12n
0310
(K12n
0310
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 5 12 11 5 7 8 9
Solving Sequence
2,5
6 3
7,9
10 11 1 4 8 12
c
5
c
2
c
6
c
9
c
10
c
1
c
4
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
18
− 4u
17
+ ··· + 2b − 1, −u
18
− 2u
17
+ ··· + 4a − 1, u
19
+ 4u
18
+ ··· + u + 1i
I
u
2
= hb, u
2
a + a
2
− 2au + u
2
+ a − 2u + 2, u
3
− u
2
+ 1i
I
u
3
= hb, −u
2
+ a + u, u
3
− u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 28 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−u
18
−4u
17
+· · ·+2b−1, −u
18
−2u
17
+· · ·+4a−1, u
19
+4u
18
+· · ·+u+1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
u
2
a
9
=
1
4
u
18
+
1
2
u
17
+ ··· −
1
2
u +
1
4
1
2
u
18
+ 2u
17
+ ··· +
1
2
u
2
+
1
2
a
10
=
−
1
4
u
18
−
3
2
u
17
+ ··· −
1
2
u −
1
4
1
2
u
18
+ 2u
17
+ ··· +
1
2
u
2
+
1
2
a
11
=
1
4
u
17
+
1
2
u
16
+ ··· −
1
2
u +
1
4
−
1
4
u
17
−
1
4
u
16
+ ··· +
3
4
u
2
−
1
4
u
a
1
=
u
3
u
5
− u
3
+ u
a
4
=
u
7
− 2u
5
+ 2u
3
− 2u
−u
7
+ u
5
− 2u
3
+ u
a
8
=
1
4
u
18
+
5
4
u
17
+ ··· −
5
4
u +
1
4
−
1
4
u
18
− u
17
+ ··· +
5
4
u
2
−
1
4
a
12
=
1
4
u
16
+
3
4
u
15
+ ··· +
7
4
u +
3
4
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
21
4
u
18
−
35
2
u
17
−
95
4
u
16
−
1
4
u
15
+ 4u
14
− 59u
13
−
497
4
u
12
−
87
2
u
11
+
193
4
u
10
−
67
2
u
9
−
659
4
u
8
− 90u
7
+
105
2
u
6
+
97
2
u
5
+
5
4
u
4
−
13
2
u
3
−
3
2
u
2
+
31
4
u −
13
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
19
+ 2u
18
+ ··· + u + 1
c
2
, c
5
u
19
+ 4u
18
+ ··· + u + 1
c
3
u
19
− 4u
18
+ ··· + 10539u + 33529
c
4
, c
9
u
19
+ u
18
+ ··· + 1024u + 512
c
7
, c
8
, c
11
u
19
+ 4u
18
+ ··· + 9u + 1
c
10
, c
12
u
19
− 4u
18
+ ··· + 19u + 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
19
+ 34y
18
+ ··· + y −1
c
2
, c
5
y
19
− 2y
18
+ ··· + y −1
c
3
y
19
+ 182y
18
+ ··· + 14622555837y −1124193841
c
4
, c
9
y
19
+ 49y
18
+ ··· + 917504y −262144
c
7
, c
8
, c
11
y
19
+ 14y
18
+ ··· + 57y −1
c
10
, c
12
y
19
− 42y
18
+ ··· + 193y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.912718 + 0.242073I
a = −0.605722 − 0.847563I
b = −0.688530 − 0.629214I
−3.61234 + 0.24752I −6.49775 − 1.24505I
u = −0.912718 − 0.242073I
a = −0.605722 + 0.847563I
b = −0.688530 + 0.629214I
−3.61234 − 0.24752I −6.49775 + 1.24505I
u = 0.815575 + 0.877596I
a = −0.144284 − 0.906002I
b = 1.173140 + 0.647275I
3.20435 − 1.33478I −0.465469 + 0.575891I
u = 0.815575 − 0.877596I
a = −0.144284 + 0.906002I
b = 1.173140 − 0.647275I
3.20435 + 1.33478I −0.465469 − 0.575891I
u = −0.314616 + 0.672401I
a = 0.306559 + 0.532855I
b = 1.170880 − 0.267606I
−1.32909 + 2.90393I −0.75358 − 3.65471I
u = −0.314616 − 0.672401I
a = 0.306559 − 0.532855I
b = 1.170880 + 0.267606I
−1.32909 − 2.90393I −0.75358 + 3.65471I
u = 1.023690 + 0.762858I
a = 0.911797 + 0.469844I
b = −0.93096 + 1.09132I
2.47335 − 4.87674I −0.88643 + 4.00302I
u = 1.023690 − 0.762858I
a = 0.911797 − 0.469844I
b = −0.93096 − 1.09132I
2.47335 + 4.87674I −0.88643 − 4.00302I
u = −0.683274
a = 0.475212
b = 0.353102
−0.926928 −11.6640
u = 0.580344 + 0.259618I
a = −0.91642 + 2.18155I
b = −0.114009 − 0.652352I
−3.71810 − 3.34728I −0.40454 + 7.55184I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.580344 − 0.259618I
a = −0.91642 − 2.18155I
b = −0.114009 + 0.652352I
−3.71810 + 3.34728I −0.40454 − 7.55184I
u = 0.217095 + 0.536747I
a = −0.01996 − 1.43982I
b = −0.369486 + 0.882626I
1.37613 − 0.79780I 4.12075 + 2.79421I
u = 0.217095 − 0.536747I
a = −0.01996 + 1.43982I
b = −0.369486 − 0.882626I
1.37613 + 0.79780I 4.12075 − 2.79421I
u = −0.95754 + 1.07145I
a = 1.23432 + 1.19343I
b = −0.47811 − 3.01384I
16.9874 − 2.3515I −0.326728 + 0.639267I
u = −0.95754 − 1.07145I
a = 1.23432 − 1.19343I
b = −0.47811 + 3.01384I
16.9874 + 2.3515I −0.326728 − 0.639267I
u = −1.07089 + 0.97401I
a = 1.36374 + 1.53260I
b = 0.88181 − 2.70786I
16.5659 + 9.8325I −0.83634 − 4.64715I
u = −1.07089 − 0.97401I
a = 1.36374 − 1.53260I
b = 0.88181 + 2.70786I
16.5659 − 9.8325I −0.83634 + 4.64715I
u = −1.03929 + 1.04464I
a = −1.36765 − 1.36775I
b = −0.32129 + 3.19553I
−18.3247 + 3.8317I 1.88207 − 2.04996I
u = −1.03929 − 1.04464I
a = −1.36765 + 1.36775I
b = −0.32129 − 3.19553I
−18.3247 − 3.8317I 1.88207 + 2.04996I
6
II. I
u
2
= hb, u
2
a + a
2
− 2au + u
2
+ a − 2u + 2, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
7
=
−u
2
+ 1
u
2
a
9
=
a
0
a
10
=
a
0
a
11
=
−au
−u
2
a + au + a
a
1
=
u
2
− 1
−u
2
a
4
=
1
0
a
8
=
−u
2
a + au − u
2
+ a + 2u − 1
u
2
a + u
2
− u
a
12
=
au − a + u − 2
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
2
a + au − a + 5u − 5
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
, c
9
u
6
c
5
(u
3
− u
2
+ 1)
2
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
5
, c
10
c
12
(y
3
− y
2
+ 2y −1)
2
c
4
, c
9
y
6
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.447279 + 0.744862I
b = 0
− 5.65624I −2.97732 + 5.45590I
u = 0.877439 + 0.744862I
a = 0.092519 − 0.562280I
b = 0
4.13758 − 2.82812I 1.30443 + 3.86214I
u = 0.877439 − 0.744862I
a = 0.447279 − 0.744862I
b = 0
5.65624I −2.97732 − 5.45590I
u = 0.877439 − 0.744862I
a = 0.092519 + 0.562280I
b = 0
4.13758 + 2.82812I 1.30443 − 3.86214I
u = −0.754878
a = −1.53980 + 1.30714I
b = 0
−4.13758 − 2.82812I −7.82711 − 0.80415I
u = −0.754878
a = −1.53980 − 1.30714I
b = 0
−4.13758 + 2.82812I −7.82711 + 0.80415I
10
III. I
u
3
= hb, −u
2
+ a + u, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
7
=
−u
2
+ 1
u
2
a
9
=
u
2
− u
0
a
10
=
u
2
− u
0
a
11
=
1
u
2
− 1
a
1
=
u
2
− 1
−u
2
a
4
=
1
0
a
8
=
−u
u + 1
a
12
=
u
2
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
− u − 1
11
(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
, c
9
u
3
c
5
u
3
− u
2
+ 1
c
6
, c
7
, c
8
u
3
+ u
2
+ 2u + 1
12
(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
5
, c
10
c
12
y
3
− y
2
+ 2y −1
c
4
, c
9
y
3
13
(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 = 0
0 −1.66236 + 0.56228I
u = 0.877439 − 0.744862I
a = −0.662359 − 0.562280I
b = 0
0 −1.66236 − 0.56228I
u = −0.754878
a = 1.32472
b = 0
0 0.324720
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
3
)(u
19
+ 2u
18
+ ··· + u + 1)
c
2
((u
3
+ u
2
− 1)
3
)(u
19
+ 4u
18
+ ··· + u + 1)
c
3
((u
3
− u
2
+ 2u − 1)
3
)(u
19
− 4u
18
+ ··· + 10539u + 33529)
c
4
, c
9
u
9
(u
19
+ u
18
+ ··· + 1024u + 512)
c
5
((u
3
− u
2
+ 1)
3
)(u
19
+ 4u
18
+ ··· + u + 1)
c
6
((u
3
+ u
2
+ 2u + 1)
3
)(u
19
+ 2u
18
+ ··· + u + 1)
c
7
, c
8
((u
3
+ u
2
+ 2u + 1)
3
)(u
19
+ 4u
18
+ ··· + 9u + 1)
c
10
, c
12
((u
3
+ u
2
− 1)
3
)(u
19
− 4u
18
+ ··· + 19u + 2)
c
11
((u
3
− u
2
+ 2u − 1)
3
)(u
19
+ 4u
18
+ ··· + 9u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
((y
3
+ 3y
2
+ 2y −1)
3
)(y
19
+ 34y
18
+ ··· + y −1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
3
)(y
19
− 2y
18
+ ··· + y −1)
c
3
(y
3
+ 3y
2
+ 2y −1)
3
· (y
19
+ 182y
18
+ ··· + 14622555837y −1124193841)
c
4
, c
9
y
9
(y
19
+ 49y
18
+ ··· + 917504y −262144)
c
7
, c
8
, c
11
((y
3
+ 3y
2
+ 2y −1)
3
)(y
19
+ 14y
18
+ ··· + 57y −1)
c
10
, c
12
((y
3
− y
2
+ 2y −1)
3
)(y
19
− 42y
18
+ ··· + 193y −4)
16
