12n
0019
(K12n
0019
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 9 4 11 6 12 8 10
Solving Sequence
2,6
5 3 4
1,10
9 7 8 12 11
c
5
c
2
c
3
c
1
c
9
c
6
c
7
c
12
c
11
c
4
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h222u
17
+ 1807u
16
+ ··· + 536b − 1177, −19u
17
− 156u
16
+ ··· + 8a + 89, u
18
+ 8u
17
+ ··· − 8u + 1i
I
u
2
= hb, −u
4
a − 2u
3
a + u
4
− 3u
2
a − u
3
+ a
2
− 2au − 2u
2
− a − 5u − 3, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
I
u
3
= h−a
3
u − a
3
− 3a
2
− au + 3b + 2a + u + 4, a
4
− a
3
u + 3a
3
− a
2
u + a
2
− 4a − u − 3, u
2
− u + 1i
* 3 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
= h222u
17
+ 1807u
16
+ · · · + 536b − 1177, −19u
17
− 156u
16
+ · · · + 8a +
89, u
18
+ 8u
17
+ · · · − 8u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
10
=
19
8
u
17
+
39
2
u
16
+ ··· +
79
2
u −
89
8
−0.414179u
17
− 3.37127u
16
+ ··· − 6.08769u + 2.19590
a
9
=
2.78918u
17
+ 22.8713u
16
+ ··· + 45.5877u − 13.3209
−0.414179u
17
− 3.37127u
16
+ ··· − 6.08769u + 2.19590
a
7
=
−1.65112u
17
− 13.6642u
16
+ ··· − 27.3918u + 7.50560
0.430970u
17
+ 3.58396u
16
+ ··· + 6.21455u − 1.77985
a
8
=
−1.69590u
17
− 13.9813u
16
+ ··· − 27.3134u + 7.47948
0.345149u
17
+ 2.83022u
16
+ ··· + 4.67724u − 1.35075
a
12
=
0.447761u
17
+ 3.79664u
16
+ ··· + 7.84142u − 0.613806
−0.345149u
17
− 2.83022u
16
+ ··· − 4.67724u + 1.35075
a
11
=
1.43657u
17
+ 11.9049u
16
+ ··· + 24.4235u − 6.30784
−0.468284u
17
− 3.88993u
16
+ ··· − 6.77425u + 2.21642
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1867
536
u
17
+
15631
536
u
16
+ ··· +
28407
536
u −
3763
268
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
18
+ 2u
17
+ ··· − 34u + 1
c
2
, c
5
u
18
+ 8u
17
+ ··· − 8u + 1
c
3
u
18
− 8u
17
+ ··· − 16496u + 1921
c
4
, c
7
u
18
+ 2u
17
+ ··· − 384u + 256
c
6
, c
9
u
18
+ 2u
17
+ ··· + 1024u
2
+ 1024
c
8
, c
11
u
18
− 9u
17
+ ··· + 5u + 1
c
10
, c
12
u
18
+ u
17
+ ··· + 7u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
18
+ 34y
17
+ ··· − 706y + 1
c
2
, c
5
y
18
+ 2y
17
+ ··· − 34y + 1
c
3
y
18
+ 42y
17
+ ··· − 77040466y + 3690241
c
4
, c
7
y
18
+ 30y
17
+ ··· + 409600y + 65536
c
6
, c
9
y
18
+ 50y
17
+ ··· + 2097152y + 1048576
c
8
, c
11
y
18
− y
17
+ ··· − 7y + 1
c
10
, c
12
y
18
+ 47y
17
+ ··· − 199y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.489678 + 0.809386I
a = −3.46446 + 1.37116I
b = −0.367948 − 0.217959I
−0.02354 + 3.71255I 2.6622 − 33.9545I
u = 0.489678 − 0.809386I
a = −3.46446 − 1.37116I
b = −0.367948 + 0.217959I
−0.02354 − 3.71255I 2.6622 + 33.9545I
u = −0.528473 + 1.113200I
a = 0.884253 − 0.149684I
b = −0.293472 − 1.150100I
−6.98798 − 6.29888I −7.63956 + 6.18005I
u = −0.528473 − 1.113200I
a = 0.884253 + 0.149684I
b = −0.293472 + 1.150100I
−6.98798 + 6.29888I −7.63956 − 6.18005I
u = 0.402685 + 0.640215I
a = −0.600704 − 0.110262I
b = 0.079711 + 0.564353I
−0.176698 + 1.378410I −2.62845 − 4.45652I
u = 0.402685 − 0.640215I
a = −0.600704 + 0.110262I
b = 0.079711 − 0.564353I
−0.176698 − 1.378410I −2.62845 + 4.45652I
u = 0.166779 + 0.714203I
a = −0.576482 + 0.150196I
b = 0.406152 + 0.438776I
−0.194005 + 1.320020I −1.40154 − 3.97468I
u = 0.166779 − 0.714203I
a = −0.576482 − 0.150196I
b = 0.406152 − 0.438776I
−0.194005 − 1.320020I −1.40154 + 3.97468I
u = −0.79804 + 1.31718I
a = −1.30880 − 1.44991I
b = −1.88686 + 2.04182I
14.5520 − 13.1732I −2.14093 + 5.47150I
u = −0.79804 − 1.31718I
a = −1.30880 + 1.44991I
b = −1.88686 − 2.04182I
14.5520 + 13.1732I −2.14093 − 5.47150I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.48576 + 0.43889I
a = −0.636632 − 0.933823I
b = −2.70328 − 3.24263I
17.4544 + 5.4859I −0.93598 − 1.55559I
u = −1.48576 − 0.43889I
a = −0.636632 + 0.933823I
b = −2.70328 + 3.24263I
17.4544 − 5.4859I −0.93598 + 1.55559I
u = −1.39001 + 1.00947I
a = −1.31713 + 0.69280I
b = −0.33733 + 5.04758I
−5.22275 + 0.41218I −1.70669 + 0.I
u = −1.39001 − 1.00947I
a = −1.31713 − 0.69280I
b = −0.33733 − 5.04758I
−5.22275 − 0.41218I −1.70669 + 0.I
u = −1.06945 + 1.38280I
a = 1.27493 + 1.39729I
b = 3.58701 − 2.69224I
11.82800 − 4.92111I −2.64479 + 1.56009I
u = −1.06945 − 1.38280I
a = 1.27493 − 1.39729I
b = 3.58701 + 2.69224I
11.82800 + 4.92111I −2.64479 − 1.56009I
u = 0.212586 + 0.037327I
a = −0.25498 + 2.94572I
b = 0.516021 − 0.465095I
0.024368 − 1.375910I 0.93572 + 4.18536I
u = 0.212586 − 0.037327I
a = −0.25498 − 2.94572I
b = 0.516021 + 0.465095I
0.024368 + 1.375910I 0.93572 − 4.18536I
6
II. I
u
2
= hb, −u
4
a + u
4
+ · · · − a − 3, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
1
=
u
3
−u
4
− u
3
− u
2
− 1
a
10
=
a
0
a
9
=
a
0
a
7
=
1
0
a
8
=
−u
3
u
4
+ u
3
+ u
2
+ 1
a
12
=
−u
4
− u
3
− 3u
2
+ a − 2u − 1
−u
4
− u
3
− u
2
− 1
a
11
=
2u
3
a − u
4
− 2u
3
+ au − 3u
2
+ 2a − 2u − 1
−2u
4
a − 2u
3
a − 2u
2
a − au − 2a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
4
a − 3u
3
a + u
4
− 4u
2
a − 5u
3
− 6u
2
− 2a − 9u − 5
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
2
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
3
, c
4
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
5
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
6
, c
9
u
10
c
7
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
8
, c
12
(u
2
− u + 1)
5
c
10
, c
11
(u
2
+ u + 1)
5
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
2
c
2
, c
5
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
c
3
, c
4
, c
7
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
c
6
, c
9
y
10
c
8
, c
10
, c
11
c
12
(y
2
+ y + 1)
5
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.339110 + 0.822375I
a = 1.20942 + 2.19910I
b = 0
−0.329100 − 0.499304I 2.94328 − 6.15174I
u = 0.339110 + 0.822375I
a = −2.50919 − 0.05217I
b = 0
−0.32910 + 3.56046I −6.96704 − 8.14994I
u = 0.339110 − 0.822375I
a = 1.20942 − 2.19910I
b = 0
−0.329100 + 0.499304I 2.94328 + 6.15174I
u = 0.339110 − 0.822375I
a = −2.50919 + 0.05217I
b = 0
−0.32910 − 3.56046I −6.96704 + 8.14994I
u = −0.766826
a = 0.337181 + 0.584015I
b = 0
−2.40108 + 2.02988I −0.15429 − 1.95361I
u = −0.766826
a = 0.337181 − 0.584015I
b = 0
−2.40108 − 2.02988I −0.15429 + 1.95361I
u = −0.455697 + 1.200150I
a = 0.358089 + 0.327409I
b = 0
−5.87256 − 2.37095I −5.14480 + 4.03066I
u = −0.455697 + 1.200150I
a = 0.104500 − 0.473819I
b = 0
−5.87256 − 6.43072I −0.67715 + 5.27500I
u = −0.455697 − 1.200150I
a = 0.358089 − 0.327409I
b = 0
−5.87256 + 2.37095I −5.14480 − 4.03066I
u = −0.455697 − 1.200150I
a = 0.104500 + 0.473819I
b = 0
−5.87256 + 6.43072I −0.67715 − 5.27500I
10
III. I
u
3
= h−a
3
u − a
3
− 3a
2
− au + 3b + 2a + u + 4, a
4
− a
3
u + 3a
3
− a
2
u +
a
2
− 4a − u − 3, u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u − 1
a
3
=
u
u − 1
a
4
=
1
u − 1
a
1
=
−1
0
a
10
=
a
1
3
a
3
u +
1
3
au + ··· −
2
3
a −
4
3
a
9
=
−
1
3
a
3
u −
1
3
au + ··· +
5
3
a +
4
3
1
3
a
3
u +
1
3
au + ··· −
2
3
a −
4
3
a
7
=
−
1
3
a
3
u −
4
3
a
2
u + ··· − a −
4
3
2
3
a
3
u +
2
3
a
2
u + ··· + a +
5
3
a
8
=
−
1
3
a
3
u −
4
3
a
2
u + ··· − a −
4
3
2
3
a
3
u +
2
3
a
2
u + ··· + a +
5
3
a
12
=
1
3
a
3
u −
2
3
a
2
u + ··· +
4
3
a
2
−
5
3
2
3
a
3
u +
2
3
a
2
u + ··· + a +
5
3
a
11
=
4
3
a
3
u +
4
3
a
2
u + ··· +
1
3
a
2
+
1
3
−
1
3
a
3
u −
1
3
a
2
u + ··· + a +
2
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
7
3
a
3
u +
11
3
a
3
− 5a
2
u + 4a
2
+
11
3
au −
25
3
a +
25
3
u −
44
3
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
4
, c
7
u
8
c
6
, c
10
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
8
(u
4
+ u
3
+ u
2
+ 1)
2
c
9
, c
12
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
11
(u
4
− u
3
+ u
2
+ 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
4
c
4
, c
7
y
8
c
6
, c
9
, c
10
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
8
, c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = −0.715307 − 0.631577I
b = −0.395123 + 0.506844I
0.211005 + 0.614778I 0.01166 + 7.13374I
u = 0.500000 + 0.866025I
a = 1.248740 + 0.225872I
b = −0.10488 + 1.55249I
−6.79074 − 1.13408I −8.12668 + 3.09304I
u = 0.500000 + 0.866025I
a = −1.44025 − 0.04422I
b = −0.10488 − 1.55249I
−6.79074 + 5.19385I −5.34148 − 0.51945I
u = 0.500000 + 0.866025I
a = −1.59319 + 1.31595I
b = −0.395123 − 0.506844I
0.21101 + 3.44499I 4.95650 − 5.37720I
u = 0.500000 − 0.866025I
a = −0.715307 + 0.631577I
b = −0.395123 − 0.506844I
0.211005 − 0.614778I 0.01166 − 7.13374I
u = 0.500000 − 0.866025I
a = 1.248740 − 0.225872I
b = −0.10488 − 1.55249I
−6.79074 + 1.13408I −8.12668 − 3.09304I
u = 0.500000 − 0.866025I
a = −1.44025 + 0.04422I
b = −0.10488 + 1.55249I
−6.79074 − 5.19385I −5.34148 + 0.51945I
u = 0.500000 − 0.866025I
a = −1.59319 − 1.31595I
b = −0.395123 + 0.506844I
0.21101 − 3.44499I 4.95650 + 5.37720I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
4
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
· (u
18
+ 2u
17
+ ··· − 34u + 1)
c
2
((u
2
+ u + 1)
4
)(u
5
− u
4
+ ··· + u − 1)
2
(u
18
+ 8u
17
+ ··· − 8u + 1)
c
3
(u
2
− u + 1)
4
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
· (u
18
− 8u
17
+ ··· − 16496u + 1921)
c
4
u
8
(u
5
+ u
4
+ ··· + u − 1)
2
(u
18
+ 2u
17
+ ··· − 384u + 256)
c
5
((u
2
− u + 1)
4
)(u
5
+ u
4
+ ··· + u + 1)
2
(u
18
+ 8u
17
+ ··· − 8u + 1)
c
6
u
10
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
(u
18
+ 2u
17
+ ··· + 1024u
2
+ 1024)
c
7
u
8
(u
5
− u
4
+ ··· + u + 1)
2
(u
18
+ 2u
17
+ ··· − 384u + 256)
c
8
((u
2
− u + 1)
5
)(u
4
+ u
3
+ u
2
+ 1)
2
(u
18
− 9u
17
+ ··· + 5u + 1)
c
9
u
10
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
(u
18
+ 2u
17
+ ··· + 1024u
2
+ 1024)
c
10
((u
2
+ u + 1)
5
)(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
(u
18
+ u
17
+ ··· + 7u + 1)
c
11
((u
2
+ u + 1)
5
)(u
4
− u
3
+ u
2
+ 1)
2
(u
18
− 9u
17
+ ··· + 5u + 1)
c
12
((u
2
− u + 1)
5
)(u
4
− u
3
+ 3u
2
− 2u + 1)
2
(u
18
+ u
17
+ ··· + 7u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)
4
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
2
· (y
18
+ 34y
17
+ ··· − 706y + 1)
c
2
, c
5
(y
2
+ y + 1)
4
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
· (y
18
+ 2y
17
+ ··· − 34y + 1)
c
3
(y
2
+ y + 1)
4
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
· (y
18
+ 42y
17
+ ··· − 77040466y + 3690241)
c
4
, c
7
y
8
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
· (y
18
+ 30y
17
+ ··· + 409600y + 65536)
c
6
, c
9
y
10
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
18
+ 50y
17
+ ··· + 2097152y + 1048576)
c
8
, c
11
((y
2
+ y + 1)
5
)(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
(y
18
− y
17
+ ··· − 7y + 1)
c
10
, c
12
(y
2
+ y + 1)
5
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
18
+ 47y
17
+ ··· − 199y + 1)
16
