12a
0286
(K12a
0286
)
A knot diagram
1
Linearized knot diagam
3 6 8 9 11 2 10 1 12 7 5 4
Solving Sequence
4,9 5,12
10 1 8 3 7 11 6 2
c
4
c
9
c
12
c
8
c
3
c
7
c
11
c
5
c
2
c
1
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.08457 × 10
28
u
31
− 9.68465 × 10
27
u
30
+ ··· + 3.30621 × 10
28
b + 6.26853 × 10
28
,
1.00919 × 10
29
u
31
− 4.41562 × 10
28
u
30
+ ··· + 3.30621 × 10
28
a + 4.42115 × 10
28
, u
32
− 2u
30
+ ··· + 2u + 1i
I
u
2
= h−u
3
+ u
2
+ 4b − 5u + 2, a, u
4
− u
3
+ 5u
2
− 2u + 4i
* 2 irreducible components of dim
C
= 0, with total 36 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−1.08×10
28
u
31
−9.68×10
27
u
30
+· · ·+3.31×10
28
b+6.27×10
28
, 1.01×
10
29
u
31
−4.42×10
28
u
30
+· · ·+3.31×10
28
a+4.42×10
28
, u
32
−2u
30
+· · ·+2u+1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
5
=
1
−u
2
a
12
=
−3.05243u
31
+ 1.33556u
30
+ ··· − 5.33264u − 1.33723
0.328041u
31
+ 0.292923u
30
+ ··· + 0.733300u − 1.89599
a
10
=
6.38763u
31
− 3.08529u
30
+ ··· + 7.92825u + 8.64307
−0.227028u
31
+ 0.570345u
30
+ ··· − 1.64514u + 1.43963
a
1
=
−2.72438u
31
+ 1.62848u
30
+ ··· − 4.59934u − 3.23322
0.328041u
31
+ 0.292923u
30
+ ··· + 0.733300u − 1.89599
a
8
=
5.67422u
31
− 2.13572u
30
+ ··· + 1.41383u + 9.56776
−0.486384u
31
+ 0.379216u
30
+ ··· − 2.86928u − 0.514941
a
3
=
−3.24515u
31
+ 1.67422u
30
+ ··· − 3.02690u − 4.97838
0.304612u
31
− 0.193228u
30
+ ··· + 3.65994u − 0.679133
a
7
=
7.00844u
31
− 0.808937u
30
+ ··· − 7.09680u + 17.1331
−2.34735u
31
− 0.0978616u
30
+ ··· − 1.67702u − 2.05955
a
11
=
−3.95240u
31
+ 1.64688u
30
+ ··· − 4.21803u − 4.56877
0.349246u
31
+ 0.484634u
30
+ ··· + 0.455964u − 1.58467
a
6
=
−2.33177u
31
+ 0.938856u
30
+ ··· + 3.06552u − 9.72970
2.18533u
31
− 0.590773u
30
+ ··· + 3.26730u + 2.12120
a
2
=
−8.37149u
31
+ 5.01296u
30
+ ··· − 12.6435u − 10.7929
−1.24677u
31
− 0.611740u
30
+ ··· + 8.72740u − 2.92118
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4.80290u
31
− 0.110659u
30
+ ··· − 8.67744u + 4.28329
2
(iv) u-Polynomials at the component
3
Crossings u-Polynomials at each crossing
c
1
u
32
− 16u
31
+ ··· − 10u + 1
c
2
u
32
− 4u
31
+ ··· − 6u + 1
c
3
u
32
+ 8u
30
+ ··· + 6u + 1
c
4
u
32
− 2u
30
+ ··· + 2u + 1
c
5
u
32
− 4u
31
+ ··· + 8u + 1
c
6
u
32
+ 4u
31
+ ··· + 6u + 1
c
7
u
32
− 10u
31
+ ··· − 2u + 1
c
8
u
32
+ 6u
31
+ ··· + 4u + 1
c
9
u
32
+ 10u
31
+ ··· + 538u + 73
c
10
u
32
+ 10u
31
+ ··· + 2u + 1
c
11
u
32
+ 4u
31
+ ··· − 8u + 1
c
12
u
32
+ 4u
31
+ ··· − 2u + 1
4
5
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
32
+ 4y
31
+ ··· + 10y + 1
c
2
, c
6
y
32
+ 16y
31
+ ··· + 10y + 1
c
3
y
32
+ 16y
31
+ ··· − 12y + 1
c
4
y
32
− 4y
31
+ ··· − 16y + 1
c
5
, c
11
y
32
+ 20y
31
+ ··· + 16y + 1
c
7
, c
10
y
32
+ 18y
31
+ ··· + 22y + 1
c
8
y
32
− 12y
31
+ ··· + 14y + 1
c
9
y
32
− 14y
31
+ ··· + 77162y + 5329
c
12
y
32
− 4y
31
+ ··· + 20y + 1
6
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.011430 + 0.076540I
a = −0.299553 − 1.057440I
b = 0.066450 + 1.277390I
1.95636 + 4.51992I −2.62248 − 7.58475I
u = −1.011430 − 0.076540I
a = −0.299553 + 1.057440I
b = 0.066450 − 1.277390I
1.95636 − 4.51992I −2.62248 + 7.58475I
u = −0.195501 + 0.960625I
a = −0.54851 − 1.43505I
b = −0.037707 − 0.482351I
−1.46529 − 0.97885I −11.84592 − 4.46110I
u = −0.195501 − 0.960625I
a = −0.54851 + 1.43505I
b = −0.037707 + 0.482351I
−1.46529 + 0.97885I −11.84592 + 4.46110I
u = 0.963349 + 0.082588I
a = 0.874463 + 0.489100I
b = −1.056860 − 0.880917I
0.52518 − 1.70904I 1.303152 + 0.227505I
u = 0.963349 − 0.082588I
a = 0.874463 − 0.489100I
b = −1.056860 + 0.880917I
0.52518 + 1.70904I 1.303152 − 0.227505I
u = 0.897933 + 0.134537I
a = −0.079600 − 1.057370I
b = −0.417549 + 0.423958I
−1.93841 − 3.89093I −2.89352 + 9.56275I
u = 0.897933 − 0.134537I
a = −0.079600 + 1.057370I
b = −0.417549 − 0.423958I
−1.93841 + 3.89093I −2.89352 − 9.56275I
u = −0.790801 + 0.178174I
a = −1.07264 + 1.31010I
b = 0.144360 − 0.486258I
−1.66071 − 1.88565I 0.611617 + 0.405112I
u = −0.790801 − 0.178174I
a = −1.07264 − 1.31010I
b = 0.144360 + 0.486258I
−1.66071 + 1.88565I 0.611617 − 0.405112I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.725473 + 0.306578I
a = 2.59490 + 0.04955I
b = −1.370940 − 0.282852I
−0.233393 + 0.291097I −5.20473 + 2.81677I
u = 0.725473 − 0.306578I
a = 2.59490 − 0.04955I
b = −1.370940 + 0.282852I
−0.233393 − 0.291097I −5.20473 − 2.81677I
u = −0.358787 + 1.159020I
a = 0.506557 + 0.930883I
b = −0.576105 + 0.212433I
7.03446 − 10.43960I 3.24799 + 7.92017I
u = −0.358787 − 1.159020I
a = 0.506557 − 0.930883I
b = −0.576105 − 0.212433I
7.03446 + 10.43960I 3.24799 − 7.92017I
u = −1.236250 + 0.423599I
a = −1.23017 + 0.89428I
b = 1.08185 − 1.47183I
4.04750 − 6.51115I 7.4827 + 17.7604I
u = −1.236250 − 0.423599I
a = −1.23017 − 0.89428I
b = 1.08185 + 1.47183I
4.04750 + 6.51115I 7.4827 − 17.7604I
u = 0.754987 + 1.099260I
a = −0.851878 + 0.547278I
b = 0.676413 + 0.328413I
8.54048 + 4.66147I 5.60896 − 3.18468I
u = 0.754987 − 1.099260I
a = −0.851878 − 0.547278I
b = 0.676413 − 0.328413I
8.54048 − 4.66147I 5.60896 + 3.18468I
u = −0.541157 + 0.373122I
a = −0.611032 − 0.446290I
b = 1.48390 − 0.85410I
2.56066 − 1.39415I 8.59786 − 1.04596I
u = −0.541157 − 0.373122I
a = −0.611032 + 0.446290I
b = 1.48390 + 0.85410I
2.56066 + 1.39415I 8.59786 + 1.04596I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.078990 + 0.816307I
a = 1.071980 − 0.021161I
b = −1.00062 − 1.06785I
0.04554 + 9.55039I 0.17757 − 7.82747I
u = 1.078990 − 0.816307I
a = 1.071980 + 0.021161I
b = −1.00062 + 1.06785I
0.04554 − 9.55039I 0.17757 + 7.82747I
u = 0.349670 + 0.386213I
a = −0.448966 − 0.257825I
b = 1.82694 + 1.31311I
3.77595 + 1.55640I −10.9090 − 16.7964I
u = 0.349670 − 0.386213I
a = −0.448966 + 0.257825I
b = 1.82694 − 1.31311I
3.77595 − 1.55640I −10.9090 + 16.7964I
u = 1.21893 + 0.86163I
a = −0.997262 − 0.051859I
b = 0.766581 + 0.643238I
7.91161 + 4.63893I 7.21423 − 8.93330I
u = 1.21893 − 0.86163I
a = −0.997262 + 0.051859I
b = 0.766581 − 0.643238I
7.91161 − 4.63893I 7.21423 + 8.93330I
u = −1.27730 + 0.81643I
a = −0.841795 + 0.203609I
b = 0.93799 − 1.07457I
2.06970 − 6.21261I −3.30852 + 0.I
u = −1.27730 − 0.81643I
a = −0.841795 − 0.203609I
b = 0.93799 + 1.07457I
2.06970 + 6.21261I −3.30852 + 0.I
u = −0.435914 + 0.191346I
a = 3.01618 − 0.43588I
b = −0.516431 − 0.791322I
1.57422 − 2.11475I 0.389556 + 1.338497I
u = −0.435914 − 0.191346I
a = 3.01618 + 0.43588I
b = −0.516431 + 0.791322I
1.57422 + 2.11475I 0.389556 − 1.338497I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.14221 + 1.69421I
a = −0.082679 − 0.594170I
b = −0.008276 − 0.433413I
1.44470 + 2.66235I 7.15055 − 3.68105I
u = −0.14221 − 1.69421I
a = −0.082679 + 0.594170I
b = −0.008276 + 0.433413I
1.44470 − 2.66235I 7.15055 + 3.68105I
10
II. I
u
2
= h−u
3
+ u
2
+ 4b − 5u + 2, a, u
4
− u
3
+ 5u
2
− 2u + 4i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
5
=
1
−u
2
a
12
=
0
1
4
u
3
−
1
4
u
2
+
5
4
u −
1
2
a
10
=
0
u
a
1
=
1
4
u
3
−
1
4
u
2
+
5
4
u −
1
2
1
4
u
3
−
1
4
u
2
+
5
4
u −
1
2
a
8
=
−
1
4
u
3
+
1
4
u
2
−
5
4
u +
1
2
−
1
4
u
3
+
1
4
u
2
−
1
4
u +
1
2
a
3
=
1
8
u
3
+
1
8
u
2
+
3
8
u + 1
1
8
u
3
−
7
8
u
2
+
3
8
u − 1
a
7
=
−
1
4
u
3
+
1
4
u
2
−
5
4
u +
1
2
−
1
4
u
3
+
1
4
u
2
−
5
4
u +
1
2
a
11
=
1
4
u
3
−
1
4
u
2
+
5
4
u −
1
2
1
4
u
3
−
1
4
u
2
+
9
4
u −
1
2
a
6
=
−
1
8
u
3
−
1
8
u
2
−
3
8
u
−
1
8
u
3
−
9
8
u
2
−
3
8
u − 2
a
2
=
1
4
u
2
+
1
4
u +
3
4
−
3
4
u
2
+
5
4
u −
5
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
103
32
u
3
+
223
32
u
2
−
279
32
u +
21
16
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
2
− u + 1)
2
c
2
(u
2
+ u + 1)
2
c
3
4(4u
4
− 6u
3
+ 11u
2
− 6u + 1)
c
4
u
4
− u
3
+ 5u
2
− 2u + 4
c
5
4(4u
4
+ 2u
3
+ 5u
2
+ u + 1)
c
7
, c
8
(u − 1)
4
c
9
u
4
c
10
(u + 1)
4
c
11
, c
12
4(4u
4
− 2u
3
+ 5u
2
− u + 1)
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
(y
2
+ y + 1)
2
c
3
16(16y
4
+ 52y
3
+ 57y
2
− 14y + 1)
c
4
y
4
+ 9y
3
+ 29y
2
+ 36y + 16
c
5
, c
11
, c
12
16(16y
4
+ 36y
3
+ 29y
2
+ 9y + 1)
c
7
, c
8
, c
10
(y −1)
4
c
9
y
4
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.175835 + 1.026610I
a = 0
b = −0.162083 + 0.946318I
−1.64493 + 2.02988I −5.57770 − 3.25874I
u = 0.175835 − 1.026610I
a = 0
b = −0.162083 − 0.946318I
−1.64493 − 2.02988I −5.57770 + 3.25874I
u = 0.32417 + 1.89264I
a = 0
b = −0.087917 + 0.513305I
−1.64493 − 2.02988I −14.6411 + 11.9508I
u = 0.32417 − 1.89264I
a = 0
b = −0.087917 − 0.513305I
−1.64493 + 2.02988I −14.6411 − 11.9508I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
2
)(u
32
− 16u
31
+ ··· − 10u + 1)
c
2
((u
2
+ u + 1)
2
)(u
32
− 4u
31
+ ··· − 6u + 1)
c
3
4(4u
4
− 6u
3
+ ··· − 6u + 1)(u
32
+ 8u
30
+ ··· + 6u + 1)
c
4
(u
4
− u
3
+ 5u
2
− 2u + 4)(u
32
− 2u
30
+ ··· + 2u + 1)
c
5
4(4u
4
+ 2u
3
+ ··· + u + 1)(u
32
− 4u
31
+ ··· + 8u + 1)
c
6
((u
2
− u + 1)
2
)(u
32
+ 4u
31
+ ··· + 6u + 1)
c
7
((u − 1)
4
)(u
32
− 10u
31
+ ··· − 2u + 1)
c
8
((u − 1)
4
)(u
32
+ 6u
31
+ ··· + 4u + 1)
c
9
u
4
(u
32
+ 10u
31
+ ··· + 538u + 73)
c
10
((u + 1)
4
)(u
32
+ 10u
31
+ ··· + 2u + 1)
c
11
4(4u
4
− 2u
3
+ ··· − u + 1)(u
32
+ 4u
31
+ ··· − 8u + 1)
c
12
4(4u
4
− 2u
3
+ ··· − u + 1)(u
32
+ 4u
31
+ ··· − 2u + 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
2
)(y
32
+ 4y
31
+ ··· + 10y + 1)
c
2
, c
6
((y
2
+ y + 1)
2
)(y
32
+ 16y
31
+ ··· + 10y + 1)
c
3
16(16y
4
+ 52y
3
+ ··· − 14y + 1)(y
32
+ 16y
31
+ ··· − 12y + 1)
c
4
(y
4
+ 9y
3
+ 29y
2
+ 36y + 16)(y
32
− 4y
31
+ ··· − 16y + 1)
c
5
, c
11
16(16y
4
+ 36y
3
+ ··· + 9y + 1)(y
32
+ 20y
31
+ ··· + 16y + 1)
c
7
, c
10
((y −1)
4
)(y
32
+ 18y
31
+ ··· + 22y + 1)
c
8
((y −1)
4
)(y
32
− 12y
31
+ ··· + 14y + 1)
c
9
y
4
(y
32
− 14y
31
+ ··· + 77162y + 5329)
c
12
16(16y
4
+ 36y
3
+ ··· + 9y + 1)(y
32
− 4y
31
+ ··· + 20y + 1)
16
