12n
0192
(K12n
0192
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 10 11 4 12 5 6 8 9
Solving Sequence
5,9
10 6
3,11
2 1 4 12 8 7
c
9
c
5
c
10
c
2
c
1
c
4
c
12
c
8
c
7
c
3
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−3.17034 × 10
19
u
22
+ 2.06027 × 10
19
u
21
+ ··· + 1.07970 × 10
20
b + 1.66107 × 10
20
,
− 9.79843 × 10
18
u
22
− 2.69589 × 10
19
u
21
+ ··· + 2.15940 × 10
20
a − 7.11427 × 10
20
,
u
23
− 2u
22
+ ··· − 24u + 8i
I
u
2
= h−2a
2
− au + b − 2a − u − 1, 4a
3
+ 2a
2
u − u, u
2
− 2i
I
u
3
= hb + u − 1, u
2
+ a − u − 2, u
3
− u
2
− 2u + 1i
I
v
1
= ha, b + v + 2, v
3
+ 3v
2
+ 2v −1i
* 4 irreducible components of dim
C
= 0, with total 35 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−3.17 × 10
19
u
22
+ 2.06 × 10
19
u
21
+ · · · + 1.08 × 10
20
b + 1.66 ×
10
20
, −9.80 × 10
18
u
22
− 2.70 × 10
19
u
21
+ · · · + 2.16 × 10
20
a − 7.11 ×
10
20
, u
23
− 2u
22
+ · · · − 24u + 8i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
6
=
u
−u
3
+ u
a
3
=
0.0453756u
22
+ 0.124844u
21
+ ··· + 6.96611u + 3.29455
0.293631u
22
− 0.190818u
21
+ ··· + 6.03607u − 1.53845
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
0.0453756u
22
+ 0.124844u
21
+ ··· + 6.96611u + 3.29455
−0.0606713u
22
+ 0.0493494u
21
+ ··· + 1.22479u + 0.186315
a
1
=
0.00815407u
22
+ 0.0581561u
21
+ ··· + 0.816899u + 0.942300
0.167916u
22
− 0.134620u
21
+ ··· + 3.19264u − 1.46520
a
4
=
−0.318134u
22
+ 0.323483u
21
+ ··· + 1.70680u + 4.47989
0.354835u
22
− 0.251401u
21
+ ··· + 7.02928u − 1.98143
a
12
=
−0.159762u
22
+ 0.192776u
21
+ ··· − 2.37574u + 2.40750
0.167916u
22
− 0.134620u
21
+ ··· + 3.19264u − 1.46520
a
8
=
−0.231610u
22
+ 0.238881u
21
+ ··· − 3.80455u + 2.85872
0.190322u
22
− 0.136611u
21
+ ··· + 2.10246u − 1.34349
a
7
=
u
3
− 2u
−u
5
+ 3u
3
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
158891265015344169787
107970245740225855892
u
22
−
49683941378257544770
26992561435056463973
u
21
+ ··· −
765639597674599940106
26992561435056463973
u −
395673949272391412502
26992561435056463973
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
23
+ 23u
22
+ ··· + 431u + 1
c
2
, c
4
u
23
− 7u
22
+ ··· − 25u − 1
c
3
, c
7
u
23
+ 2u
22
+ ··· − 92u + 8
c
5
, c
6
, c
9
c
10
u
23
+ 2u
22
+ ··· − 24u − 8
c
8
, c
11
, c
12
u
23
− 5u
22
+ ··· + 105u − 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
23
− 39y
22
+ ··· + 167215y −1
c
2
, c
4
y
23
− 23y
22
+ ··· + 431y −1
c
3
, c
7
y
23
− 12y
22
+ ··· + 4048y −64
c
5
, c
6
, c
9
c
10
y
23
− 18y
22
+ ··· + 1984y −64
c
8
, c
11
, c
12
y
23
− 3y
22
+ ··· + 3857y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.524450 + 0.823406I
a = 1.198540 − 0.301680I
b = −0.0820455 − 0.0681602I
−2.40419 − 0.36830I 2.14910 − 0.07440I
u = 0.524450 − 0.823406I
a = 1.198540 + 0.301680I
b = −0.0820455 + 0.0681602I
−2.40419 + 0.36830I 2.14910 + 0.07440I
u = 0.647880 + 0.361661I
a = −1.60944 − 0.06303I
b = 0.610235 + 0.499402I
5.21554 − 2.25150I 8.85155 − 0.03890I
u = 0.647880 − 0.361661I
a = −1.60944 + 0.06303I
b = 0.610235 − 0.499402I
5.21554 + 2.25150I 8.85155 + 0.03890I
u = −0.968334 + 0.805177I
a = −0.791369 − 0.993332I
b = 0.30258 − 1.86935I
−3.71088 − 3.05913I 3.40896 + 2.62935I
u = −0.968334 − 0.805177I
a = −0.791369 + 0.993332I
b = 0.30258 + 1.86935I
−3.71088 + 3.05913I 3.40896 − 2.62935I
u = 1.348180 + 0.047266I
a = −0.707662 − 0.573551I
b = 0.028918 − 1.035860I
7.94226 − 2.99119I 5.45880 + 3.25887I
u = 1.348180 − 0.047266I
a = −0.707662 + 0.573551I
b = 0.028918 + 1.035860I
7.94226 + 2.99119I 5.45880 − 3.25887I
u = 1.37411
a = 0.0360037
b = 1.16320
6.50526 14.0870
u = −0.596550 + 0.120314I
a = −0.457301 − 0.724116I
b = −0.799393 − 0.727747I
0.931592 − 0.038203I 9.32722 + 1.98466I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.596550 − 0.120314I
a = −0.457301 + 0.724116I
b = −0.799393 + 0.727747I
0.931592 + 0.038203I 9.32722 − 1.98466I
u = 1.253460 + 0.611260I
a = −0.257061 − 0.819047I
b = −0.178277 − 0.996927I
−0.01572 + 5.94333I 6.39784 − 4.46809I
u = 1.253460 − 0.611260I
a = −0.257061 + 0.819047I
b = −0.178277 + 0.996927I
−0.01572 − 5.94333I 6.39784 + 4.46809I
u = −1.42929
a = −0.686494
b = −12.0652
4.96770 −112.550
u = −0.25726 + 1.43341I
a = −0.014716 + 1.238780I
b = −0.02116 + 1.93499I
−11.10700 − 5.35109I 2.30683 + 2.56727I
u = −0.25726 − 1.43341I
a = −0.014716 − 1.238780I
b = −0.02116 − 1.93499I
−11.10700 + 5.35109I 2.30683 − 2.56727I
u = −0.444315
a = −0.534871
b = −0.856360
0.878779 12.7060
u = 1.59666 + 0.55985I
a = −0.723361 + 0.626398I
b = 0.55500 + 2.06165I
−5.20585 + 12.38690I 5.26515 − 5.63238I
u = 1.59666 − 0.55985I
a = −0.723361 − 0.626398I
b = 0.55500 − 2.06165I
−5.20585 − 12.38690I 5.26515 + 5.63238I
u = −1.49695 + 0.90558I
a = 0.809749 + 0.607559I
b = −0.42702 + 1.59069I
−7.48548 − 2.83924I 3.01997 + 1.35778I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.49695 − 0.90558I
a = 0.809749 − 0.607559I
b = −0.42702 − 1.59069I
−7.48548 + 2.83924I 3.01997 − 1.35778I
u = 0.244699
a = 3.66585
b = 0.520617
−1.28182 −11.3970
u = −1.84826
a = 0.624762
b = −0.739954
15.6748 −4.21640
7
II. I
u
2
= h−2a
2
− au + b − 2a − u − 1, 4a
3
+ 2a
2
u − u, u
2
− 2i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−2
a
6
=
u
−u
a
3
=
a
2a
2
+ au + 2a + u + 1
a
11
=
−1
0
a
2
=
a
2a
2
+ au + u + 1
a
1
=
a
2
u + a −
1
2
u
1
a
4
=
a
2
u
au + 2a + u + 1
a
12
=
a
2
u + a −
1
2
u − 1
1
a
8
=
a
2
u + a −
1
2
u
1
a
7
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4au + 8
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
3
(u
3
+ u
2
+ 2u + 1)
2
c
4
(u
3
− u
2
+ 1)
2
c
5
, c
6
, c
9
c
10
(u
2
− 2)
3
c
8
(u − 1)
6
c
11
, c
12
(u + 1)
6
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
(y
3
− y
2
+ 2y −1)
2
c
5
, c
6
, c
9
c
10
(y −2)
6
c
8
, c
11
, c
12
(y −1)
6
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = −0.620443 + 0.526697I
b = 0.510969 + 0.491114I
9.60386 + 2.82812I 11.50976 − 2.97945I
u = 1.41421
a = −0.620443 − 0.526697I
b = 0.510969 − 0.491114I
9.60386 − 2.82812I 11.50976 + 2.97945I
u = 1.41421
a = 0.533779
b = 4.80649
5.46628 4.98050
u = −1.41421
a = 0.620443 + 0.526697I
b = 0.16431 + 1.61567I
9.60386 − 2.82812I 11.50976 + 2.97945I
u = −1.41421
a = 0.620443 − 0.526697I
b = 0.16431 − 1.61567I
9.60386 + 2.82812I 11.50976 − 2.97945I
u = −1.41421
a = −0.533779
b = −0.157054
5.46628 4.98050
11
III. I
u
3
= hb + u − 1, u
2
+ a − u − 2, u
3
− u
2
− 2u + 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
6
=
u
−u
2
− u + 1
a
3
=
−u
2
+ u + 2
−u + 1
a
11
=
−u
2
+ 1
u
2
+ u − 1
a
2
=
−u
2
+ u + 2
−2u + 1
a
1
=
0
−u
a
4
=
−u
2
+ u + 2
−u + 1
a
12
=
u
−u
a
8
=
−u
2
+ 1
u
2
a
7
=
−u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
+ 4u + 12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
7
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
8
u
3
+ u
2
− 2u − 1
c
9
, c
10
, c
11
c
12
u
3
− u
2
− 2u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
7
y
3
c
5
, c
6
, c
8
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = −0.801938
b = 2.24698
4.69981 8.56700
u = 0.445042
a = 2.24698
b = 0.554958
−0.939962 13.9780
u = 1.80194
a = 0.554958
b = −0.801938
15.9794 22.4550
15
IV. I
v
1
= ha, b + v + 2, v
3
+ 3v
2
+ 2v − 1i
(i) Arc colorings
a
5
=
v
0
a
9
=
1
0
a
10
=
1
0
a
6
=
v
0
a
3
=
0
−v − 2
a
11
=
1
0
a
2
=
v
2
+ 2v − 1
−v − 2
a
1
=
v
2
+ 2v − 1
−1
a
4
=
−2v
2
− 2v + 1
−v
2
− 2v − 1
a
12
=
v
2
+ 2v
−1
a
8
=
−v
2
− 2v + 1
1
a
7
=
v
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −10v
2
− 22v − 10
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
4
u
3
− u
2
+ 1
c
5
, c
6
, c
9
c
10
u
3
c
7
u
3
+ u
2
+ 2u + 1
c
8
(u + 1)
3
c
11
, c
12
(u − 1)
3
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
y
3
+ 3y
2
+ 2y − 1
c
2
, c
4
y
3
− y
2
+ 2y − 1
c
5
, c
6
, c
9
c
10
y
3
c
8
, c
11
, c
12
(y − 1)
3
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.324718
a = 0
b = −2.32472
0.531480 −18.1980
v = −1.66236 + 0.56228I
a = 0
b = −0.337641 − 0.562280I
4.66906 − 2.82812I 2.09911 + 6.32406I
v = −1.66236 − 0.56228I
a = 0
b = −0.337641 + 0.562280I
4.66906 + 2.82812I 2.09911 − 6.32406I
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
3
− u
2
+ 2u − 1)
3
(u
23
+ 23u
22
+ ··· + 431u + 1)
c
2
((u − 1)
3
)(u
3
+ u
2
− 1)
3
(u
23
− 7u
22
+ ··· − 25u − 1)
c
3
u
3
(u
3
− u
2
+ 2u − 1)(u
3
+ u
2
+ 2u + 1)
2
(u
23
+ 2u
22
+ ··· − 92u + 8)
c
4
((u + 1)
3
)(u
3
− u
2
+ 1)
3
(u
23
− 7u
22
+ ··· − 25u − 1)
c
5
, c
6
u
3
(u
2
− 2)
3
(u
3
+ u
2
− 2u − 1)(u
23
+ 2u
22
+ ··· − 24u − 8)
c
7
u
3
(u
3
− u
2
+ 2u − 1)
2
(u
3
+ u
2
+ 2u + 1)(u
23
+ 2u
22
+ ··· − 92u + 8)
c
8
((u − 1)
6
)(u + 1)
3
(u
3
+ u
2
− 2u − 1)(u
23
− 5u
22
+ ··· + 105u − 7)
c
9
, c
10
u
3
(u
2
− 2)
3
(u
3
− u
2
− 2u + 1)(u
23
+ 2u
22
+ ··· − 24u − 8)
c
11
, c
12
((u − 1)
3
)(u + 1)
6
(u
3
− u
2
− 2u + 1)(u
23
− 5u
22
+ ··· + 105u − 7)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
3
)(y
3
+ 3y
2
+ 2y − 1)
3
(y
23
− 39y
22
+ ··· + 167215y − 1)
c
2
, c
4
((y − 1)
3
)(y
3
− y
2
+ 2y − 1)
3
(y
23
− 23y
22
+ ··· + 431y − 1)
c
3
, c
7
y
3
(y
3
+ 3y
2
+ 2y − 1)
3
(y
23
− 12y
22
+ ··· + 4048y − 64)
c
5
, c
6
, c
9
c
10
y
3
(y − 2)
6
(y
3
− 5y
2
+ 6y − 1)(y
23
− 18y
22
+ ··· + 1984y − 64)
c
8
, c
11
, c
12
((y − 1)
9
)(y
3
− 5y
2
+ 6y − 1)(y
23
− 3y
22
+ ··· + 3857y − 49)
21
