12n
0851
(K12n
0851
)
A knot diagram
1
Linearized knot diagam
4 8 10 8 11 12 3 1 4 6 5 10
Solving Sequence
6,10
11 5 12
1,8
4 2 3 7 9
c
10
c
5
c
11
c
12
c
4
c
1
c
3
c
7
c
9
c
2
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
13
− 4u
12
+ 14u
11
− 32u
10
+ 61u
9
− 92u
8
+ 114u
7
− 113u
6
+ 90u
5
− 49u
4
+ 18u
3
+ 2u
2
+ 2b − 5u,
− u
13
+ 5u
12
+ ··· + 2a − 5, u
14
− 6u
13
+ ··· + 10u − 4i
I
u
2
= hu
11
+ 5u
9
+ 9u
7
+ 5u
5
+ u
4
− 2u
3
+ 2u
2
+ b − 2u,
− u
11
+ u
10
− 5u
9
+ 5u
8
− 9u
7
+ 9u
6
− 5u
5
+ 4u
4
+ 3u
3
− 4u
2
+ a + 4u − 2,
u
12
+ 6u
10
+ 13u
8
+ 10u
6
+ u
5
− 2u
4
+ 3u
3
− 4u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 26 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
=
hu
13
−4u
12
+· · ·+2b−5u, −u
13
+5u
12
+· · ·+2a−5, u
14
−6u
13
+· · ·+10u−4i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
5
=
u
u
3
+ u
a
12
=
u
2
+ 1
u
4
+ 2u
2
a
1
=
−u
4
− u
2
+ 1
u
4
+ 2u
2
a
8
=
1
2
u
13
−
5
2
u
12
+ ··· −
7
2
u +
5
2
−
1
2
u
13
+ 2u
12
+ ··· − u
2
+
5
2
u
a
4
=
1
4
u
13
− u
12
+ ··· +
3
4
u − 1
1
2
u
13
− 3u
12
+ ··· −
9
2
u + 3
a
2
=
−u
13
+
11
2
u
12
+ ··· + 6u −
5
2
−
1
2
u
13
+ 3u
12
+ ··· +
11
2
u − 4
a
3
=
3
4
u
13
− 4u
12
+ ··· −
15
4
u + 2
1
2
u
13
− 3u
12
+ ··· −
9
2
u + 3
a
7
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
9
=
−u
13
+
9
2
u
12
+ ··· + 4u −
3
2
1
2
u
13
− 2u
12
+ ··· + u
2
−
3
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
13
+ 6u
12
−24u
11
+ 68u
10
−151u
9
+ 270u
8
−394u
7
+ 469u
6
−
448u
5
+ 327u
4
− 164u
3
+ 34u
2
+ 24u − 26
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
− 5u
13
+ ··· − 2u + 1
c
2
, c
3
, c
7
c
9
u
14
− u
13
+ ··· − 2u − 1
c
4
, c
8
u
14
+ 2u
13
+ ··· − 3u − 1
c
5
, c
10
, c
11
u
14
− 6u
13
+ ··· + 10u − 4
c
6
u
14
+ 6u
13
+ ··· − 462u − 180
c
12
u
14
− 4u
13
+ ··· + 1960u + 1216
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 29y
13
+ ··· − 38y + 1
c
2
, c
3
, c
7
c
9
y
14
+ 39y
13
+ ··· + 14y + 1
c
4
, c
8
y
14
+ 30y
13
+ ··· − 33y + 1
c
5
, c
10
, c
11
y
14
+ 12y
13
+ ··· − 156y + 16
c
6
y
14
− 4y
13
+ ··· − 178524y + 32400
c
12
y
14
+ 76y
13
+ ··· − 36622528y + 1478656
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.858928
a = −0.109294
b = −0.665442
−6.60134 −10.5000
u = −0.057867 + 1.252230I
a = 0.540753 − 0.387507I
b = −0.147291 + 0.486948I
3.16264 + 1.26178I −8.12371 − 5.07005I
u = −0.057867 − 1.252230I
a = 0.540753 + 0.387507I
b = −0.147291 − 0.486948I
3.16264 − 1.26178I −8.12371 + 5.07005I
u = 0.606137 + 0.416811I
a = −0.101897 − 1.053150I
b = 1.314670 − 0.337317I
2.36646 − 1.94252I −10.80294 + 5.02822I
u = 0.606137 − 0.416811I
a = −0.101897 + 1.053150I
b = 1.314670 + 0.337317I
2.36646 + 1.94252I −10.80294 − 5.02822I
u = 1.116120 + 0.614225I
a = 0.611664 + 0.699390I
b = −1.89280 + 0.05423I
14.8430 − 3.5814I −10.19251 + 1.82347I
u = 1.116120 − 0.614225I
a = 0.611664 − 0.699390I
b = −1.89280 − 0.05423I
14.8430 + 3.5814I −10.19251 − 1.82347I
u = 0.396990 + 1.275890I
a = 0.477782 + 0.520657I
b = −0.666680 − 0.056910I
−2.63737 − 4.50416I −7.28522 + 5.75195I
u = 0.396990 − 1.275890I
a = 0.477782 − 0.520657I
b = −0.666680 + 0.056910I
−2.63737 + 4.50416I −7.28522 − 5.75195I
u = 0.22659 + 1.45890I
a = −2.00625 − 0.57060I
b = 1.74247 − 0.66474I
8.40238 − 5.00923I −8.98141 + 6.39703I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.22659 − 1.45890I
a = −2.00625 + 0.57060I
b = 1.74247 + 0.66474I
8.40238 + 5.00923I −8.98141 − 6.39703I
u = 0.43989 + 1.60061I
a = 1.72225 + 0.99994I
b = −1.94337 + 0.15344I
−17.6523 − 9.3227I −8.42289 + 3.11986I
u = 0.43989 − 1.60061I
a = 1.72225 − 0.99994I
b = −1.94337 − 0.15344I
−17.6523 + 9.3227I −8.42289 − 3.11986I
u = −0.314654
a = 0.620672
b = −0.148554
−0.498688 −19.8820
6
II.
I
u
2
= hu
11
+5u
9
+· · ·+b−2u, −u
11
+u
10
+· · ·+a−2, u
12
+6u
10
+· · ·+2u+1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
5
=
u
u
3
+ u
a
12
=
u
2
+ 1
u
4
+ 2u
2
a
1
=
−u
4
− u
2
+ 1
u
4
+ 2u
2
a
8
=
u
11
− u
10
+ 5u
9
− 5u
8
+ 9u
7
− 9u
6
+ 5u
5
− 4u
4
− 3u
3
+ 4u
2
− 4u + 2
−u
11
− 5u
9
− 9u
7
− 5u
5
− u
4
+ 2u
3
− 2u
2
+ 2u
a
4
=
u
10
− u
9
+ 5u
8
− 5u
7
+ 8u
6
− 8u
5
+ 3u
4
− u
3
− 4u
2
+ 6u − 3
−u
10
− 5u
8
− 8u
6
− 3u
4
− u
3
+ 2u
2
− 2u
a
2
=
−u
11
− u
10
− 7u
9
− 4u
8
− 17u
7
− 4u
6
− 15u
5
+ u
4
+ u
3
− 2u
2
+ 6u − 5
u
9
+ 4u
7
+ 5u
5
+ u
3
+ u
2
− u + 1
a
3
=
−u
9
− 5u
7
− 8u
5
− 2u
3
− 2u
2
+ 4u − 3
−u
10
− 5u
8
− 8u
6
− 3u
4
− u
3
+ 2u
2
− 2u
a
7
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
9
=
2u
11
− u
10
+ ··· − 4u + 3
−u
11
− 5u
9
− 8u
7
− 2u
5
− u
4
+ 4u
3
− 2u
2
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
10
+ 8u
8
+ 2u
7
+ 8u
6
+ 10u
5
− 4u
4
+ 14u
3
− 6u
2
+ 5u − 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
− 10u
11
+ ··· − 3u + 1
c
2
, c
9
u
12
+ u
10
− u
9
− 3u
8
− 2u
6
+ 2u
5
+ 3u
4
+ u
3
+ u
2
− u − 1
c
3
, c
7
u
12
+ u
10
+ u
9
− 3u
8
− 2u
6
− 2u
5
+ 3u
4
− u
3
+ u
2
+ u − 1
c
4
, c
8
u
12
+ u
11
− u
10
− u
9
− 3u
8
− 2u
7
+ 2u
6
+ 3u
4
+ u
3
− u
2
− 1
c
5
u
12
+ 6u
10
+ 13u
8
+ 10u
6
− u
5
− 2u
4
− 3u
3
− 4u
2
− 2u + 1
c
6
u
12
− 2u
10
+ 2u
9
+ 5u
8
+ u
7
− 8u
6
− 13u
5
+ 2u
4
+ 6u
3
+ u
2
− 4u + 1
c
10
, c
11
u
12
+ 6u
10
+ 13u
8
+ 10u
6
+ u
5
− 2u
4
+ 3u
3
− 4u
2
+ 2u + 1
c
12
u
12
− 4u
11
+ ··· + 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
− 22y
11
+ ··· + 13y + 1
c
2
, c
3
, c
7
c
9
y
12
+ 2y
11
+ ··· − 3y + 1
c
4
, c
8
y
12
− 3y
11
+ ··· + 2y + 1
c
5
, c
10
, c
11
y
12
+ 12y
11
+ ··· − 12y + 1
c
6
y
12
− 4y
11
+ ··· − 14y + 1
c
12
y
12
− 4y
11
+ ··· − 32y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.166667 + 1.154930I
a = −1.18491 − 1.41896I
b = 1.50605 + 0.16129I
5.94732 − 1.25425I −6.73308 − 0.07102I
u = 0.166667 − 1.154930I
a = −1.18491 + 1.41896I
b = 1.50605 − 0.16129I
5.94732 + 1.25425I −6.73308 + 0.07102I
u = −0.810172
a = −0.514676
b = 0.230352
−7.30066 −23.1150
u = 0.607258 + 0.278363I
a = 0.486854 − 1.232570I
b = 1.81887 − 0.27158I
3.51234 − 1.43480I −5.19843 + 3.96866I
u = 0.607258 − 0.278363I
a = 0.486854 + 1.232570I
b = 1.81887 + 0.27158I
3.51234 + 1.43480I −5.19843 − 3.96866I
u = −0.361738 + 1.288800I
a = −0.494619 + 0.090187I
b = 0.269953 − 0.236589I
−3.28068 + 4.21532I −18.1322 − 1.6804I
u = −0.361738 − 1.288800I
a = −0.494619 − 0.090187I
b = 0.269953 + 0.236589I
−3.28068 − 4.21532I −18.1322 + 1.6804I
u = −0.101870 + 1.358190I
a = 1.389980 + 0.057997I
b = −0.915023 + 0.580086I
0.36145 + 1.41595I −9.15778 − 0.19766I
u = −0.101870 − 1.358190I
a = 1.389980 − 0.057997I
b = −0.915023 − 0.580086I
0.36145 − 1.41595I −9.15778 + 0.19766I
u = 0.23985 + 1.43128I
a = −2.20239 − 0.98427I
b = 2.10135 − 0.50082I
9.05494 − 4.57784I −0.04990 + 1.52761I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.23985 − 1.43128I
a = −2.20239 + 0.98427I
b = 2.10135 + 0.50082I
9.05494 + 4.57784I −0.04990 − 1.52761I
u = −0.290167
a = 3.52484
b = −0.792760
−4.15087 −8.34210
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
12
− 10u
11
+ ··· − 3u + 1)(u
14
− 5u
13
+ ··· − 2u + 1)
c
2
, c
9
(u
12
+ u
10
− u
9
− 3u
8
− 2u
6
+ 2u
5
+ 3u
4
+ u
3
+ u
2
− u − 1)
· (u
14
− u
13
+ ··· − 2u − 1)
c
3
, c
7
(u
12
+ u
10
+ u
9
− 3u
8
− 2u
6
− 2u
5
+ 3u
4
− u
3
+ u
2
+ u − 1)
· (u
14
− u
13
+ ··· − 2u − 1)
c
4
, c
8
(u
12
+ u
11
− u
10
− u
9
− 3u
8
− 2u
7
+ 2u
6
+ 3u
4
+ u
3
− u
2
− 1)
· (u
14
+ 2u
13
+ ··· − 3u − 1)
c
5
(u
12
+ 6u
10
+ 13u
8
+ 10u
6
− u
5
− 2u
4
− 3u
3
− 4u
2
− 2u + 1)
· (u
14
− 6u
13
+ ··· + 10u − 4)
c
6
(u
12
− 2u
10
+ 2u
9
+ 5u
8
+ u
7
− 8u
6
− 13u
5
+ 2u
4
+ 6u
3
+ u
2
− 4u + 1)
· (u
14
+ 6u
13
+ ··· − 462u − 180)
c
10
, c
11
(u
12
+ 6u
10
+ 13u
8
+ 10u
6
+ u
5
− 2u
4
+ 3u
3
− 4u
2
+ 2u + 1)
· (u
14
− 6u
13
+ ··· + 10u − 4)
c
12
(u
12
− 4u
11
+ ··· + 2u + 1)(u
14
− 4u
13
+ ··· + 1960u + 1216)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
12
− 22y
11
+ ··· + 13y + 1)(y
14
− 29y
13
+ ··· − 38y + 1)
c
2
, c
3
, c
7
c
9
(y
12
+ 2y
11
+ ··· − 3y + 1)(y
14
+ 39y
13
+ ··· + 14y + 1)
c
4
, c
8
(y
12
− 3y
11
+ ··· + 2y + 1)(y
14
+ 30y
13
+ ··· − 33y + 1)
c
5
, c
10
, c
11
(y
12
+ 12y
11
+ ··· − 12y + 1)(y
14
+ 12y
13
+ ··· − 156y + 16)
c
6
(y
12
− 4y
11
+ ··· − 14y + 1)(y
14
− 4y
13
+ ··· − 178524y + 32400)
c
12
(y
12
− 4y
11
+ ··· − 32y + 1)
· (y
14
+ 76y
13
+ ··· − 36622528y + 1478656)
13
