12n
0723
(K12n
0723
)
A knot diagram
1
Linearized knot diagam
4 6 7 9 2 3 12 11 4 7 8 10
Solving Sequence
7,12 4,8
3 6 2 5 11 10 1 9
c
7
c
3
c
6
c
2
c
5
c
11
c
10
c
12
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
14
+ 2u
13
+ ··· + 2b − 1, −u
14
+ 4u
13
+ ··· + 2a + 4, u
15
− 3u
14
+ ··· − 3u + 1i
I
u
2
= h−au + b, u
2
a + a
2
+ au − 2u
2
+ 2a − u − 3, u
3
+ u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 21 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
14
+2u
13
+· · ·+2b −1, −u
14
+4u
13
+· · ·+2a +4, u
15
−3u
14
+· · ·−3u +1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
1
2
u
14
− 2u
13
+ ··· + 9u − 2
1
2
u
14
− u
13
+ ··· +
1
2
u +
1
2
a
8
=
1
−u
2
a
3
=
u
14
− 3u
13
+ ··· +
19
2
u −
3
2
1
2
u
14
− u
13
+ ··· +
1
2
u +
1
2
a
6
=
−
1
2
u
12
+ u
11
+ ··· −
7
2
u +
1
2
1
2
u
14
− u
13
+ ··· +
3
2
u −
1
2
a
2
=
1
2
u
14
− u
13
+ ··· −
7
2
u
2
+ 5u
−
1
2
u
14
+ u
13
+ ··· −
1
2
u +
1
2
a
5
=
−
1
2
u
14
+ u
13
+ ··· − 7u + 1
1
2
u
14
− 2u
13
+ ··· +
5
2
u −
3
2
a
11
=
−u
u
3
+ u
a
10
=
−u
3
− 2u
u
3
+ u
a
1
=
u
7
+ 4u
5
+ 4u
3
−u
7
− 3u
5
− 2u
3
+ u
a
9
=
u
2
+ 1
−u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
−
1
2
u
13
+
1
2
u
12
−
7
2
u
11
+
5
2
u
10
−
17
2
u
9
+
11
2
u
8
−13u
7
+
27
2
u
6
−
53
2
u
5
+26u
4
−
69
2
u
3
+18u
2
−11u−
9
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
− 18u
14
+ ··· + 4668u + 207
c
2
, c
3
, c
5
c
6
u
15
+ 4u
14
+ ··· + 6u − 1
c
4
, c
9
u
15
+ u
14
+ ··· − 96u − 64
c
7
, c
8
, c
11
u
15
+ 3u
14
+ ··· − 3u − 1
c
10
u
15
− 3u
14
+ ··· − 37u − 41
c
12
u
15
− u
14
+ ··· − 15u − 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
− 110y
14
+ ··· + 14260806y −42849
c
2
, c
3
, c
5
c
6
y
15
− 26y
14
+ ··· + 46y −1
c
4
, c
9
y
15
+ 35y
14
+ ··· + 33792y −4096
c
7
, c
8
, c
11
y
15
+ 17y
14
+ ··· − 15y −1
c
10
y
15
+ 21y
14
+ ··· − 37007y −1681
c
12
y
15
+ 41y
14
+ ··· + 273y −9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.900691 + 0.591172I
a = −1.46534 + 1.01047I
b = 1.91718 − 0.04386I
17.6730 + 2.9604I −3.47231 − 2.17330I
u = 0.900691 − 0.591172I
a = −1.46534 − 1.01047I
b = 1.91718 + 0.04386I
17.6730 − 2.9604I −3.47231 + 2.17330I
u = 0.508960 + 0.560149I
a = 1.18891 − 1.58128I
b = −1.49086 + 0.13884I
−8.38938 + 1.78426I −4.62403 − 2.81000I
u = 0.508960 − 0.560149I
a = 1.18891 + 1.58128I
b = −1.49086 − 0.13884I
−8.38938 − 1.78426I −4.62403 + 2.81000I
u = −0.184001 + 1.341680I
a = −0.211499 − 0.179174I
b = −0.279310 + 0.250795I
−3.45010 − 2.42340I 3.50251 + 0.27987I
u = −0.184001 − 1.341680I
a = −0.211499 + 0.179174I
b = −0.279310 − 0.250795I
−3.45010 + 2.42340I 3.50251 − 0.27987I
u = −0.03840 + 1.49525I
a = 0.358226 + 0.638671I
b = 0.968727 − 0.511112I
−7.29450 + 0.31744I −5.62657 − 1.11275I
u = −0.03840 − 1.49525I
a = 0.358226 − 0.638671I
b = 0.968727 + 0.511112I
−7.29450 − 0.31744I −5.62657 + 1.11275I
u = −0.498442
a = −0.328877
b = −0.163926
0.854874 12.4950
u = 0.14640 + 1.58860I
a = −0.162813 − 1.015850I
b = −1.58994 + 0.40736I
−15.7392 + 4.1680I −6.43676 − 2.09201I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.14640 − 1.58860I
a = −0.162813 + 1.015850I
b = −1.58994 − 0.40736I
−15.7392 − 4.1680I −6.43676 + 2.09201I
u = 0.33131 + 1.59485I
a = −0.156693 + 1.177500I
b = 1.92985 − 0.14022I
10.55240 + 7.55501I −5.88870 − 2.72866I
u = 0.33131 − 1.59485I
a = −0.156693 − 1.177500I
b = 1.92985 + 0.14022I
10.55240 − 7.55501I −5.88870 + 2.72866I
u = 0.084263 + 0.319070I
a = 0.11365 + 1.99296I
b = 0.626317 − 0.204194I
−1.181970 + 0.424054I −6.20164 − 1.93342I
u = 0.084263 − 0.319070I
a = 0.11365 − 1.99296I
b = 0.626317 + 0.204194I
−1.181970 − 0.424054I −6.20164 + 1.93342I
6
II. I
u
2
= h−au + b, u
2
a + a
2
+ au − 2u
2
+ 2a − u − 3, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
a
au
a
8
=
1
−u
2
a
3
=
au + a
au
a
6
=
−au + u
2
− a + u + 2
−au − 1
a
2
=
u
2
− a + u + 1
−au − 1
a
5
=
a
au
a
11
=
−u
−u
2
− u − 1
a
10
=
u
2
+ 1
−u
2
− u − 1
a
1
=
−1
0
a
9
=
u
2
+ 1
−u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
a + 3u
2
+ a + 5u
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
(u
2
+ u − 1)
3
c
4
, c
9
u
6
c
5
, c
6
(u
2
− u − 1)
3
c
7
, c
8
(u
3
+ u
2
+ 2u + 1)
2
c
10
, c
12
(u
3
+ u
2
− 1)
2
c
11
(u
3
− u
2
+ 2u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
(y
2
− 3y + 1)
3
c
4
, c
9
y
6
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
10
, c
12
(y
3
− y
2
+ 2y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −0.198308 − 1.205210I
b = 1.61803
−11.90680 − 2.82812I −5.91278 + 1.52866I
u = −0.215080 + 1.307140I
a = 0.075747 + 0.460350I
b = −0.618034
−4.01109 − 2.82812I −6.11966 + 6.11708I
u = −0.215080 − 1.307140I
a = −0.198308 + 1.205210I
b = 1.61803
−11.90680 + 2.82812I −5.91278 − 1.52866I
u = −0.215080 − 1.307140I
a = 0.075747 − 0.460350I
b = −0.618034
−4.01109 + 2.82812I −6.11966 − 6.11708I
u = −0.569840
a = 1.08457
b = −0.618034
0.126494 −1.14270
u = −0.569840
a = −2.83945
b = 1.61803
−7.76919 −3.79250
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u − 1)
3
)(u
15
− 18u
14
+ ··· + 4668u + 207)
c
2
, c
3
((u
2
+ u − 1)
3
)(u
15
+ 4u
14
+ ··· + 6u − 1)
c
4
, c
9
u
6
(u
15
+ u
14
+ ··· − 96u − 64)
c
5
, c
6
((u
2
− u − 1)
3
)(u
15
+ 4u
14
+ ··· + 6u − 1)
c
7
, c
8
((u
3
+ u
2
+ 2u + 1)
2
)(u
15
+ 3u
14
+ ··· − 3u − 1)
c
10
((u
3
+ u
2
− 1)
2
)(u
15
− 3u
14
+ ··· − 37u − 41)
c
11
((u
3
− u
2
+ 2u − 1)
2
)(u
15
+ 3u
14
+ ··· − 3u − 1)
c
12
((u
3
+ u
2
− 1)
2
)(u
15
− u
14
+ ··· − 15u − 3)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
− 3y + 1)
3
)(y
15
− 110y
14
+ ··· + 1.42608 × 10
7
y −42849)
c
2
, c
3
, c
5
c
6
((y
2
− 3y + 1)
3
)(y
15
− 26y
14
+ ··· + 46y −1)
c
4
, c
9
y
6
(y
15
+ 35y
14
+ ··· + 33792y −4096)
c
7
, c
8
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
15
+ 17y
14
+ ··· − 15y −1)
c
10
((y
3
− y
2
+ 2y −1)
2
)(y
15
+ 21y
14
+ ··· − 37007y −1681)
c
12
((y
3
− y
2
+ 2y −1)
2
)(y
15
+ 41y
14
+ ··· + 273y −9)
12
