31
+ ··· + 10u − 3
−8u
32
+ 16u
31
+ ··· + 27u − 9
a
10
=
4u
31
− 8u
30
+ ··· + 19u − 8
−4u
32
+ 6u
31
+ ··· + 14u − 4
a
5
=
3u
32
− 12u
31
+ ··· − 24u + 9
2u
32
− 3u
31
+ ··· − 15u + 4
a
8
=
12u
32
− 35u
31
+ ··· − 48u + 17
−6u
32
+ 11u
31
+ ··· + 10u − 3
a
11
=
12u
32
− 15u
31
+ ··· + 3u − 14
3u
32
− 11u
31
+ ··· − 22u + 12
a
4
=
u
4
+ u
2
+ 1
u
6
+ 2u
4
+ u
2
a
12
=
14u
32
− 23u
31
+ ··· − 9u − 8
2u
32
− 9u
31
+ ··· − 20u + 11
(ii) Obstruction class = 1
(iii) Cusp Shapes = 20u
32
− 13u
31
+ 159u
30
− 58u
29
+ 575u
28
− 25u
27
+ 1135u
26
+
552u
25
+ 1074u
24
+ 2154u
23
− 267u
22
+ 4203u
21
− 1896u
20
+ 4776u
19
− 1567u
18
+
2737u
17
+ 1151u
16
− 350u
15
+ 3593u
14
− 1677u
13
+ 3251u
12
− 624u
11
+ 989u
10
+
1026u
9
− 732u
8
+ 1548u
7
− 975u
6
+ 992u
5
− 556u
4
+ 318u
3
− 196u
2
+ 35u − 42
2
(iv) u-Polynomials at the component
3
Crossings u-Polynomials at each crossing
c
1
u
33
− 18u
32
+ ··· − 12u + 1
c
2
u
33
− 2u
32
+ ··· + 2u − 1
c
3
u
33
+ 2u
32
+ ··· − 2u − 1
c
4
u
33
+ u
32
+ ··· − u − 1
c
5
u
33
+ 3u
32
+ ··· + 3u − 1
c
6
u
33
+ 2u
32
+ ··· + 2u + 1
c
7
u
33
+ 2u
32
+ ··· + 2u + 1
c
8
u
33
− 3u
32
+ ··· + 3u + 1
c
9
u
33
+ 5u
32
+ ··· + 5u + 1
c
10
u
33
+ 6u
32
+ ··· + 12u + 1
c
11
u
33
− u
32
+ ··· − u + 1
c
12
u
33
− 5u
32
+ ··· + 5u − 1
4
5
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
33
− 2y
32
+ ··· + 8y −1
c
2
, c
6
y
33
+ 18y
32
+ ··· − 12y −1
c
3
y
33
− 22y
32
+ ··· − 26y −1
c
4
, c
11
y
33
+ 23y
32
+ ··· + 13y −1
c
5
, c
8
y
33
− 37y
32
+ ··· + 13y −1
c
7
y
33
− 10y
32
+ ··· + 34y −1
c
9
, c
12
y
33
+ 21y
32
+ ··· − 13y −1
c
10
y
33
− 10y
32
+ ··· + 30y −1
6
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.415757 + 0.879714I
a = −0.310927 + 0.116953I
b = 1.245100 + 0.447794I
−0.28049 − 3.92650I −4.28046 + 8.45501I
u = −0.415757 − 0.879714I
a = −0.310927 − 0.116953I
b = 1.245100 − 0.447794I
−0.28049 + 3.92650I −4.28046 − 8.45501I
u = 0.889089 + 0.341493I
a = −1.334450 + 0.044982I
b = −0.835473 − 0.925109I
−6.37279 − 2.89979I −7.35601 + 2.74561I
u = 0.889089 − 0.341493I
a = −1.334450 − 0.044982I
b = −0.835473 + 0.925109I
−6.37279 + 2.89979I −7.35601 − 2.74561I
u = −0.925009 + 0.139920I
a = 1.47738 + 0.02503I
b = 0.640026 − 0.713052I
−7.59997 + 1.16632I −5.62381 + 0.53430I
u = −0.925009 − 0.139920I
a = 1.47738 − 0.02503I
b = 0.640026 + 0.713052I
−7.59997 − 1.16632I −5.62381 − 0.53430I
u = 0.362523 + 1.024310I
a = 0.746326 − 0.488114I
b = 2.02961 + 1.06595I
−6.88884 − 1.47204I −7.24631 − 0.05525I
u = 0.362523 − 1.024310I
a = 0.746326 + 0.488114I
b = 2.02961 − 1.06595I
−6.88884 + 1.47204I −7.24631 + 0.05525I
u = −0.450656 + 0.786575I
a = 0.026429 + 0.336633I
b = −0.047044 + 1.053090I
0.012283 + 0.311664I 3.64526 − 0.95783I
u = −0.450656 − 0.786575I
a = 0.026429 − 0.336633I
b = −0.047044 − 1.053090I
0.012283 − 0.311664I 3.64526 + 0.95783I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.699677 + 0.526180I
a = −0.998558 + 0.221434I
b = −0.310310 + 0.195387I
−3.61554 − 3.10551I −5.05153 + 3.45408I
u = 0.699677 − 0.526180I
a = −0.998558 − 0.221434I
b = −0.310310 − 0.195387I
−3.61554 + 3.10551I −5.05153 − 3.45408I
u = −0.461345 + 1.031380I
a = −0.309219 − 0.530900I
b = −0.588642 + 0.498459I
−1.18585 − 3.17980I −8.28309 + 4.12212I
u = −0.461345 − 1.031380I
a = −0.309219 + 0.530900I
b = −0.588642 − 0.498459I
−1.18585 + 3.17980I −8.28309 − 4.12212I
u = 0.319392 + 1.139410I
a = 0.801591 − 1.128270I
b = 0.645311 − 0.002396I
−10.97950 + 0.06682I −10.54672 − 0.14143I
u = 0.319392 − 1.139410I
a = 0.801591 + 1.128270I
b = 0.645311 + 0.002396I
−10.97950 − 0.06682I −10.54672 + 0.14143I
u = 0.567598 + 1.073240I
a = −0.024768 − 0.681279I
b = 0.029381 − 0.576170I
−5.33194 + 8.00776I −7.58671 − 6.67740I
u = 0.567598 − 1.073240I
a = −0.024768 + 0.681279I
b = 0.029381 + 0.576170I
−5.33194 − 8.00776I −7.58671 + 6.67740I
u = 0.213741 + 0.747529I
a = 0.44378 + 1.36082I
b = −2.65599 − 0.82648I
−5.67246 + 4.09164I −1.47347 − 8.31889I
u = 0.213741 − 0.747529I
a = 0.44378 − 1.36082I
b = −2.65599 + 0.82648I
−5.67246 − 4.09164I −1.47347 + 8.31889I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.044555 + 0.763029I
a = 0.26951 + 2.29758I
b = −1.035990 + 0.506927I
−8.91211 + 1.56145I −7.74657 − 2.97681I
u = 0.044555 − 0.763029I
a = 0.26951 − 2.29758I
b = −1.035990 − 0.506927I
−8.91211 − 1.56145I −7.74657 + 2.97681I
u = 0.454510 + 1.153340I
a = 0.284320 − 0.958356I
b = 0.99905 − 1.93065I
−3.43511 + 4.06164I −1.84707 − 3.29080I
u = 0.454510 − 1.153340I
a = 0.284320 + 0.958356I
b = 0.99905 + 1.93065I
−3.43511 − 4.06164I −1.84707 + 3.29080I
u = −0.369754 + 1.204530I
a = −0.469785 − 1.239170I
b = −0.478866 − 1.069080I
−11.94570 − 2.89125I −9.51390 + 3.40460I
u = −0.369754 − 1.204530I
a = −0.469785 + 1.239170I
b = −0.478866 + 1.069080I
−11.94570 + 2.89125I −9.51390 − 3.40460I
u = −0.491464 + 1.224780I
a = −0.115049 − 1.120650I
b = −1.80600 − 1.71244I
−11.05150 − 6.20149I −8.51895 + 3.78482I
u = −0.491464 − 1.224780I
a = −0.115049 + 1.120650I
b = −1.80600 + 1.71244I
−11.05150 + 6.20149I −8.51895 − 3.78482I
u = 0.570176 + 1.190950I
a = −0.053100 − 0.985645I
b = 1.63829 − 1.28317I
−9.06173 + 8.29388I −8.92407 − 7.11050I
u = 0.570176 − 1.190950I
a = −0.053100 + 0.985645I
b = 1.63829 + 1.28317I
−9.06173 − 8.29388I −8.92407 + 7.11050I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.316420 + 0.565399I
a = 0.460695 + 1.226470I
b = 1.380710 + 0.272067I
0.381875 − 0.429677I −0.97944 − 3.23646I
u = −0.316420 − 0.565399I
a = 0.460695 − 1.226470I
b = 1.380710 − 0.272067I
0.381875 + 0.429677I −0.97944 + 3.23646I
u = 0.618289
a = −1.78836
b = 0.301684
−0.353891 2.66570
10
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
33
− 18u
32
+ ··· − 12u + 1
c
2
u
33
− 2u
32
+ ··· + 2u − 1
c
3
u
33
+ 2u
32
+ ··· − 2u − 1
c
4
u
33
+ u
32
+ ··· − u − 1
c
5
u
33
+ 3u
32
+ ··· + 3u − 1
c
6
u
33
+ 2u
32
+ ··· + 2u + 1
c
7
u
33
+ 2u
32
+ ··· + 2u + 1
c
8
u
33
− 3u
32
+ ··· + 3u + 1
c
9
u
33
+ 5u
32
+ ··· + 5u + 1
c
10
u
33
+ 6u
32
+ ··· + 12u + 1
c
11
u
33
− u
32
+ ··· − u + 1
c
12
u
33
− 5u
32
+ ··· + 5u − 1
11
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
33
− 2y
32
+ ··· + 8y −1
c
2
, c
6
y
33
+ 18y
32
+ ··· − 12y −1
c
3
y
33
− 22y
32
+ ··· − 26y −1
c
4
, c
11
y
33
+ 23y
32
+ ··· + 13y −1
c
5
, c
8
y
33
− 37y
32
+ ··· + 13y −1
c
7
y
33
− 10y
32
+ ··· + 34y −1
c
9
, c
12
y
33
+ 21y
32
+ ··· − 13y −1
c
10
y
33
− 10y
32
+ ··· + 30y −1
12
