12n
0347
(K12n
0347
)
A knot diagram
1
Linearized knot diagam
3 6 8 7 2 11 3 4 12 6 10 9
Solving Sequence
6,10
11
3,7
8 12 2 1 5 4 9
c
10
c
6
c
7
c
11
c
2
c
1
c
5
c
4
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−90767u
23
+ 130788u
22
+ ··· + 115958b − 256556,
78760u
23
− 139684u
22
+ ··· + 57979a + 292292, u
24
− 2u
23
+ ··· + 4u − 1i
I
u
2
= h−u
2
+ b, −u
2
+ a − u, u
3
+ u
2
− 1i
I
u
3
= h−2u
2
b + b
2
+ u
2
+ u − 3, −u
2
+ a + u, u
3
− u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 33 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−9.08 × 10
4
u
23
+ 1.31 × 10
5
u
22
+ · · · + 1.16 × 10
5
b − 2.57 ×
10
5
, 78760u
23
− 139684u
22
+ · · · + 57979a + 292292, u
24
− 2u
23
+ · · · + 4u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
3
=
−1.35842u
23
+ 2.40922u
22
+ ··· − 0.463478u − 5.04134
0.782758u
23
− 1.12789u
22
+ ··· − 0.287630u + 2.21249
a
7
=
u
−u
3
+ u
a
8
=
2.13475u
23
− 3.53621u
22
+ ··· + 0.957097u + 7.64951
−0.392478u
23
+ 1.12576u
22
+ ··· − 0.0678349u − 2.52723
a
12
=
−u
2
+ 1
−u
2
a
2
=
−1.35842u
23
+ 2.40922u
22
+ ··· − 0.463478u − 5.04134
0.687568u
23
− 1.07441u
22
+ ··· − 0.159722u + 2.52012
a
1
=
−u
6
+ u
4
− 2u
2
+ 1
−u
6
− u
2
a
5
=
1.53130u
23
− 2.66924u
22
+ ··· − 0.730765u + 6.01754
−0.254842u
23
+ 0.829093u
22
+ ··· + 0.396342u − 2.41867
a
4
=
1.46724u
23
− 2.41500u
22
+ ··· − 0.384769u + 6.42115
−0.745253u
23
+ 1.22722u
22
+ ··· + 1.31089u − 2.14118
a
9
=
u
4
− u
2
+ 1
u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
202771
57979
u
23
+
271376
57979
u
22
+ ··· −
44531
57979
u +
50669
57979
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
24
+ 38u
23
+ ··· + 19329u + 529
c
2
, c
5
u
24
+ 4u
23
+ ··· + 187u − 23
c
3
, c
7
, c
8
u
24
− u
23
+ ··· + 40u − 8
c
4
u
24
+ 3u
23
+ ··· + 1848u − 392
c
6
, c
10
u
24
+ 2u
23
+ ··· − 4u − 1
c
9
, c
11
, c
12
u
24
− 4u
23
+ ··· − 22u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
− 94y
23
+ ··· − 151108609y + 279841
c
2
, c
5
y
24
− 38y
23
+ ··· − 19329y + 529
c
3
, c
7
, c
8
y
24
− 17y
23
+ ··· − 704y + 64
c
4
y
24
+ 67y
23
+ ··· − 4042304y + 153664
c
6
, c
10
y
24
− 4y
23
+ ··· − 22y + 1
c
9
, c
11
, c
12
y
24
+ 36y
23
+ ··· − 238y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.835116 + 0.528467I
a = 0.069686 + 0.738784I
b = 0.547462 + 1.048480I
2.94268 − 4.01681I 9.35710 + 7.00457I
u = −0.835116 − 0.528467I
a = 0.069686 − 0.738784I
b = 0.547462 − 1.048480I
2.94268 + 4.01681I 9.35710 − 7.00457I
u = 0.963205
a = 1.31465
b = 0.839556
0.222515 11.2950
u = −0.597222 + 0.889562I
a = −0.58903 − 1.37270I
b = 0.50973 − 1.55938I
−5.52645 + 0.12674I 2.10877 − 0.40561I
u = −0.597222 − 0.889562I
a = −0.58903 + 1.37270I
b = 0.50973 + 1.55938I
−5.52645 − 0.12674I 2.10877 + 0.40561I
u = 0.864539
a = −0.402491
b = −1.66775
5.76082 17.2780
u = −1.060560 + 0.611058I
a = −1.131470 − 0.597921I
b = −0.47456 − 1.66985I
−3.87887 − 5.71265I 4.95306 + 5.47972I
u = −1.060560 − 0.611058I
a = −1.131470 + 0.597921I
b = −0.47456 + 1.66985I
−3.87887 + 5.71265I 4.95306 − 5.47972I
u = 0.862448 + 0.906754I
a = −0.104076 + 1.054660I
b = 0.67129 + 1.47789I
−5.38835 + 5.03722I 3.61138 − 5.42024I
u = 0.862448 − 0.906754I
a = −0.104076 − 1.054660I
b = 0.67129 − 1.47789I
−5.38835 − 5.03722I 3.61138 + 5.42024I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.513421 + 0.536855I
a = 1.139640 + 0.211368I
b = −0.097677 − 0.235228I
2.07401 + 0.04231I 5.83295 + 0.03083I
u = −0.513421 − 0.536855I
a = 1.139640 − 0.211368I
b = −0.097677 + 0.235228I
2.07401 − 0.04231I 5.83295 − 0.03083I
u = 0.967380 + 0.818919I
a = −0.960942 + 0.174387I
b = −0.274739 + 0.939682I
−4.99363 + 1.35928I 3.66855 + 1.22776I
u = 0.967380 − 0.818919I
a = −0.960942 − 0.174387I
b = −0.274739 − 0.939682I
−4.99363 − 1.35928I 3.66855 − 1.22776I
u = 0.506125 + 0.470764I
a = 0.180480 − 1.065620I
b = 0.251752 − 0.534035I
−1.13375 + 1.36766I 0.54938 − 4.80618I
u = 0.506125 − 0.470764I
a = 0.180480 + 1.065620I
b = 0.251752 + 0.534035I
−1.13375 − 1.36766I 0.54938 + 4.80618I
u = 0.890899 + 1.015170I
a = 1.11515 − 0.92750I
b = −0.64029 − 1.99328I
−15.6478 − 3.5499I 3.29333 + 0.72039I
u = 0.890899 − 1.015170I
a = 1.11515 + 0.92750I
b = −0.64029 + 1.99328I
−15.6478 + 3.5499I 3.29333 − 0.72039I
u = −0.616175
a = 0.345774
b = −0.317535
0.783360 13.8780
u = 1.042290 + 0.914631I
a = 0.827306 − 1.092810I
b = −0.33967 − 2.78227I
−15.1291 + 10.6100I 3.94704 − 4.98017I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.042290 − 0.914631I
a = 0.827306 + 1.092810I
b = −0.33967 + 2.78227I
−15.1291 − 10.6100I 3.94704 + 4.98017I
u = −0.994437 + 0.996281I
a = 1.00204 + 1.07454I
b = −0.42318 + 2.42093I
19.4048 − 3.6460I 1.66218 + 2.14478I
u = −0.994437 − 0.996281I
a = 1.00204 − 1.07454I
b = −0.42318 − 2.42093I
19.4048 + 3.6460I 1.66218 − 2.14478I
u = 0.251655
a = −4.35547
b = 1.68547
3.37303 1.58050
7
II. I
u
2
= h−u
2
+ b, −u
2
+ a − u, u
3
+ u
2
− 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
3
=
u
2
+ u
u
2
a
7
=
u
u
2
+ u − 1
a
8
=
u
u
2
+ u − 1
a
12
=
−u
2
+ 1
−u
2
a
2
=
u
2
+ u
u
2
− u
a
1
=
0
−u
a
5
=
u
2
+ u
u
2
a
4
=
u
2
+ u
u
2
a
9
=
u
u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 2u + 4
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
7
c
8
u
3
c
5
(u + 1)
3
c
6
u
3
− u
2
+ 1
c
9
u
3
+ u
2
+ 2u + 1
c
10
u
3
+ u
2
− 1
c
11
, c
12
u
3
− u
2
+ 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
3
c
3
, c
4
, c
7
c
8
y
3
c
6
, c
10
y
3
− y
2
+ 2y −1
c
9
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.662359 − 0.562280I
b = 0.215080 − 1.307140I
−4.66906 − 2.82812I 4.89456 + 3.73884I
u = −0.877439 − 0.744862I
a = −0.662359 + 0.562280I
b = 0.215080 + 1.307140I
−4.66906 + 2.82812I 4.89456 − 3.73884I
u = 0.754878
a = 1.32472
b = 0.569840
−0.531480 0.210880
11
III. I
u
3
= h−2u
2
b + b
2
+ u
2
+ u − 3, −u
2
+ a + u, u
3
− u
2
+ 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
3
=
u
2
− u
b
a
7
=
u
−u
2
+ u + 1
a
8
=
−u
2
+ b + u
−u
2
b + u
2
+ b − 2u
a
12
=
−u
2
+ 1
−u
2
a
2
=
u
2
− u
b + u
a
1
=
0
u
a
5
=
−u
2
+ u
−b
a
4
=
u
2
b − 2u
2
+ 2u + 1
bu − 2u
2
+ 1
a
9
=
−u
u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u + 8
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
6
c
2
(u + 1)
6
c
3
, c
4
, c
7
c
8
(u
2
− 2)
3
c
6
(u
3
+ u
2
− 1)
2
c
9
(u
3
+ u
2
+ 2u + 1)
2
c
10
(u
3
− u
2
+ 1)
2
c
11
, c
12
(u
3
− u
2
+ 2u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
7
c
8
(y −2)
6
c
6
, c
10
(y
3
− y
2
+ 2y −1)
2
c
9
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.662359 + 0.562280I
b = 1.151800 + 0.511958I
0.26574 + 2.82812I 4.49024 − 2.97945I
u = 0.877439 + 0.744862I
a = −0.662359 + 0.562280I
b = −0.72164 + 2.10232I
0.26574 + 2.82812I 4.49024 − 2.97945I
u = 0.877439 − 0.744862I
a = −0.662359 − 0.562280I
b = 1.151800 − 0.511958I
0.26574 − 2.82812I 4.49024 + 2.97945I
u = 0.877439 − 0.744862I
a = −0.662359 − 0.562280I
b = −0.72164 − 2.10232I
0.26574 − 2.82812I 4.49024 + 2.97945I
u = −0.754878
a = 1.32472
b = −1.30359
4.40332 11.0200
u = −0.754878
a = 1.32472
b = 2.44327
4.40332 11.0200
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
24
+ 38u
23
+ ··· + 19329u + 529)
c
2
((u − 1)
3
)(u + 1)
6
(u
24
+ 4u
23
+ ··· + 187u − 23)
c
3
, c
7
, c
8
u
3
(u
2
− 2)
3
(u
24
− u
23
+ ··· + 40u − 8)
c
4
u
3
(u
2
− 2)
3
(u
24
+ 3u
23
+ ··· + 1848u − 392)
c
5
((u − 1)
6
)(u + 1)
3
(u
24
+ 4u
23
+ ··· + 187u − 23)
c
6
(u
3
− u
2
+ 1)(u
3
+ u
2
− 1)
2
(u
24
+ 2u
23
+ ··· − 4u − 1)
c
9
((u
3
+ u
2
+ 2u + 1)
3
)(u
24
− 4u
23
+ ··· − 22u + 1)
c
10
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
24
+ 2u
23
+ ··· − 4u − 1)
c
11
, c
12
((u
3
− u
2
+ 2u − 1)
3
)(u
24
− 4u
23
+ ··· − 22u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
24
− 94y
23
+ ··· − 1.51109 × 10
8
y + 279841)
c
2
, c
5
((y −1)
9
)(y
24
− 38y
23
+ ··· − 19329y + 529)
c
3
, c
7
, c
8
y
3
(y −2)
6
(y
24
− 17y
23
+ ··· − 704y + 64)
c
4
y
3
(y −2)
6
(y
24
+ 67y
23
+ ··· − 4042304y + 153664)
c
6
, c
10
((y
3
− y
2
+ 2y −1)
3
)(y
24
− 4y
23
+ ··· − 22y + 1)
c
9
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
24
+ 36y
23
+ ··· − 238y + 1)
17
