12a
0369
(K12a
0369
)
A knot diagram
1
Linearized knot diagam
3 6 9 10 2 11 12 1 4 5 7 8
Solving Sequence
7,11
12 8 1 9
2,6
3 5 10 4
c
11
c
7
c
12
c
8
c
6
c
2
c
5
c
10
c
4
c
1
, c
3
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−478134295627u
38
+ 149778839704u
37
+ ··· + 783747150866b − 1669436053919,
− 19258298275u
38
− 10323980760u
37
+ ··· + 783747150866a + 8187780738725,
u
39
− 2u
38
+ ··· + 10u − 1i
I
u
2
= hb + a + u − 1, a
2
+ 4au − 2a + 3, u
2
− u − 1i
I
u
3
= hb + u, a − 2u − 1, u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 45 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−4.78 × 10
11
u
38
+ 1.50 × 10
11
u
37
+ · · · + 7.84 × 10
11
b − 1.67 ×
10
12
, −1.93 × 10
10
u
38
− 1.03 × 10
10
u
37
+ · · · + 7.84 × 10
11
a + 8.19 ×
10
12
, u
39
− 2u
38
+ · · · + 10u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
8
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
9
=
−u
3
+ 2u
u
5
− 3u
3
+ u
a
2
=
0.0245721u
38
+ 0.0131726u
37
+ ··· − 11.4436u − 10.4470
0.610062u
38
− 0.191106u
37
+ ··· − 5.42326u + 2.13007
a
6
=
−u
u
a
3
=
−0.819512u
38
+ 1.59433u
37
+ ··· − 1.16490u − 11.5383
1.45415u
38
− 1.77227u
37
+ ··· − 15.7020u + 3.22140
a
5
=
4.72658u
38
− 5.95208u
37
+ ··· − 44.4581u + 17.4015
−2.05862u
38
+ 2.32509u
37
+ ··· + 22.0842u − 3.94672
a
10
=
−2.46589u
38
+ 3.88914u
37
+ ··· + 4.27839u − 17.1124
−0.582048u
38
+ 0.868548u
37
+ ··· + 13.4013u + 0.570488
a
4
=
0.883550u
38
− 0.877719u
37
+ ··· − 20.4637u − 8.99829
1.11038u
38
− 0.900266u
37
+ ··· − 10.7236u + 2.76872
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
627190064793
391873575433
u
38
+
1064333701951
391873575433
u
37
+ ··· −
1173686425629
391873575433
u +
611656068405
391873575433
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
39
+ 15u
38
+ ··· + 111u + 1
c
2
, c
5
u
39
+ 3u
38
+ ··· + 3u + 1
c
3
, c
4
, c
9
c
10
u
39
− u
38
+ ··· + 4u − 4
c
6
, c
7
, c
8
c
11
, c
12
u
39
+ 2u
38
+ ··· + 10u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
39
+ 25y
38
+ ··· + 6327y −1
c
2
, c
5
y
39
− 15y
38
+ ··· + 111y −1
c
3
, c
4
, c
9
c
10
y
39
− 49y
38
+ ··· + 368y −16
c
6
, c
7
, c
8
c
11
, c
12
y
39
− 54y
38
+ ··· + 102y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.020110 + 0.131184I
a = −1.009270 + 0.527233I
b = 0.286806 − 1.184210I
2.30767 − 2.27298I 10.27248 + 3.27626I
u = −1.020110 − 0.131184I
a = −1.009270 − 0.527233I
b = 0.286806 + 1.184210I
2.30767 + 2.27298I 10.27248 − 3.27626I
u = −1.04641
a = 3.22469
b = −1.92904
7.09140 13.7060
u = 1.073560 + 0.109186I
a = −0.557430 − 0.305343I
b = −0.051344 − 0.282062I
5.55859 + 1.07065I 15.4669 − 1.4412I
u = 1.073560 − 0.109186I
a = −0.557430 + 0.305343I
b = −0.051344 + 0.282062I
5.55859 − 1.07065I 15.4669 + 1.4412I
u = 1.062930 + 0.284051I
a = −1.64402 − 0.62028I
b = 0.93533 + 1.43760I
4.46872 + 6.29211I 12.7084 − 7.5348I
u = 1.062930 − 0.284051I
a = −1.64402 + 0.62028I
b = 0.93533 − 1.43760I
4.46872 − 6.29211I 12.7084 + 7.5348I
u = −1.109570 + 0.405465I
a = −2.09516 + 0.56202I
b = 1.43927 − 1.50328I
12.9852 − 8.7765I 14.1494 + 5.9356I
u = −1.109570 − 0.405465I
a = −2.09516 − 0.56202I
b = 1.43927 + 1.50328I
12.9852 + 8.7765I 14.1494 − 5.9356I
u = 0.801500
a = 2.55587
b = −1.10388
−0.106375 16.7010
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.476965 + 0.626572I
a = 1.09357 + 0.93891I
b = 0.323319 − 1.133120I
9.07010 − 0.79273I 12.42094 − 0.46386I
u = 0.476965 − 0.626572I
a = 1.09357 − 0.93891I
b = 0.323319 + 1.133120I
9.07010 + 0.79273I 12.42094 + 0.46386I
u = 0.783899
a = 0.595351
b = −1.19130
5.58284 18.2860
u = 0.311989 + 0.683555I
a = 0.474675 − 0.108825I
b = −0.86174 − 1.47607I
8.56307 + 5.06640I 11.02566 − 5.09114I
u = 0.311989 − 0.683555I
a = 0.474675 + 0.108825I
b = −0.86174 + 1.47607I
8.56307 − 5.06640I 11.02566 + 5.09114I
u = −1.213320 + 0.300422I
a = −0.625507 + 1.072000I
b = −0.039879 − 0.445738I
14.4735 − 2.4041I 15.9891 + 0.I
u = −1.213320 − 0.300422I
a = −0.625507 − 1.072000I
b = −0.039879 + 0.445738I
14.4735 + 2.4041I 15.9891 + 0.I
u = −0.268654 + 0.522744I
a = 0.312554 + 0.528929I
b = −0.417052 + 1.216500I
0.30789 − 3.54473I 8.22087 + 8.01132I
u = −0.268654 − 0.522744I
a = 0.312554 − 0.528929I
b = −0.417052 − 1.216500I
0.30789 + 3.54473I 8.22087 − 8.01132I
u = −0.437639 + 0.390994I
a = 1.41608 − 0.38982I
b = −0.040008 + 0.660042I
0.876008 + 0.412132I 11.41445 + 0.11588I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.437639 − 0.390994I
a = 1.41608 + 0.38982I
b = −0.040008 − 0.660042I
0.876008 − 0.412132I 11.41445 − 0.11588I
u = −0.409866
a = 0.871979
b = −0.319381
0.600340 16.7330
u = 1.59193
a = −1.99806
b = 1.49492
7.67890 0
u = −1.60820
a = −1.60477
b = 1.53392
13.8022 0
u = 0.198091 + 0.297215I
a = 0.72132 − 1.56787I
b = 0.125739 − 0.795889I
−1.45987 + 0.82313I −0.72620 − 2.45968I
u = 0.198091 − 0.297215I
a = 0.72132 + 1.56787I
b = 0.125739 + 0.795889I
−1.45987 − 0.82313I −0.72620 + 2.45968I
u = −1.67666
a = −2.65337
b = 1.89462
8.75767 0
u = 1.73600 + 0.03045I
a = 1.06886 + 1.00878I
b = −0.56637 − 1.46214I
12.24790 + 2.91315I 0
u = 1.73600 − 0.03045I
a = 1.06886 − 1.00878I
b = −0.56637 + 1.46214I
12.24790 − 2.91315I 0
u = 1.74422
a = −3.10810
b = 2.23888
17.2035 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.74314 + 0.07213I
a = 1.88603 − 1.11976I
b = −1.29148 + 1.58672I
14.5204 − 7.7607I 0
u = −1.74314 − 0.07213I
a = 1.88603 + 1.11976I
b = −1.29148 − 1.58672I
14.5204 + 7.7607I 0
u = −1.74624 + 0.02105I
a = 0.433151 + 0.263501I
b = −0.044228 − 0.773960I
15.7407 − 1.5655I 0
u = −1.74624 − 0.02105I
a = 0.433151 − 0.263501I
b = −0.044228 + 0.773960I
15.7407 + 1.5655I 0
u = 1.75579 + 0.11113I
a = 2.51996 + 0.96207I
b = −1.85527 − 1.45871I
−16.2979 + 10.9754I 0
u = 1.75579 − 0.11113I
a = 2.51996 − 0.96207I
b = −1.85527 + 1.45871I
−16.2979 − 10.9754I 0
u = 1.78103 + 0.07205I
a = 0.318355 + 0.567328I
b = 0.0141751 + 0.0635167I
−14.1664 + 4.0147I 0
u = 1.78103 − 0.07205I
a = 0.318355 − 0.567328I
b = 0.0141751 − 0.0635167I
−14.1664 − 4.0147I 0
u = 0.104256
a = −11.5099
b = 1.46674
3.32525 1.86210
8
II. I
u
2
= hb + a + u − 1, a
2
+ 4au − 2a + 3, u
2
− u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u − 1
a
8
=
u
−u − 1
a
1
=
−u
u
a
9
=
−1
0
a
2
=
a
−a − u + 1
a
6
=
−u
u
a
3
=
a + u
−a − 2u + 1
a
5
=
−a − u
a + 2u − 1
a
10
=
au − a − u + 2
2
a
4
=
−u + 1
−a − 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
9
c
10
(u
2
− 2)
2
c
6
, c
7
, c
8
(u
2
+ u − 1)
2
c
11
, c
12
(u
2
− u − 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
9
c
10
(y −2)
4
c
6
, c
7
, c
8
c
11
, c
12
(y
2
− 3y + 1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.821854
b = 0.796180
4.27683 12.0000
u = −0.618034
a = 3.65028
b = −2.03225
4.27683 12.0000
u = 1.61803
a = −0.821854
b = 0.203820
12.1725 12.0000
u = 1.61803
a = −3.65028
b = 3.03225
12.1725 12.0000
12
III. I
u
3
= hb + u, a − 2u − 1, u
2
+ u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
u − 1
a
8
=
u
−u + 1
a
1
=
u
−u
a
9
=
1
0
a
2
=
2u + 1
−u
a
6
=
−u
u
a
3
=
u + 1
0
a
5
=
u + 1
0
a
10
=
1
0
a
4
=
u + 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
9
c
10
u
2
c
5
(u + 1)
2
c
6
, c
7
, c
8
u
2
− u − 1
c
11
, c
12
u
2
+ u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
2
c
3
, c
4
, c
9
c
10
y
2
c
6
, c
7
, c
8
c
11
, c
12
y
2
− 3y + 1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 2.23607
b = −0.618034
−0.657974 2.00000
u = −1.61803
a = −2.23607
b = 1.61803
7.23771 2.00000
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
39
+ 15u
38
+ ··· + 111u + 1)
c
2
((u − 1)
2
)(u + 1)
4
(u
39
+ 3u
38
+ ··· + 3u + 1)
c
3
, c
4
, c
9
c
10
u
2
(u
2
− 2)
2
(u
39
− u
38
+ ··· + 4u − 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
39
+ 3u
38
+ ··· + 3u + 1)
c
6
, c
7
, c
8
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
39
+ 2u
38
+ ··· + 10u + 1)
c
11
, c
12
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
39
+ 2u
38
+ ··· + 10u + 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
39
+ 25y
38
+ ··· + 6327y −1)
c
2
, c
5
((y −1)
6
)(y
39
− 15y
38
+ ··· + 111y −1)
c
3
, c
4
, c
9
c
10
y
2
(y −2)
4
(y
39
− 49y
38
+ ··· + 368y −16)
c
6
, c
7
, c
8
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
39
− 54y
38
+ ··· + 102y −1)
18
