12n
0472
(K12n
0472
)
A knot diagram
1
Linearized knot diagam
3 6 9 10 2 11 12 3 4 6 7 8
Solving Sequence
6,11
7 12
3,8
2 1 5 10 4 9
c
6
c
11
c
7
c
2
c
1
c
5
c
10
c
4
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h59u
11
+ 169u
10
+ ··· + 377b + 257, −861u
11
− 1495u
10
+ ··· + 754a − 3284,
u
12
+ 2u
11
− 4u
10
− 8u
9
+ 7u
8
+ 9u
7
− 10u
6
+ u
5
+ 8u
4
− 17u
3
− 15u
2
+ 2u + 1i
I
u
2
= hb − 1, a
2
− 2a + 2u − 3, u
2
− u − 1i
I
u
3
= hb + 1, a + 1, u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 18 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
= h59u
11
+ 169u
10
+ · · · + 377b + 257, −861u
11
− 1495u
10
+ · · · +
754a − 3284, u
12
+ 2u
11
+ · · · + 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
12
=
−u
−u
3
+ u
a
3
=
1.14191u
11
+ 1.98276u
10
+ ··· − 12.8289u + 4.35544
−0.156499u
11
− 0.448276u
10
+ ··· + 2.06366u − 0.681698
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
0.985411u
11
+ 1.53448u
10
+ ··· − 10.7653u + 3.67374
−0.156499u
11
− 0.448276u
10
+ ··· + 2.06366u − 0.681698
a
1
=
u
3
− 2u
u
5
− 3u
3
+ u
a
5
=
1.32493u
11
+ 1.91379u
10
+ ··· − 7.68302u + 5.08488
0.196286u
11
+ 0.172414u
10
+ ··· + 1.75066u − 0.246684
a
10
=
u
u
a
4
=
0.982759u
11
+ 1.58621u
10
+ ··· − 7.58621u + 4.56897
−0.145889u
11
− 0.155172u
10
+ ··· + 1.84748u − 0.762599
a
9
=
−1.44430u
11
− 2.58621u
10
+ ··· + 15.7401u − 5.79973
0.301061u
11
+ 0.379310u
10
+ ··· − 2.07162u + 1.14191
(ii) Obstruction class = −1
(iii) Cusp Shapes =
805
377
u
11
+
83
29
u
10
−
3610
377
u
9
−
296
29
u
8
+
6558
377
u
7
+
2345
377
u
6
−
510
29
u
5
+
4614
377
u
4
+
1112
377
u
3
−
11957
377
u
2
−
4385
377
u −
5337
377
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
− 3u
11
+ ··· + 59u + 1
c
2
, c
5
u
12
+ 3u
11
+ ··· + 11u + 1
c
3
, c
4
, c
8
c
9
u
12
+ u
11
+ ··· − 12u − 4
c
6
, c
7
, c
10
c
11
, c
12
u
12
− 2u
11
+ ··· − 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
+ 11y
11
+ ··· − 1875y + 1
c
2
, c
5
y
12
+ 3y
11
+ ··· − 59y + 1
c
3
, c
4
, c
8
c
9
y
12
− 7y
11
+ ··· − 176y + 16
c
6
, c
7
, c
10
c
11
, c
12
y
12
− 12y
11
+ ··· − 34y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.327774 + 1.008010I
a = 0.125111 − 1.395870I
b = 0.62225 + 1.49572I
3.00343 − 3.59147I −14.5630 + 3.1404I
u = 0.327774 − 1.008010I
a = 0.125111 + 1.395870I
b = 0.62225 − 1.49572I
3.00343 + 3.59147I −14.5630 − 3.1404I
u = −1.049180 + 0.328681I
a = 0.227437 + 0.113345I
b = 1.043330 + 0.669500I
−3.63748 + 0.95206I −18.5953 − 2.4083I
u = −1.049180 − 0.328681I
a = 0.227437 − 0.113345I
b = 1.043330 − 0.669500I
−3.63748 − 0.95206I −18.5953 + 2.4083I
u = 1.196290 + 0.592671I
a = −0.801458 + 0.561608I
b = −0.45032 − 1.39377I
0.38150 − 2.09841I −15.3966 + 1.5939I
u = 1.196290 − 0.592671I
a = −0.801458 − 0.561608I
b = −0.45032 + 1.39377I
0.38150 + 2.09841I −15.3966 − 1.5939I
u = −1.50943
a = −1.21747
b = −1.31818
−11.5846 −22.3790
u = −1.56587 + 0.38910I
a = 0.939600 + 0.740035I
b = 1.19135 − 1.12676I
−3.19868 + 8.72155I −18.7407 − 4.5394I
u = −1.56587 − 0.38910I
a = 0.939600 − 0.740035I
b = 1.19135 + 1.12676I
−3.19868 − 8.72155I −18.7407 + 4.5394I
u = 1.63097
a = 0.940425
b = 0.160276
−13.8390 −17.9400
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.285895
a = −0.821672
b = 0.229843
−0.495710 −19.9520
u = −0.225459
a = 6.11734
b = −0.885138
−6.65661 −13.1380
6
II. I
u
2
= hb − 1, a
2
− 2a + 2u − 3, u
2
− u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u + 1
a
12
=
−u
−u − 1
a
3
=
a
1
a
8
=
−u
−u
a
2
=
a + 1
1
a
1
=
1
0
a
5
=
−a
−1
a
10
=
u
u
a
4
=
au − u − 1
au + a − u − 2
a
9
=
au − 2
au − 2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −24
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
8
c
9
(u
2
− 2)
2
c
6
, c
7
(u
2
− u − 1)
2
c
10
, c
11
, c
12
(u
2
+ u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
8
c
9
(y −2)
4
c
6
, c
7
, c
10
c
11
, c
12
(y
2
− 3y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −1.28825
b = 1.00000
−7.56670 −24.0000
u = −0.618034
a = 3.28825
b = 1.00000
−7.56670 −24.0000
u = 1.61803
a = 0.125968
b = 1.00000
−15.4624 −24.0000
u = 1.61803
a = 1.87403
b = 1.00000
−15.4624 −24.0000
10
III. I
u
3
= hb + 1, a + 1, u
2
+ u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u + 1
a
12
=
−u
−u + 1
a
3
=
−1
−1
a
8
=
u
u
a
2
=
−2
−1
a
1
=
−1
0
a
5
=
−1
−1
a
10
=
u
u
a
4
=
−1
−1
a
9
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −14
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
8
c
9
u
2
c
5
(u + 1)
2
c
6
, c
7
u
2
+ u − 1
c
10
, c
11
, c
12
u
2
− u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
2
c
3
, c
4
, c
8
c
9
y
2
c
6
, c
7
, c
10
c
11
, c
12
y
2
− 3y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.00000
b = −1.00000
−2.63189 −14.0000
u = −1.61803
a = −1.00000
b = −1.00000
−10.5276 −14.0000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
12
− 3u
11
+ ··· + 59u + 1)
c
2
((u − 1)
2
)(u + 1)
4
(u
12
+ 3u
11
+ ··· + 11u + 1)
c
3
, c
4
, c
8
c
9
u
2
(u
2
− 2)
2
(u
12
+ u
11
+ ··· − 12u − 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
12
+ 3u
11
+ ··· + 11u + 1)
c
6
, c
7
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
12
− 2u
11
+ ··· − 2u + 1)
c
10
, c
11
, c
12
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
12
− 2u
11
+ ··· − 2u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
12
+ 11y
11
+ ··· − 1875y + 1)
c
2
, c
5
((y −1)
6
)(y
12
+ 3y
11
+ ··· − 59y + 1)
c
3
, c
4
, c
8
c
9
y
2
(y −2)
4
(y
12
− 7y
11
+ ··· − 176y + 16)
c
6
, c
7
, c
10
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
12
− 12y
11
+ ··· − 34y + 1)
16
