12n
0115
(K12n
0115
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 4 11 12 5 7 8 9
Solving Sequence
7,10
11 8
4,12
3 6 5 2 1 9
c
10
c
7
c
11
c
3
c
6
c
5
c
2
c
1
c
9
c
4
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h152u
13
− 2309u
12
+ ··· + 4348b + 7399, −4859u
13
+ 22201u
12
+ ··· + 4348a − 33463,
u
14
− 5u
13
+ 5u
12
+ 10u
11
− 13u
10
− 15u
9
− 5u
8
+ 77u
7
− 45u
6
− 64u
5
+ 60u
4
+ 21u
3
− 41u
2
+ 14u − 1i
I
u
2
= h2a
2
u − a
2
+ au + b − a + 2u, a
3
− a
2
u − a
2
+ 2au + 4a − 2u − 3, u
2
+ u − 1i
I
u
3
= hu
2
+ b − u − 2, a, u
3
− u
2
− 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 23 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
= h152u
13
− 2309u
12
+ · · · + 4348b + 7399, −4859u
13
+ 22201u
12
+
· · · + 4348a − 33463, u
14
− 5u
13
+ · · · + 14u − 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
8
=
u
−u
3
+ u
a
4
=
1.11753u
13
− 5.10603u
12
+ ··· − 32.0340u + 7.69618
−0.0349586u
13
+ 0.531049u
12
+ ··· + 9.42157u − 1.70170
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
1.11753u
13
− 5.10603u
12
+ ··· − 32.0340u + 7.69618
−0.393514u
13
+ 2.11431u
12
+ ··· + 15.0465u − 2.18330
a
6
=
0.332567u
13
− 1.32498u
12
+ ··· − 10.1125u + 3.24448
−0.299908u
13
+ 1.56900u
12
+ ··· + 9.10350u − 1.05934
a
5
=
0.632475u
13
− 2.89397u
12
+ ··· − 19.2160u + 4.30382
−0.299908u
13
+ 1.56900u
12
+ ··· + 9.10350u − 1.05934
a
2
=
0.632475u
13
− 2.89397u
12
+ ··· − 19.2160u + 4.30382
0.00666973u
13
+ 0.252300u
12
+ ··· + 7.25345u − 1.30198
a
1
=
−u
4
+ 3u
2
− 1
u
6
− 4u
4
+ 3u
2
a
9
=
−u
3
+ 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
11021
2174
u
13
+
53061
2174
u
12
+ ··· +
369277
2174
u −
54785
2174
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
+ 18u
13
+ ··· + 1086u + 1
c
2
, c
4
u
14
− 6u
13
+ ··· − 34u − 1
c
3
, c
6
u
14
− 3u
13
+ ··· − 28u + 8
c
5
, c
9
u
14
+ 2u
13
+ ··· + 352u + 64
c
7
, c
8
, c
10
c
11
, c
12
u
14
− 5u
13
+ ··· + 14u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 38y
13
+ ··· − 1163634y + 1
c
2
, c
4
y
14
− 18y
13
+ ··· − 1086y + 1
c
3
, c
6
y
14
− 15y
13
+ ··· − 2512y + 64
c
5
, c
9
y
14
+ 30y
13
+ ··· − 87040y + 4096
c
7
, c
8
, c
10
c
11
, c
12
y
14
− 15y
13
+ ··· − 114y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.847247 + 0.340274I
a = −1.07919 + 1.10825I
b = −0.214655 + 0.076845I
4.35404 + 2.18891I 7.66563 + 1.41199I
u = −0.847247 − 0.340274I
a = −1.07919 − 1.10825I
b = −0.214655 − 0.076845I
4.35404 − 2.18891I 7.66563 − 1.41199I
u = 0.683451 + 0.439394I
a = −0.372092 − 1.119180I
b = −0.54173 − 1.61181I
−1.117970 + 0.457834I 6.03602 − 2.75865I
u = 0.683451 − 0.439394I
a = −0.372092 + 1.119180I
b = −0.54173 + 1.61181I
−1.117970 − 0.457834I 6.03602 + 2.75865I
u = 1.293890 + 0.440522I
a = 0.947194 − 0.804395I
b = −0.03997 − 1.45852I
1.31890 + 3.26489I 6.50646 − 2.86357I
u = 1.293890 − 0.440522I
a = 0.947194 + 0.804395I
b = −0.03997 + 1.45852I
1.31890 − 3.26489I 6.50646 + 2.86357I
u = −0.79945 + 1.23640I
a = 0.51219 + 1.76331I
b = −0.10570 + 1.90296I
−13.41460 − 4.06288I 3.37939 + 1.99626I
u = −0.79945 − 1.23640I
a = 0.51219 − 1.76331I
b = −0.10570 − 1.90296I
−13.41460 + 4.06288I 3.37939 − 1.99626I
u = 0.485579
a = −0.359180
b = 0.426392
0.739738 13.5200
u = −1.59630
a = 0.385525
b = −1.79529
7.97868 20.7070
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.74684 + 0.37580I
a = −0.709908 + 0.913168I
b = 0.34316 + 1.93469I
−5.06340 + 10.16720I 5.53186 − 3.95031I
u = 1.74684 − 0.37580I
a = −0.709908 − 0.913168I
b = 0.34316 − 1.93469I
−5.06340 − 10.16720I 5.53186 + 3.95031I
u = 1.85818
a = 0.512193
b = 0.340829
15.4110 1.86040
u = 0.0975686
a = 4.86506
b = −0.854160
−1.21825 −10.3270
6
II.
I
u
2
= h2a
2
u−a
2
+au+b −a + 2u, a
3
−a
2
u−a
2
+2au+4a −2u − 3, u
2
+u−1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u − 1
a
8
=
u
−u + 1
a
4
=
a
−2a
2
u + a
2
− au + a − 2u
a
12
=
u
−u
a
3
=
a
−2a
2
u + a
2
− 2u
a
6
=
a
2
u
0
a
5
=
a
2
u
0
a
2
=
a
2
u
−2a
2
u + a
2
− 2u
a
1
=
0
−u
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 19a
2
u − 13a
2
+ 9au − a + 8u − 9
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
(u
3
− u
2
+ 1)
2
c
5
, c
9
u
6
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
(u
2
− u − 1)
3
c
10
, c
11
, c
12
(u
2
+ u − 1)
3
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
(y
3
− y
2
+ 2y −1)
2
c
5
, c
9
y
6
c
7
, c
8
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0.922021
b = −1.08457
−0.126494 −0.918090
u = 0.618034
a = 0.34801 + 2.11500I
b = −0.075747 + 0.460350I
4.01109 − 2.82812I 3.00413 + 7.79836I
u = 0.618034
a = 0.34801 − 2.11500I
b = −0.075747 − 0.460350I
4.01109 + 2.82812I 3.00413 − 7.79836I
u = −1.61803
a = −0.132927 + 0.807858I
b = 0.198308 + 1.205210I
11.90680 + 2.82812I 7.89941 − 3.17745I
u = −1.61803
a = −0.132927 − 0.807858I
b = 0.198308 − 1.205210I
11.90680 − 2.82812I 7.89941 + 3.17745I
u = −1.61803
a = −0.352181
b = 2.83945
7.76919 −21.8890
10
III. I
u
3
= hu
2
+ b − u − 2, a, u
3
− u
2
− 2u + 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
8
=
u
−u
2
− u + 1
a
4
=
0
−u
2
+ u + 2
a
12
=
−u
2
+ 1
u
2
+ u − 1
a
3
=
0
−u
2
+ u + 2
a
6
=
0
u
a
5
=
−u
u
a
2
=
u
−u
2
+ 2
a
1
=
u
−u
a
9
=
−u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u
2
+ 7u + 30
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
6
u
3
c
4
(u + 1)
3
c
5
, c
7
, c
8
u
3
+ u
2
− 2u − 1
c
9
, c
10
, c
11
c
12
u
3
− u
2
− 2u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
6
y
3
c
5
, c
7
, c
8
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = 0
b = −0.801938
4.69981 8.83150
u = 0.445042
a = 0
b = 2.24698
−0.939962 31.5310
u = 1.80194
a = 0
b = 0.554958
15.9794 16.6380
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
3
− u
2
+ 2u − 1)
2
(u
14
+ 18u
13
+ ··· + 1086u + 1)
c
2
((u − 1)
3
)(u
3
+ u
2
− 1)
2
(u
14
− 6u
13
+ ··· − 34u − 1)
c
3
u
3
(u
3
− u
2
+ 2u − 1)
2
(u
14
− 3u
13
+ ··· − 28u + 8)
c
4
((u + 1)
3
)(u
3
− u
2
+ 1)
2
(u
14
− 6u
13
+ ··· − 34u − 1)
c
5
u
6
(u
3
+ u
2
− 2u − 1)(u
14
+ 2u
13
+ ··· + 352u + 64)
c
6
u
3
(u
3
+ u
2
+ 2u + 1)
2
(u
14
− 3u
13
+ ··· − 28u + 8)
c
7
, c
8
((u
2
− u − 1)
3
)(u
3
+ u
2
− 2u − 1)(u
14
− 5u
13
+ ··· + 14u − 1)
c
9
u
6
(u
3
− u
2
− 2u + 1)(u
14
+ 2u
13
+ ··· + 352u + 64)
c
10
, c
11
, c
12
((u
2
+ u − 1)
3
)(u
3
− u
2
− 2u + 1)(u
14
− 5u
13
+ ··· + 14u − 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
3
)(y
3
+ 3y
2
+ 2y −1)
2
(y
14
− 38y
13
+ ··· − 1163634y + 1)
c
2
, c
4
((y −1)
3
)(y
3
− y
2
+ 2y −1)
2
(y
14
− 18y
13
+ ··· − 1086y + 1)
c
3
, c
6
y
3
(y
3
+ 3y
2
+ 2y −1)
2
(y
14
− 15y
13
+ ··· − 2512y + 64)
c
5
, c
9
y
6
(y
3
− 5y
2
+ 6y −1)(y
14
+ 30y
13
+ ··· − 87040y + 4096)
c
7
, c
8
, c
10
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
3
− 5y
2
+ 6y −1)(y
14
− 15y
13
+ ··· − 114y + 1)
16
