12n
0215
(K12n
0215
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 9 3 12 6 5 8 11
Solving Sequence
5,10
6
3,11
2 1 4 9 7 8 12
c
5
c
10
c
2
c
1
c
4
c
9
c
6
c
7
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h3.05834 × 10
15
u
34
− 7.90175 × 10
15
u
33
+ ··· + 2.09912 × 10
16
b + 1.74277 × 10
16
,
7.98092 × 10
16
u
34
− 1.40145 × 10
17
u
33
+ ··· + 6.29736 × 10
16
a + 1.13113 × 10
17
, u
35
− 2u
34
+ ··· + 2u − 1i
I
u
2
= hb + 1, 4u
4
− 3u
3
+ 16u
2
+ 3a − 8u + 10, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
* 2 irreducible components of dim
C
= 0, with total 40 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
= h3.06 × 10
15
u
34
− 7.90 × 10
15
u
33
+ · · · + 2.10 × 10
16
b + 1.74 × 10
16
, 7.98 ×
10
16
u
34
−1.40×10
17
u
33
+· · ·+6.30×10
16
a+1.13×10
17
, u
35
−2u
34
+· · ·+2u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
3
=
−1.26734u
34
+ 2.22546u
33
+ ··· − 0.810928u − 1.79619
−0.145696u
34
+ 0.376432u
33
+ ··· − 1.05037u − 0.830238
a
11
=
−u
u
a
2
=
−1.41304u
34
+ 2.60189u
33
+ ··· − 1.86130u − 2.62643
−0.145696u
34
+ 0.376432u
33
+ ··· − 1.05037u − 0.830238
a
1
=
−0.375032u
34
+ 0.960289u
33
+ ··· − 0.228111u − 1.08728
0.0296343u
34
− 0.108392u
33
+ ··· − 0.0371536u − 0.0245560
a
4
=
−1.24493u
34
+ 2.18658u
33
+ ··· − 1.42273u − 1.62947
−0.161215u
34
+ 0.434551u
33
+ ··· − 1.04853u − 0.806713
a
9
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
−0.367678u
34
+ 0.886943u
33
+ ··· − 0.895712u − 0.926179
0.0213029u
34
− 0.0716902u
33
+ ··· − 1.30130u + 0.337244
a
12
=
−0.367678u
34
+ 0.886943u
33
+ ··· − 0.895712u − 0.926179
0.0222806u
34
− 0.0350461u
33
+ ··· + 0.630447u − 0.185658
(ii) Obstruction class = −1
(iii) Cusp Shapes =
111368899656244763
188920669770572085
u
34
−
254954201870553542
188920669770572085
u
33
+···+
234691530598543088
37784133954114417
u−
1174662391668223564
188920669770572085
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
35
+ 40u
34
+ ··· + 11466u + 81
c
2
, c
4
u
35
− 6u
34
+ ··· + 102u − 9
c
3
, c
7
u
35
− 3u
34
+ ··· + 768u + 288
c
5
, c
6
, c
9
c
10
u
35
− 2u
34
+ ··· + 2u − 1
c
8
, c
11
u
35
+ 2u
34
+ ··· − 2u − 1
c
12
u
35
+ 24u
34
+ ··· + 12u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
35
− 84y
34
+ ··· + 90434070y −6561
c
2
, c
4
y
35
− 40y
34
+ ··· + 11466y −81
c
3
, c
7
y
35
+ 33y
34
+ ··· + 935424y −82944
c
5
, c
6
, c
9
c
10
y
35
+ 36y
34
+ ··· + 12y −1
c
8
, c
11
y
35
− 24y
34
+ ··· + 12y −1
c
12
y
35
− 24y
34
+ ··· + 308y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.823932 + 0.577808I
a = −0.155370 − 0.797556I
b = 1.65857 − 0.14735I
−11.98490 + 3.28428I −11.99097 − 0.69750I
u = 0.823932 − 0.577808I
a = −0.155370 + 0.797556I
b = 1.65857 + 0.14735I
−11.98490 − 3.28428I −11.99097 + 0.69750I
u = −0.845135 + 0.556282I
a = 0.064458 + 0.876271I
b = 1.57251 − 0.09566I
−7.58978 + 2.78007I −9.93716 − 2.70438I
u = −0.845135 − 0.556282I
a = 0.064458 − 0.876271I
b = 1.57251 + 0.09566I
−7.58978 − 2.78007I −9.93716 + 2.70438I
u = 0.826909 + 0.535223I
a = 0.160633 − 1.094940I
b = 1.67277 + 0.32231I
−12.1055 − 8.7330I −11.74675 + 5.60158I
u = 0.826909 − 0.535223I
a = 0.160633 + 1.094940I
b = 1.67277 − 0.32231I
−12.1055 + 8.7330I −11.74675 − 5.60158I
u = −0.219192 + 0.760543I
a = 0.413862 + 0.000591I
b = 0.381823 + 0.140239I
1.26028 + 2.02659I −0.48961 − 4.26135I
u = −0.219192 − 0.760543I
a = 0.413862 − 0.000591I
b = 0.381823 − 0.140239I
1.26028 − 2.02659I −0.48961 + 4.26135I
u = −0.107525 + 1.257610I
a = 1.49280 − 1.31844I
b = −1.77830 + 0.61280I
−2.63751 + 2.36122I −11.77236 − 2.92931I
u = −0.107525 − 1.257610I
a = 1.49280 + 1.31844I
b = −1.77830 − 0.61280I
−2.63751 − 2.36122I −11.77236 + 2.92931I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.048161 + 1.339420I
a = 0.150468 + 1.217930I
b = −1.223220 − 0.264994I
2.04451 − 1.19616I −3.22654 + 0.68114I
u = 0.048161 − 1.339420I
a = 0.150468 − 1.217930I
b = −1.223220 + 0.264994I
2.04451 + 1.19616I −3.22654 − 0.68114I
u = 0.537614 + 0.278615I
a = −0.194970 + 0.880512I
b = −0.777018 − 1.038680I
−4.08309 − 3.66975I −13.2607 + 6.9860I
u = 0.537614 − 0.278615I
a = −0.194970 − 0.880512I
b = −0.777018 + 1.038680I
−4.08309 + 3.66975I −13.2607 − 6.9860I
u = 0.07172 + 1.41763I
a = 1.40227 + 1.58971I
b = −0.568896 − 0.136236I
2.34289 − 0.25916I −8.00000 − 1.73801I
u = 0.07172 − 1.41763I
a = 1.40227 − 1.58971I
b = −0.568896 + 0.136236I
2.34289 + 0.25916I −8.00000 + 1.73801I
u = 0.18444 + 1.40762I
a = 0.09162 + 2.08933I
b = −0.46414 − 1.41482I
1.31340 − 6.28579I −8.00000 + 6.17936I
u = 0.18444 − 1.40762I
a = 0.09162 − 2.08933I
b = −0.46414 + 1.41482I
1.31340 + 6.28579I −8.00000 − 6.17936I
u = −0.13397 + 1.43446I
a = 0.24966 − 1.49398I
b = −0.263513 + 0.834456I
5.13989 + 3.03126I 0. − 3.33629I
u = −0.13397 − 1.43446I
a = 0.24966 + 1.49398I
b = −0.263513 − 0.834456I
5.13989 − 3.03126I 0. + 3.33629I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.544097
a = −0.567908
b = −1.72921
−6.34482 −17.4700
u = −0.383769 + 0.315386I
a = 0.415262 − 1.043400I
b = −0.449868 + 0.371769I
−0.534568 + 1.131870I −6.35643 − 6.10930I
u = −0.383769 − 0.315386I
a = 0.415262 + 1.043400I
b = −0.449868 − 0.371769I
−0.534568 − 1.131870I −6.35643 + 6.10930I
u = 0.273233 + 0.399953I
a = 3.06116 + 1.12201I
b = −0.847017 + 0.385254I
−3.28792 + 0.94378I −10.44072 + 4.60691I
u = 0.273233 − 0.399953I
a = 3.06116 − 1.12201I
b = −0.847017 − 0.385254I
−3.28792 − 0.94378I −10.44072 − 4.60691I
u = 0.30180 + 1.54032I
a = −0.90061 − 1.57197I
b = 1.62707 + 0.48300I
−5.36857 − 12.88450I 0
u = 0.30180 − 1.54032I
a = −0.90061 + 1.57197I
b = 1.62707 − 0.48300I
−5.36857 + 12.88450I 0
u = −0.31717 + 1.54997I
a = −0.84603 + 1.20335I
b = 1.51007 − 0.28880I
−0.76082 + 7.07720I 0
u = −0.31717 − 1.54997I
a = −0.84603 − 1.20335I
b = 1.51007 + 0.28880I
−0.76082 − 7.07720I 0
u = 0.30705 + 1.57204I
a = −1.108790 − 0.870354I
b = 1.56930 + 0.03692I
−4.96905 − 0.92386I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.30705 − 1.57204I
a = −1.108790 + 0.870354I
b = 1.56930 − 0.03692I
−4.96905 + 0.92386I 0
u = −0.393303
a = 1.22750
b = 0.0923140
−0.995121 −10.5770
u = −0.06492 + 1.66171I
a = −0.365988 − 0.087264I
b = 0.749270 + 0.127452I
9.76537 + 3.12159I 0
u = −0.06492 − 1.66171I
a = −0.365988 + 0.087264I
b = 0.749270 − 0.127452I
9.76537 − 3.12159I 0
u = 0.331033
a = −3.18710
b = −1.10194
−2.12637 −0.370270
8
II.
I
u
2
= hb + 1, 4u
4
− 3u
3
+ 16u
2
+ 3a − 8u + 10, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
3
=
−
4
3
u
4
+ u
3
+ ··· +
8
3
u −
10
3
−1
a
11
=
−u
u
a
2
=
−
4
3
u
4
+ u
3
+ ··· +
8
3
u −
13
3
−1
a
1
=
−1
0
a
4
=
−
4
3
u
4
+ u
3
+ ··· +
8
3
u −
10
3
−1
a
9
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
u
2
+ 1
u
4
+ 2u
2
a
12
=
−u
2
− 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
58
9
u
4
+
13
3
u
3
−
211
9
u
2
+
128
9
u −
223
9
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
7
u
5
c
4
(u + 1)
5
c
5
, c
6
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
8
u
5
− u
4
+ u
2
+ u − 1
c
9
, c
10
, c
12
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
11
u
5
+ u
4
− u
2
+ u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
5
c
3
, c
7
y
5
c
5
, c
6
, c
9
c
10
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1
c
8
, c
11
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = 0.162657 + 0.410020I
b = −1.00000
0.17487 − 2.21397I −9.00284 + 4.40290I
u = 0.233677 − 0.885557I
a = 0.162657 − 0.410020I
b = −1.00000
0.17487 + 2.21397I −9.00284 − 4.40290I
u = 0.416284
a = −3.11537
b = −1.00000
−2.52712 −22.8010
u = 0.05818 + 1.69128I
a = 0.728361 + 0.139255I
b = −1.00000
9.31336 − 3.33174I −11.48557 + 5.79761I
u = 0.05818 − 1.69128I
a = 0.728361 − 0.139255I
b = −1.00000
9.31336 + 3.33174I −11.48557 − 5.79761I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
35
+ 40u
34
+ ··· + 11466u + 81)
c
2
((u − 1)
5
)(u
35
− 6u
34
+ ··· + 102u − 9)
c
3
, c
7
u
5
(u
35
− 3u
34
+ ··· + 768u + 288)
c
4
((u + 1)
5
)(u
35
− 6u
34
+ ··· + 102u − 9)
c
5
, c
6
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
35
− 2u
34
+ ··· + 2u − 1)
c
8
(u
5
− u
4
+ u
2
+ u − 1)(u
35
+ 2u
34
+ ··· − 2u − 1)
c
9
, c
10
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
35
− 2u
34
+ ··· + 2u − 1)
c
11
(u
5
+ u
4
− u
2
+ u + 1)(u
35
+ 2u
34
+ ··· − 2u − 1)
c
12
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
35
+ 24u
34
+ ··· + 12u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
5
)(y
35
− 84y
34
+ ··· + 90434070y − 6561)
c
2
, c
4
((y − 1)
5
)(y
35
− 40y
34
+ ··· + 11466y − 81)
c
3
, c
7
y
5
(y
35
+ 33y
34
+ ··· + 935424y − 82944)
c
5
, c
6
, c
9
c
10
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)(y
35
+ 36y
34
+ ··· + 12y − 1)
c
8
, c
11
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1)(y
35
− 24y
34
+ ··· + 12y − 1)
c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)(y
35
− 24y
34
+ ··· + 308y − 1)
14
