12n
0118
(K12n
0118
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 4 12 11 5 7 8 9
Solving Sequence
7,12 4,8
3 6 11 9 1 10 5 2
c
7
c
3
c
6
c
11
c
8
c
12
c
10
c
5
c
2
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−647u
17
+ 2375u
16
+ ··· + 9396b − 4651, −6142u
17
+ 31873u
16
+ ··· + 9396a − 63899,
u
18
− 5u
17
+ ··· + 9u + 1i
I
u
2
= hau − u
2
+ b + a + u + 1, −2u
2
a + a
2
− 2au + 4u
2
− a + 8, u
3
+ u
2
+ 2u + 1i
I
u
3
= hb, u
5
− 2u
4
+ 4u
3
− 4u
2
+ a + 3u − 2, u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
I
u
4
= hb − u, u
2
+ a + u + 1, u
3
+ u
2
+ 2u + 1i
* 4 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−647u
17
+ 2375u
16
+ · · · + 9396b − 4651, −6142u
17
+ 31873u
16
+
· · · + 9396a − 63899, u
18
− 5u
17
+ · · · + 9u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
0.653682u
17
− 3.39219u
16
+ ··· − 10.7699u + 6.80066
0.0688591u
17
− 0.252767u
16
+ ··· + 0.207322u + 0.494998
a
8
=
1
−u
2
a
3
=
0.722542u
17
− 3.64496u
16
+ ··· − 10.5626u + 7.29566
0.0688591u
17
− 0.252767u
16
+ ··· + 0.207322u + 0.494998
a
6
=
0.257450u
17
− 0.780438u
16
+ ··· − 0.469455u − 2.80502
−0.357812u
17
+ 1.57620u
16
+ ··· + 3.73297u + 0.156982
a
11
=
−u
u
3
+ u
a
9
=
u
2
+ 1
−u
4
− 2u
2
a
1
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
10
=
−u
3
− 2u
u
3
+ u
a
5
=
−0.00500213u
17
+ 0.206152u
16
+ ··· + 1.80449u − 2.50234
−0.181141u
17
+ 0.997233u
16
+ ··· + 2.45732u − 0.00500213
a
2
=
0.232546u
17
− 1.30726u
16
+ ··· − 5.17156u + 5.55034
0.181141u
17
− 0.997233u
16
+ ··· − 2.45732u + 0.00500213
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
2501
1566
u
17
+
12545
1566
u
16
+ ··· +
44011
3132
u −
28247
3132
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
18
+ 10u
17
+ ··· + 526u + 1
c
2
, c
4
u
18
− 10u
17
+ ··· + 14u + 1
c
3
, c
6
u
18
− 4u
17
+ ··· + 256u − 64
c
5
, c
9
u
18
+ 9u
17
+ ··· − 2048u − 512
c
7
, c
8
, c
11
u
18
+ 5u
17
+ ··· − 9u + 1
c
10
, c
12
u
18
− 5u
17
+ ··· − 497u + 49
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
18
+ 134y
17
+ ··· − 240078y + 1
c
2
, c
4
y
18
− 10y
17
+ ··· − 526y + 1
c
3
, c
6
y
18
+ 48y
17
+ ··· − 81920y + 4096
c
5
, c
9
y
18
− 63y
17
+ ··· − 3014656y + 262144
c
7
, c
8
, c
11
y
18
+ 13y
17
+ ··· − 109y + 1
c
10
, c
12
y
18
− 47y
17
+ ··· − 257005y + 2401
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.405109 + 0.770998I
a = −0.291015 − 0.772728I
b = 1.202570 + 0.386446I
−1.63880 − 0.39759I 1.43515 + 1.37331I
u = −0.405109 − 0.770998I
a = −0.291015 + 0.772728I
b = 1.202570 − 0.386446I
−1.63880 + 0.39759I 1.43515 − 1.37331I
u = 0.725373 + 0.450694I
a = −1.51182 − 2.82088I
b = 0.43981 + 1.86757I
4.14833 − 1.63757I 3.60384 − 0.80616I
u = 0.725373 − 0.450694I
a = −1.51182 + 2.82088I
b = 0.43981 − 1.86757I
4.14833 + 1.63757I 3.60384 + 0.80616I
u = 1.194030 + 0.232985I
a = 1.74711 − 4.55330I
b = −1.67536 + 2.69388I
−16.4086 + 6.1635I 4.05351 − 2.26793I
u = 1.194030 − 0.232985I
a = 1.74711 + 4.55330I
b = −1.67536 − 2.69388I
−16.4086 − 6.1635I 4.05351 + 2.26793I
u = 0.360607 + 1.183990I
a = 0.448733 + 1.258690I
b = 0.550074 − 0.990997I
1.63305 + 5.70935I 3.81042 − 6.18784I
u = 0.360607 − 1.183990I
a = 0.448733 − 1.258690I
b = 0.550074 + 0.990997I
1.63305 − 5.70935I 3.81042 + 6.18784I
u = −0.110512 + 1.276710I
a = −0.353124 − 0.089950I
b = −0.038877 + 0.478514I
−3.23015 − 1.96870I 3.57157 + 3.68129I
u = −0.110512 − 1.276710I
a = −0.353124 + 0.089950I
b = −0.038877 − 0.478514I
−3.23015 + 1.96870I 3.57157 − 3.68129I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.250746 + 1.287460I
a = −1.43169 + 2.15528I
b = −0.532012 − 0.309792I
−4.27182 − 2.53682I −10.79254 + 3.39126I
u = −0.250746 − 1.287460I
a = −1.43169 − 2.15528I
b = −0.532012 + 0.309792I
−4.27182 + 2.53682I −10.79254 − 3.39126I
u = −0.456369
a = −0.475415
b = −0.212377
0.789107 12.7820
u = 0.48335 + 1.48077I
a = −0.44539 − 2.81436I
b = −1.31271 + 2.12200I
17.6239 + 12.0475I 1.51476 − 4.65853I
u = 0.48335 − 1.48077I
a = −0.44539 + 2.81436I
b = −1.31271 − 2.12200I
17.6239 − 12.0475I 1.51476 + 4.65853I
u = 0.77938 + 1.42726I
a = 2.16835 + 3.72673I
b = −0.76484 − 3.84358I
19.6304 + 0.9404I 2.54501 − 0.81034I
u = 0.77938 − 1.42726I
a = 2.16835 − 3.72673I
b = −0.76484 + 3.84358I
19.6304 − 0.9404I 2.54501 + 0.81034I
u = −0.0963755
a = 7.81308
b = 0.475099
−1.21816 −10.2650
6
II.
I
u
2
= hau −u
2
+b +a + u + 1, −2u
2
a + a
2
−2au +4u
2
−a +8, u
3
+u
2
+2u +1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
a
−au + u
2
− a − u − 1
a
8
=
1
−u
2
a
3
=
−au + u
2
− u − 1
−au + u
2
− a − u − 1
a
6
=
u
2
a − 3
−u
2
a − 2au − a − 3u
a
11
=
−u
−u
2
− u − 1
a
9
=
u
2
+ 1
−u
2
− u − 1
a
1
=
−1
0
a
10
=
u
2
+ 1
−u
2
− u − 1
a
5
=
u
2
a − 3
−u
2
a − 2au − a − 3u
a
2
=
−2au − a − 3u − 5
−u
2
a − 2au − a − 3u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
a − 10au + 11u
2
− 4a
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
11
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
10
, c
12
(u
3
+ u
2
− 1)
2
c
4
(u
3
− u
2
+ 1)
2
c
5
, c
9
u
6
c
6
, c
7
, c
8
(u
3
+ u
2
+ 2u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
, c
10
c
12
(y
3
− y
2
+ 2y −1)
2
c
5
, c
9
y
6
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −1.06984 + 1.06527I
b = −0.215080 − 1.307140I
− 5.65624I −0.00556 + 4.66003I
u = −0.215080 + 1.307140I
a = −1.68504 + 0.42445I
b = −0.569840
−4.13758 − 2.82812I −6.5820 + 15.2977I
u = −0.215080 − 1.307140I
a = −1.06984 − 1.06527I
b = −0.215080 + 1.307140I
5.65624I −0.00556 − 4.66003I
u = −0.215080 − 1.307140I
a = −1.68504 − 0.42445I
b = −0.569840
−4.13758 + 2.82812I −6.5820 − 15.2977I
u = −0.569840
a = 0.25488 + 3.03873I
b = −0.215080 − 1.307140I
4.13758 − 2.82812I 4.08755 + 6.14773I
u = −0.569840
a = 0.25488 − 3.03873I
b = −0.215080 + 1.307140I
4.13758 + 2.82812I 4.08755 − 6.14773I
10
III.
I
u
3
= hb, u
5
− 2u
4
+ 4u
3
− 4u
2
+ a + 3u − 2, u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−u
5
+ 2u
4
− 4u
3
+ 4u
2
− 3u + 2
0
a
8
=
1
−u
2
a
3
=
−u
5
+ 2u
4
− 4u
3
+ 4u
2
− 3u + 2
0
a
6
=
1
0
a
11
=
−u
u
3
+ u
a
9
=
u
2
+ 1
−u
4
− 2u
2
a
1
=
u
5
+ 2u
3
+ u
−u
5
+ u
4
− 2u
3
+ u
2
− u − 1
a
10
=
−u
3
− 2u
u
3
+ u
a
5
=
−u
5
− 2u
3
− u
u
5
− u
4
+ 2u
3
− u
2
+ u + 1
a
2
=
2u
4
− 2u
3
+ 4u
2
− 2u + 2
−u
5
+ u
4
− 2u
3
+ u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
5
+ 7u
4
− 13u
3
+ 20u
2
− 15u + 13
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
6
u
6
c
4
(u + 1)
6
c
5
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
7
, c
8
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
9
, c
10
, c
12
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
11
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
6
y
6
c
5
, c
9
, c
10
c
12
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
7
, c
8
, c
11
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.873214
a = 0.422181
b = 0
6.01515 10.0580
u = −0.138835 + 1.234450I
a = −0.26610 + 1.72116I
b = 0
−4.60518 − 1.97241I −6.63014 + 2.86834I
u = −0.138835 − 1.234450I
a = −0.26610 − 1.72116I
b = 0
−4.60518 + 1.97241I −6.63014 − 2.86834I
u = 0.408802 + 1.276380I
a = 0.417699 − 0.090629I
b = 0
2.05064 + 4.59213I 5.72906 − 1.01197I
u = 0.408802 − 1.276380I
a = 0.417699 + 0.090629I
b = 0
2.05064 − 4.59213I 5.72906 + 1.01197I
u = −0.413150
a = 4.27462
b = 0
−0.906083 23.7440
14
IV. I
u
4
= hb − u, u
2
+ a + u + 1, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−u
2
− u − 1
u
a
8
=
1
−u
2
a
3
=
−u
2
− 1
u
a
6
=
−u
−u
2
a
11
=
−u
−u
2
− u − 1
a
9
=
u
2
+ 1
−u
2
− u − 1
a
1
=
−1
0
a
10
=
u
2
+ 1
−u
2
− u − 1
a
5
=
−u
−u
2
a
2
=
−u
2
− u − 2
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 3u − 4
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
11
u
3
− u
2
+ 2u − 1
c
2
, c
10
, c
12
u
3
+ u
2
− 1
c
4
u
3
− u
2
+ 1
c
5
, c
9
u
3
c
6
, c
7
, c
8
u
3
+ u
2
+ 2u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
11
y
3
+ 3y
2
+ 2y −1
c
2
, c
4
, c
10
c
12
y
3
− y
2
+ 2y −1
c
5
, c
9
y
3
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = 0.877439 − 0.744862I
b = −0.215080 + 1.307140I
0 3.29468 − 1.67231I
u = −0.215080 − 1.307140I
a = 0.877439 + 0.744862I
b = −0.215080 − 1.307140I
0 3.29468 + 1.67231I
u = −0.569840
a = −0.754878
b = −0.569840
0 −3.58940
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
3
− u
2
+ 2u − 1)
3
(u
18
+ 10u
17
+ ··· + 526u + 1)
c
2
((u − 1)
6
)(u
3
+ u
2
− 1)
3
(u
18
− 10u
17
+ ··· + 14u + 1)
c
3
u
6
(u
3
− u
2
+ 2u − 1)
3
(u
18
− 4u
17
+ ··· + 256u − 64)
c
4
((u + 1)
6
)(u
3
− u
2
+ 1)
3
(u
18
− 10u
17
+ ··· + 14u + 1)
c
5
u
9
(u
6
+ u
5
+ ··· − u − 1)(u
18
+ 9u
17
+ ··· − 2048u − 512)
c
6
u
6
(u
3
+ u
2
+ 2u + 1)
3
(u
18
− 4u
17
+ ··· + 256u − 64)
c
7
, c
8
(u
3
+ u
2
+ 2u + 1)
3
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
18
+ 5u
17
+ ··· − 9u + 1)
c
9
u
9
(u
6
− u
5
+ ··· + u − 1)(u
18
+ 9u
17
+ ··· − 2048u − 512)
c
10
, c
12
(u
3
+ u
2
− 1)
3
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
18
− 5u
17
+ ··· − 497u + 49)
c
11
(u
3
− u
2
+ 2u − 1)
3
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
18
+ 5u
17
+ ··· − 9u + 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
3
+ 3y
2
+ 2y −1)
3
(y
18
+ 134y
17
+ ··· − 240078y + 1)
c
2
, c
4
((y −1)
6
)(y
3
− y
2
+ 2y −1)
3
(y
18
− 10y
17
+ ··· − 526y + 1)
c
3
, c
6
y
6
(y
3
+ 3y
2
+ 2y −1)
3
(y
18
+ 48y
17
+ ··· − 81920y + 4096)
c
5
, c
9
y
9
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
18
− 63y
17
+ ··· − 3014656y + 262144)
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
3
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
18
+ 13y
17
+ ··· − 109y + 1)
c
10
, c
12
(y
3
− y
2
+ 2y −1)
3
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
18
− 47y
17
+ ··· − 257005y + 2401)
20
